Investigation of Crouzeixs Conjecture via Nonsmooth Optimization - PowerPoint PPT Presentation

Crouzeix’s Conjecture Let p = p ( ζ ) be a polynomial and let A be a square matrix. M. Crouzeix conjectured in “Bounds for analytical functions of Crouzeix’s Conjecture matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A , The Field of Values Examples Example, continued � p ( A ) � 2 ≤ 2 � p � W ( A ) . Crouzeix’s Conjecture Crouzeix and The left-hand side is the 2-norm (spectral norm, maximum Palencia’s Theorems Special Cases singular value) of the matrix p ( A ) . Computing the Field of Values The norm on the right-hand side is the maximum of | p ( ζ ) | Johnson’s Algorithm Finds the Extreme over ζ ∈ W ( A ) . By the maximum modulus principle, this must Points Chebfun be attained on bd W ( A ) , the boundary of W ( A ) . Example, continued The Crouzeix Ratio If p = χ ( A ) , the characteristic polynomial (or minimal Computing the Crouzeix Ratio polynomial) of A , then � p ( A ) � 2 = 0 by Cayley-Hamilton, but Nonsmooth � p � W ( A ) = 0 only if A = λI for λ ∈ C , so that W ( A ) = { λ } . Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 6 / 39

Crouzeix’s Conjecture Let p = p ( ζ ) be a polynomial and let A be a square matrix. M. Crouzeix conjectured in “Bounds for analytical functions of Crouzeix’s Conjecture matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A , The Field of Values Examples Example, continued � p ( A ) � 2 ≤ 2 � p � W ( A ) . Crouzeix’s Conjecture Crouzeix and The left-hand side is the 2-norm (spectral norm, maximum Palencia’s Theorems Special Cases singular value) of the matrix p ( A ) . Computing the Field of Values The norm on the right-hand side is the maximum of | p ( ζ ) | Johnson’s Algorithm Finds the Extreme over ζ ∈ W ( A ) . By the maximum modulus principle, this must Points Chebfun be attained on bd W ( A ) , the boundary of W ( A ) . Example, continued The Crouzeix Ratio If p = χ ( A ) , the characteristic polynomial (or minimal Computing the Crouzeix Ratio polynomial) of A , then � p ( A ) � 2 = 0 by Cayley-Hamilton, but Nonsmooth � p � W ( A ) = 0 only if A = λI for λ ∈ C , so that W ( A ) = { λ } . Optimization of the Crouzeix Ratio If p ( ζ ) = ζ and A is a 2 × 2 Jordan block with 0 on the diagonal, Nonsmooth Analysis of the Crouzeix then � p ( A ) � 2 = 1 and W ( A ) is a disk centered at 0 with radius Ratio 0 . 5 , so the left and right-hand sides are equal. Concluding Remarks 6 / 39

Crouzeix and Palencia’s Theorems Crouzeix’s Crouzeix’s theorem (2008) Conjecture � p ( A ) � 2 ≤ 11 . 08 � p � W ( A ) The Field of Values Examples Example, continued i.e., the conjecture is true if we replace 2 by 11.08. Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 7 / 39

Crouzeix and Palencia’s Theorems Crouzeix’s Crouzeix’s theorem (2008) Conjecture � p ( A ) � 2 ≤ 11 . 08 � p � W ( A ) The Field of Values Examples Example, continued i.e., the conjecture is true if we replace 2 by 11.08. Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Palencia’s theorem (2016) Special Cases Computing the Field √ of Values � � � p ( A ) � 2 ≤ 1 + 2 � p � W ( A ) Johnson’s Algorithm Finds the Extreme Points √ Chebfun i.e., the conjecture is true if we replace 2 by 1 + 2 Example, continued The Crouzeix Ratio Published in SIMAX, May 2017, with Crouzeix. Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 7 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems W ( A ) = D : ■ Special Cases • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Computing the Field • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 of Values Johnson’s Algorithm Finds the Extreme (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems W ( A ) = D : ■ Special Cases • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Computing the Field • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 of Values Johnson’s Algorithm Finds the Extreme (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Points n = 2 (Crouzeix, 2004), and, more generally, the minimum ■ Chebfun Example, continued polynomial of A has degree 2 (follows from Tso and Wu, 1999) The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems W ( A ) = D : ■ Special Cases • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Computing the Field • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 of Values Johnson’s Algorithm Finds the Extreme (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Points n = 2 (Crouzeix, 2004), and, more generally, the minimum ■ Chebfun Example, continued polynomial of A has degree 2 (follows from Tso and Wu, 1999) The Crouzeix Ratio n = 3 and A 3 = 0 (Crouzeix, 2013) ■ Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems W ( A ) = D : ■ Special Cases • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Computing the Field • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 of Values Johnson’s Algorithm Finds the Extreme (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Points n = 2 (Crouzeix, 2004), and, more generally, the minimum ■ Chebfun Example, continued polynomial of A has degree 2 (follows from Tso and Wu, 1999) The Crouzeix Ratio n = 3 and A 3 = 0 (Crouzeix, 2013) ■ Computing the Crouzeix Ratio A is an upper Jordan block with a perturbation in the bottom left ■ Nonsmooth corner (Choi and Greenbaum, 2012) or any diagonal scaling of such Optimization of the Crouzeix Ratio A (Choi, 2013) Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems W ( A ) = D : ■ Special Cases • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Computing the Field • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 of Values Johnson’s Algorithm Finds the Extreme (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Points n = 2 (Crouzeix, 2004), and, more generally, the minimum ■ Chebfun Example, continued polynomial of A has degree 2 (follows from Tso and Wu, 1999) The Crouzeix Ratio n = 3 and A 3 = 0 (Crouzeix, 2013) ■ Computing the Crouzeix Ratio A is an upper Jordan block with a perturbation in the bottom left ■ Nonsmooth corner (Choi and Greenbaum, 2012) or any diagonal scaling of such Optimization of the Crouzeix Ratio A (Choi, 2013) Nonsmooth Analysis A = TDT − 1 with D diagonal and � T �� T − 1 � ≤ 2 (easy) ■ of the Crouzeix Ratio Concluding Remarks 8 / 39

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix and Palencia’s Theorems W ( A ) = D : ■ Special Cases • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Computing the Field • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 of Values Johnson’s Algorithm Finds the Extreme (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Points n = 2 (Crouzeix, 2004), and, more generally, the minimum ■ Chebfun Example, continued polynomial of A has degree 2 (follows from Tso and Wu, 1999) The Crouzeix Ratio n = 3 and A 3 = 0 (Crouzeix, 2013) ■ Computing the Crouzeix Ratio A is an upper Jordan block with a perturbation in the bottom left ■ Nonsmooth corner (Choi and Greenbaum, 2012) or any diagonal scaling of such Optimization of the Crouzeix Ratio A (Choi, 2013) Nonsmooth Analysis A = TDT − 1 with D diagonal and � T �� T − 1 � ≤ 2 (easy) ■ of the Crouzeix AA ∗ = A ∗ A (then the constant 2 can be improved to 1 ). Ratio ■ Concluding Remarks 8 / 39

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 9 / 39

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed Example, continued Crouzeix’s that the extreme points of W ( A ) can be characterized as Conjecture Crouzeix and Palencia’s Theorems ext W ( A ) = { z θ = v ∗ θ Av θ : θ ∈ [0 , 2 π ) } Special Cases Computing the Field of Values Johnson’s Algorithm where v θ is a normalized eigenvector corresponding to the largest Finds the Extreme eigenvalue of the Hermitian matrix Points Chebfun Example, continued H θ = 1 � e iθ A + e − iθ A ∗ � The Crouzeix Ratio . Computing the 2 Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 9 / 39

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed Example, continued Crouzeix’s that the extreme points of W ( A ) can be characterized as Conjecture Crouzeix and Palencia’s Theorems ext W ( A ) = { z θ = v ∗ θ Av θ : θ ∈ [0 , 2 π ) } Special Cases Computing the Field of Values Johnson’s Algorithm where v θ is a normalized eigenvector corresponding to the largest Finds the Extreme eigenvalue of the Hermitian matrix Points Chebfun Example, continued H θ = 1 � e iθ A + e − iθ A ∗ � The Crouzeix Ratio . Computing the 2 Crouzeix Ratio Nonsmooth The proof uses a supporting hyperplane argument. Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 9 / 39

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed Example, continued Crouzeix’s that the extreme points of W ( A ) can be characterized as Conjecture Crouzeix and Palencia’s Theorems ext W ( A ) = { z θ = v ∗ θ Av θ : θ ∈ [0 , 2 π ) } Special Cases Computing the Field of Values Johnson’s Algorithm where v θ is a normalized eigenvector corresponding to the largest Finds the Extreme eigenvalue of the Hermitian matrix Points Chebfun Example, continued H θ = 1 � e iθ A + e − iθ A ∗ � The Crouzeix Ratio . Computing the 2 Crouzeix Ratio Nonsmooth The proof uses a supporting hyperplane argument. Optimization of the Crouzeix Ratio Thus, we can compute as many extreme points as we like. Nonsmooth Analysis of the Crouzeix Continuing with the previous example... Ratio Concluding Remarks 9 / 39

Johnson’s Algorithm Finds the Extreme Points Crouzeix’s Conjecture 3 θ ∈ [3.99,5.3] The Field of Values Examples Example, continued 2 Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems 1 Special Cases θ ∈ [5.3,2 π ] Computing the Field of Values 0 θ ∈ [2.29,3.99] Johnson’s Algorithm Finds the Extreme Points θ ∈ [0,0.96] Chebfun -1 Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio -2 Nonsmooth Optimization of θ ∈ [0.96,2.29] the Crouzeix Ratio -3 Nonsmooth Analysis of the Crouzeix -1 0 1 2 3 4 5 6 Ratio Concluding Remarks 10 / 39

Johnson’s Algorithm Finds the Extreme Points Crouzeix’s Conjecture 3 θ ∈ [3.99,5.3] The Field of Values Examples Example, continued 2 Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems 1 Special Cases θ ∈ [5.3,2 π ] Computing the Field of Values 0 θ ∈ [2.29,3.99] Johnson’s Algorithm Finds the Extreme Points θ ∈ [0,0.96] Chebfun -1 Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio -2 Nonsmooth Optimization of θ ∈ [0.96,2.29] the Crouzeix Ratio -3 Nonsmooth Analysis of the Crouzeix -1 0 1 2 3 4 5 6 Ratio Concluding Remarks But how can we do this accurately, automatically and efficiently? 10 / 39

Chebfun Crouzeix’s Chebfun (Trefethen et al, 2004–present) represents real- or Conjecture complex-valued functions on real intervals to machine precision The Field of Values Examples accuracy using Chebyshev interpolation. Example, continued Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 11 / 39

Chebfun Crouzeix’s Chebfun (Trefethen et al, 2004–present) represents real- or Conjecture complex-valued functions on real intervals to machine precision The Field of Values Examples accuracy using Chebyshev interpolation. Example, continued Crouzeix’s Conjecture The necessary degree of the polynomial is determined Crouzeix and Palencia’s Theorems automatically. For example, representing sin( πx ) on [ − 1 , 1] to Special Cases Computing the Field machine precision requires degree 19. of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 11 / 39

Chebfun Crouzeix’s Chebfun (Trefethen et al, 2004–present) represents real- or Conjecture complex-valued functions on real intervals to machine precision The Field of Values Examples accuracy using Chebyshev interpolation. Example, continued Crouzeix’s Conjecture The necessary degree of the polynomial is determined Crouzeix and Palencia’s Theorems automatically. For example, representing sin( πx ) on [ − 1 , 1] to Special Cases Computing the Field machine precision requires degree 19. of Values Johnson’s Algorithm Finds the Extreme Most Matlab functions are overloaded to work with chebfun’s. Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 11 / 39

Chebfun Crouzeix’s Chebfun (Trefethen et al, 2004–present) represents real- or Conjecture complex-valued functions on real intervals to machine precision The Field of Values Examples accuracy using Chebyshev interpolation. Example, continued Crouzeix’s Conjecture The necessary degree of the polynomial is determined Crouzeix and Palencia’s Theorems automatically. For example, representing sin( πx ) on [ − 1 , 1] to Special Cases Computing the Field machine precision requires degree 19. of Values Johnson’s Algorithm Finds the Extreme Most Matlab functions are overloaded to work with chebfun’s. Points Chebfun Example, continued Applying Chebfun’s fov to compute the boundary of W ( A ) for The Crouzeix Ratio Computing the the previous example... Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 11 / 39

Example, continued Crouzeix’s Conjecture 3 θ ∈ [3.99,5.3] The Field of Values Examples Example, continued 2 Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems 1 Special Cases θ ∈ [5.3,2 π ] Computing the Field of Values 0 θ ∈ [2.29,3.99] Johnson’s Algorithm Finds the Extreme Points θ ∈ [0,0.96] Chebfun -1 Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio -2 Nonsmooth Optimization of θ ∈ [0.96,2.29] the Crouzeix Ratio -3 Nonsmooth Analysis of the Crouzeix -1 0 1 2 3 4 5 6 Ratio Concluding Remarks The small circles are the interpolation points generated by Chebfun. 12 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 13 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix and order n . Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 13 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix and order n . The Crouzeix ratio f is Palencia’s Theorems Special Cases A mapping from C m +1 × C n × n to R (associating Computing the Field ■ of Values polynomials p ∈ P m with their vectors of coefficients Johnson’s Algorithm Finds the Extreme c ∈ C m +1 using the monomial basis) Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 13 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix and order n . The Crouzeix ratio f is Palencia’s Theorems Special Cases A mapping from C m +1 × C n × n to R (associating Computing the Field ■ of Values polynomials p ∈ P m with their vectors of coefficients Johnson’s Algorithm Finds the Extreme c ∈ C m +1 using the monomial basis) Points Chebfun Not convex Example, continued ■ The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 13 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix and order n . The Crouzeix ratio f is Palencia’s Theorems Special Cases A mapping from C m +1 × C n × n to R (associating Computing the Field ■ of Values polynomials p ∈ P m with their vectors of coefficients Johnson’s Algorithm Finds the Extreme c ∈ C m +1 using the monomial basis) Points Chebfun Not convex Example, continued ■ The Crouzeix Ratio Not defined if p ( A ) = 0 ■ Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 13 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix and order n . The Crouzeix ratio f is Palencia’s Theorems Special Cases A mapping from C m +1 × C n × n to R (associating Computing the Field ■ of Values polynomials p ∈ P m with their vectors of coefficients Johnson’s Algorithm Finds the Extreme c ∈ C m +1 using the monomial basis) Points Chebfun Not convex Example, continued ■ The Crouzeix Ratio Not defined if p ( A ) = 0 ■ Computing the Crouzeix Ratio Lipschitz continuous at all other points, but not necessarily ■ Nonsmooth Optimization of differentiable the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 13 / 39

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix and order n . The Crouzeix ratio f is Palencia’s Theorems Special Cases A mapping from C m +1 × C n × n to R (associating Computing the Field ■ of Values polynomials p ∈ P m with their vectors of coefficients Johnson’s Algorithm Finds the Extreme c ∈ C m +1 using the monomial basis) Points Chebfun Not convex Example, continued ■ The Crouzeix Ratio Not defined if p ( A ) = 0 ■ Computing the Crouzeix Ratio Lipschitz continuous at all other points, but not necessarily ■ Nonsmooth Optimization of differentiable the Crouzeix Ratio Semialgebraic (its graph is a finite union of sets, each of ■ Nonsmooth Analysis of the Crouzeix which is defined by a finite system of polynomial inequalities) Ratio Concluding Remarks 13 / 39

Computing the Crouzeix Ratio Crouzeix’s Numerator: use Chebfun’s fov (modified to return any line Conjecture segments in the boundary) combined with its overloaded polyval The Field of Values Examples and norm( · ,inf) . Example, continued Crouzeix’s Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 14 / 39

Computing the Crouzeix Ratio Crouzeix’s Numerator: use Chebfun’s fov (modified to return any line Conjecture segments in the boundary) combined with its overloaded polyval The Field of Values Examples and norm( · ,inf) . Example, continued Crouzeix’s Denominator: use Matlab ’s standard polyvalm and norm( · ,2) . Conjecture Crouzeix and Palencia’s Theorems Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 14 / 39

Computing the Crouzeix Ratio Crouzeix’s Numerator: use Chebfun’s fov (modified to return any line Conjecture segments in the boundary) combined with its overloaded polyval The Field of Values Examples and norm( · ,inf) . Example, continued Crouzeix’s Denominator: use Matlab ’s standard polyvalm and norm( · ,2) . Conjecture Crouzeix and Palencia’s Theorems The main cost is the construction of the chebfun defining the Special Cases field of values. Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 14 / 39

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio Nonsmoothness of the Crouzeix Ratio BFGS Experiments Nonsmooth Optimization of Optimizing over A (order n ) and p (deg ≤ n − 1 ) the Crouzeix Ratio Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 15 / 39 Concluding Remarks

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in the Crouzeix’s Conjecture Crouzeix ratio f Nonsmooth Optimization of the Crouzeix Ratio Nonsmoothness of the Crouzeix Ratio BFGS Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 16 / 39 Concluding Remarks

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in the Crouzeix’s Conjecture Crouzeix ratio f Nonsmooth Optimization of When the max value of | p ( ζ ) | on bd W ( A ) is attained at the Crouzeix Ratio ■ Nonsmoothness of more than one point ζ (the most important, as this the Crouzeix Ratio BFGS frequently occurs at apparent minimizers) Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 16 / 39 Concluding Remarks

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in the Crouzeix’s Conjecture Crouzeix ratio f Nonsmooth Optimization of When the max value of | p ( ζ ) | on bd W ( A ) is attained at the Crouzeix Ratio ■ Nonsmoothness of more than one point ζ (the most important, as this the Crouzeix Ratio BFGS frequently occurs at apparent minimizers) Experiments Optimizing over A Even if such ζ is unique, when the normalized vector v for ■ (order n ) and p (deg ≤ n − 1 ) which v ∗ Av = ζ is not unique up to a scalar, implying that Final Fields of the maximum eigenvalue of the corresponding H θ matrix has Values for Lowest Computed f multiplicity two or more (does not seem to occur at Optimizing over both p and A : Final minimizers) f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 16 / 39 Concluding Remarks

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in the Crouzeix’s Conjecture Crouzeix ratio f Nonsmooth Optimization of When the max value of | p ( ζ ) | on bd W ( A ) is attained at the Crouzeix Ratio ■ Nonsmoothness of more than one point ζ (the most important, as this the Crouzeix Ratio BFGS frequently occurs at apparent minimizers) Experiments Optimizing over A Even if such ζ is unique, when the normalized vector v for ■ (order n ) and p (deg ≤ n − 1 ) which v ∗ Av = ζ is not unique up to a scalar, implying that Final Fields of the maximum eigenvalue of the corresponding H θ matrix has Values for Lowest Computed f multiplicity two or more (does not seem to occur at Optimizing over both p and A : Final minimizers) f ( p, A ) Is the Ratio 0 . 5 When the maximum singular value of p ( A ) has multiplicity ■ Attained? Final Fields of two or more (does not seem to occur at minimizers) Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 16 / 39 Concluding Remarks

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in the Crouzeix’s Conjecture Crouzeix ratio f Nonsmooth Optimization of When the max value of | p ( ζ ) | on bd W ( A ) is attained at the Crouzeix Ratio ■ Nonsmoothness of more than one point ζ (the most important, as this the Crouzeix Ratio BFGS frequently occurs at apparent minimizers) Experiments Optimizing over A Even if such ζ is unique, when the normalized vector v for ■ (order n ) and p (deg ≤ n − 1 ) which v ∗ Av = ζ is not unique up to a scalar, implying that Final Fields of the maximum eigenvalue of the corresponding H θ matrix has Values for Lowest Computed f multiplicity two or more (does not seem to occur at Optimizing over both p and A : Final minimizers) f ( p, A ) Is the Ratio 0 . 5 When the maximum singular value of p ( A ) has multiplicity ■ Attained? Final Fields of two or more (does not seem to occur at minimizers) Values for f Closest to 1 Why is the Crouzeix In all of these cases the gradient of f is not defined. Ratio One? Results for Larger But in practice, none of these cases ever occur, except the first Dimension n and Degree n − 1 one in the limit . Nonsmooth Analysis of the Crouzeix Ratio 16 / 39 Concluding Remarks

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio Nonsmoothness of the Crouzeix Ratio BFGS Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 17 / 39 Concluding Remarks

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio It works by building an approximation to the Hessian of the Nonsmoothness of the Crouzeix Ratio function using gradient differences, and has a well known BFGS Experiments superlinear convergence property under a regularity condition. Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 17 / 39 Concluding Remarks

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio It works by building an approximation to the Hessian of the Nonsmoothness of the Crouzeix Ratio function using gradient differences, and has a well known BFGS Experiments superlinear convergence property under a regularity condition. Optimizing over A (order n ) and p Although its global convergence theory is limited to the convex (deg ≤ n − 1 ) Final Fields of case (Powell, 1976), it generally finds local minimizers efficiently Values for Lowest Computed f in the nonconvex case too, although there are pathological Optimizing over counterexamples. both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 17 / 39 Concluding Remarks

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio It works by building an approximation to the Hessian of the Nonsmoothness of the Crouzeix Ratio function using gradient differences, and has a well known BFGS Experiments superlinear convergence property under a regularity condition. Optimizing over A (order n ) and p Although its global convergence theory is limited to the convex (deg ≤ n − 1 ) Final Fields of case (Powell, 1976), it generally finds local minimizers efficiently Values for Lowest Computed f in the nonconvex case too, although there are pathological Optimizing over counterexamples. both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Remarkably, this property seems to extend to nonsmooth Attained? Final Fields of functions too, with a linear rate of local convergence, although Values for f Closest the convergence theory is extremely limited (Lewis and Overton, to 1 Why is the Crouzeix 2013). It builds a very ill conditioned “Hessian” approximation, Ratio One? Results for Larger with “infinitely large” curvature in some directions and finite Dimension n and Degree n − 1 curvature in other directions. Nonsmooth Analysis of the Crouzeix Ratio 17 / 39 Concluding Remarks

Experiments We have run many experiments searching for local minimizers of Crouzeix’s Conjecture the Crouzeix ratio using BFGS. Nonsmooth Optimization of the Crouzeix Ratio Nonsmoothness of the Crouzeix Ratio BFGS Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 18 / 39 Concluding Remarks

Experiments We have run many experiments searching for local minimizers of Crouzeix’s Conjecture the Crouzeix ratio using BFGS. Nonsmooth Optimization of the Crouzeix Ratio For fixed n , optimize over A with order n and p of deg ≤ n − 1 , Nonsmoothness of the Crouzeix Ratio running BFGS for a maximum of 1000 iterations from each of BFGS 100 randomly generated starting points. Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 18 / 39 Concluding Remarks

Experiments We have run many experiments searching for local minimizers of Crouzeix’s Conjecture the Crouzeix ratio using BFGS. Nonsmooth Optimization of the Crouzeix Ratio For fixed n , optimize over A with order n and p of deg ≤ n − 1 , Nonsmoothness of the Crouzeix Ratio running BFGS for a maximum of 1000 iterations from each of BFGS 100 randomly generated starting points. Experiments Optimizing over A (order n ) and p We restrict p to have real coefficients and A to be real, in (deg ≤ n − 1 ) Hessenberg form (all but one superdiagonal is zero). Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 18 / 39 Concluding Remarks

Experiments We have run many experiments searching for local minimizers of Crouzeix’s Conjecture the Crouzeix ratio using BFGS. Nonsmooth Optimization of the Crouzeix Ratio For fixed n , optimize over A with order n and p of deg ≤ n − 1 , Nonsmoothness of the Crouzeix Ratio running BFGS for a maximum of 1000 iterations from each of BFGS 100 randomly generated starting points. Experiments Optimizing over A (order n ) and p We restrict p to have real coefficients and A to be real, in (deg ≤ n − 1 ) Hessenberg form (all but one superdiagonal is zero). Final Fields of Values for Lowest Computed f We have obtained similar results for p with complex coefficients Optimizing over both p and A : Final and complex A (then can take A to be triangular). f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 18 / 39 Concluding Remarks

Experiments We have run many experiments searching for local minimizers of Crouzeix’s Conjecture the Crouzeix ratio using BFGS. Nonsmooth Optimization of the Crouzeix Ratio For fixed n , optimize over A with order n and p of deg ≤ n − 1 , Nonsmoothness of the Crouzeix Ratio running BFGS for a maximum of 1000 iterations from each of BFGS 100 randomly generated starting points. Experiments Optimizing over A (order n ) and p We restrict p to have real coefficients and A to be real, in (deg ≤ n − 1 ) Hessenberg form (all but one superdiagonal is zero). Final Fields of Values for Lowest Computed f We have obtained similar results for p with complex coefficients Optimizing over both p and A : Final and complex A (then can take A to be triangular). f ( p, A ) Is the Ratio 0 . 5 Attained? We have also obtained similar results using Gradient Sampling Final Fields of Values for f Closest (Burke, Lewis and Overton, 2005; Kiwiel 2007) instead of BFGS. to 1 Why is the Crouzeix This method has a very satisfactory convergence theory, but it is Ratio One? Results for Larger much slower. Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 18 / 39 Concluding Remarks

Optimizing over A (order n ) and p (deg ≤ n − 1 ) n=3 n=4 n=5 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of 0.7 0.7 0.7 the Crouzeix Ratio Nonsmoothness of 0.6 0.6 0.6 the Crouzeix Ratio 0.5 0.5 0.5 BFGS 0.4 0.4 0.4 Experiments 0 50 100 0 50 100 0 50 100 Optimizing over A (order n ) and p n=6 n=7 n=8 (deg ≤ n − 1 ) 1.1 1.1 1.1 Final Fields of 1 1 1 Values for Lowest 0.9 0.9 0.9 Computed f 0.8 0.8 0.8 Optimizing over both p and A : Final 0.7 0.7 0.7 f ( p, A ) 0.6 0.6 0.6 Is the Ratio 0 . 5 0.5 0.5 0.5 Attained? Final Fields of 0.4 0.4 0.4 0 50 100 0 50 100 0 50 100 Values for f Closest to 1 Why is the Crouzeix Sorted final values of the Crouzeix ratio f Ratio One? Results for Larger found starting from 100 randomly generated initial points. Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 19 / 39 Concluding Remarks

Optimizing over A (order n ) and p (deg ≤ n − 1 ) n=3 n=4 n=5 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of 0.7 0.7 0.7 the Crouzeix Ratio Nonsmoothness of 0.6 0.6 0.6 the Crouzeix Ratio 0.5 0.5 0.5 BFGS 0.4 0.4 0.4 Experiments 0 50 100 0 50 100 0 50 100 Optimizing over A (order n ) and p n=6 n=7 n=8 (deg ≤ n − 1 ) 1.1 1.1 1.1 Final Fields of 1 1 1 Values for Lowest 0.9 0.9 0.9 Computed f 0.8 0.8 0.8 Optimizing over both p and A : Final 0.7 0.7 0.7 f ( p, A ) 0.6 0.6 0.6 Is the Ratio 0 . 5 0.5 0.5 0.5 Attained? Final Fields of 0.4 0.4 0.4 0 50 100 0 50 100 0 50 100 Values for f Closest to 1 Why is the Crouzeix Sorted final values of the Crouzeix ratio f Ratio One? Results for Larger found starting from 100 randomly generated initial points. Dimension n and Degree n − 1 Suggests that only locally optimal values of f are 0.5 and 1. Nonsmooth Analysis of the Crouzeix Ratio 19 / 39 Concluding Remarks

Final Fields of Values for Lowest Computed f n=3 n=4 n=5 2 1.5 Crouzeix’s 0.8 1.5 Conjecture 0.6 1 1 0.4 0.5 Nonsmooth 0.5 0.2 Optimization of 0 0 0 the Crouzeix Ratio -0.2 -0.5 -0.5 Nonsmoothness of -0.4 -1 -1 -0.6 the Crouzeix Ratio -1.5 -0.8 -1.5 BFGS -2 -1 0 1 2 -1 -0.5 0 -1 0 1 Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) n=6 Final Fields of n=7 n=8 Values for Lowest 6 6 5 Computed f 4 4 Optimizing over 2 2 both p and A : Final f ( p, A ) 0 0 0 Is the Ratio 0 . 5 -2 -2 Attained? -4 -4 Final Fields of -5 -6 -6 Values for f Closest -5 0 5 -6 -4 -2 0 2 4 -5 0 5 to 1 Why is the Crouzeix Solid blue curve is boundary of field of values of final computed A Ratio One? Results for Larger Blue asterisks are eigenvalues of final computed A Dimension n and Degree n − 1 Small red circles are roots of final computed p Nonsmooth Analysis of the Crouzeix Ratio 20 / 39 Concluding Remarks

Final Fields of Values for Lowest Computed f n=3 n=4 n=5 2 1.5 Crouzeix’s 0.8 1.5 Conjecture 0.6 1 1 0.4 0.5 Nonsmooth 0.5 0.2 Optimization of 0 0 0 the Crouzeix Ratio -0.2 -0.5 -0.5 Nonsmoothness of -0.4 -1 -1 -0.6 the Crouzeix Ratio -1.5 -0.8 -1.5 BFGS -2 -1 0 1 2 -1 -0.5 0 -1 0 1 Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) n=6 Final Fields of n=7 n=8 Values for Lowest 6 6 5 Computed f 4 4 Optimizing over 2 2 both p and A : Final f ( p, A ) 0 0 0 Is the Ratio 0 . 5 -2 -2 Attained? -4 -4 Final Fields of -5 -6 -6 Values for f Closest -5 0 5 -6 -4 -2 0 2 4 -5 0 5 to 1 Why is the Crouzeix Solid blue curve is boundary of field of values of final computed A Ratio One? Results for Larger Blue asterisks are eigenvalues of final computed A Dimension n and Degree n − 1 Small red circles are roots of final computed p Nonsmooth Analysis of the Crouzeix n = 3 , 4 , 5 : two eigenvalues of A and one root of p nearly coincident Ratio 20 / 39 Concluding Remarks

Optimizing over both p and A : Final f ( p, A ) Crouzeix’s Conjecture n f Nonsmooth Optimization of 3 0 . 500000000000000 the Crouzeix Ratio Nonsmoothness of 4 0 . 500000000000000 the Crouzeix Ratio BFGS 5 0 . 500000000000014 Experiments 6 0 . 500000017156953 Optimizing over A (order n ) and p 7 0 . 500000746246673 (deg ≤ n − 1 ) Final Fields of 8 0 . 500000206563813 Values for Lowest Computed f Optimizing over f is the lowest value f ( p, A ) found over 100 runs both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 21 / 39 Concluding Remarks

Is the Ratio 0 . 5 Attained? Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio Nonsmoothness of the Crouzeix Ratio BFGS Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 22 / 39 Concluding Remarks

Is the Ratio 0 . 5 Attained? Independently, Crabb, Choi and Crouzeix showed that the ratio 0 . 5 is attained if p ( ζ ) = ζ n − 1 and A is the n by n matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio   · · Nonsmoothness of   � � 0 2 the Crouzeix Ratio   if n = 2 , or · · if n > 2   BFGS 0 0   · 1 Experiments √   Optimizing over A   · 2 (order n ) and p   (deg ≤ n − 1 ) 0 Final Fields of Values for Lowest for which W ( A ) is the unit disk. Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 22 / 39 Concluding Remarks

Is the Ratio 0 . 5 Attained? Independently, Crabb, Choi and Crouzeix showed that the ratio 0 . 5 is attained if p ( ζ ) = ζ n − 1 and A is the n by n matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio   · · Nonsmoothness of   � � 0 2 the Crouzeix Ratio   if n = 2 , or · · if n > 2   BFGS 0 0   · 1 Experiments √   Optimizing over A   · 2 (order n ) and p   (deg ≤ n − 1 ) 0 Final Fields of Values for Lowest for which W ( A ) is the unit disk. Computed f Optimizing over Our computed minimizers are nearly equivalent to such pairs ( p , A ) both p and A : Final (with A changed via unitary similarity transformations, multiplication f ( p, A ) Is the Ratio 0 . 5 by a scalar, by shifting the root of p and eigenvalue of A by the same Attained? Final Fields of scalar, and by appending another diagonal block whose field of values Values for f Closest is contained in that of the first block) to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 22 / 39 Concluding Remarks

Is the Ratio 0 . 5 Attained? Independently, Crabb, Choi and Crouzeix showed that the ratio 0 . 5 is attained if p ( ζ ) = ζ n − 1 and A is the n by n matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio   · · Nonsmoothness of   � � 0 2 the Crouzeix Ratio   if n = 2 , or · · if n > 2   BFGS 0 0   · 1 Experiments √   Optimizing over A   · 2 (order n ) and p   (deg ≤ n − 1 ) 0 Final Fields of Values for Lowest for which W ( A ) is the unit disk. Computed f Optimizing over Our computed minimizers are nearly equivalent to such pairs ( p , A ) both p and A : Final (with A changed via unitary similarity transformations, multiplication f ( p, A ) Is the Ratio 0 . 5 by a scalar, by shifting the root of p and eigenvalue of A by the same Attained? Final Fields of scalar, and by appending another diagonal block whose field of values Values for f Closest is contained in that of the first block) to 1 Why is the Crouzeix Conjecture: these are the only cases where f ( p, A ) = 0 . 5 . Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 22 / 39 Concluding Remarks

Is the Ratio 0 . 5 Attained? Independently, Crabb, Choi and Crouzeix showed that the ratio 0 . 5 is attained if p ( ζ ) = ζ n − 1 and A is the n by n matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio   · · Nonsmoothness of   � � 0 2 the Crouzeix Ratio   if n = 2 , or · · if n > 2   BFGS 0 0   · 1 Experiments √   Optimizing over A   · 2 (order n ) and p   (deg ≤ n − 1 ) 0 Final Fields of Values for Lowest for which W ( A ) is the unit disk. Computed f Optimizing over Our computed minimizers are nearly equivalent to such pairs ( p , A ) both p and A : Final (with A changed via unitary similarity transformations, multiplication f ( p, A ) Is the Ratio 0 . 5 by a scalar, by shifting the root of p and eigenvalue of A by the same Attained? Final Fields of scalar, and by appending another diagonal block whose field of values Values for f Closest is contained in that of the first block) to 1 Why is the Crouzeix Conjecture: these are the only cases where f ( p, A ) = 0 . 5 . Ratio One? Results for Larger Dimension n and f is nonsmooth at these pairs ( p, A ) because | p | is constant on the Degree n − 1 boundary of W ( A ) . Nonsmooth Analysis of the Crouzeix Ratio 22 / 39 Concluding Remarks

Final Fields of Values for f Closest to 1 n=3 n=4 n=5 1.5 4 5 Crouzeix’s 4 3 1 Conjecture 3 2 2 0.5 Nonsmooth 1 1 Optimization of 0 0 0 the Crouzeix Ratio -1 Nonsmoothness of -1 -0.5 -2 the Crouzeix Ratio -2 -3 -1 BFGS -3 -4 Experiments -5 -4 -1.5 -1.5 -1 -0.5 0 0.5 -4 -2 0 2 -8 -6 -4 -2 0 Optimizing over A (order n ) and p (deg ≤ n − 1 ) n=6 n=7 n=8 Final Fields of 4 5 Values for Lowest 10 4 3 Computed f 3 Optimizing over 2 5 2 both p and A : Final 1 1 f ( p, A ) 0 0 0 Is the Ratio 0 . 5 -1 -1 Attained? -5 -2 -2 Final Fields of -3 Values for f Closest -10 -3 -4 to 1 -5 -4 Why is the Crouzeix -15 -10 -5 0 5 0 2 4 -6 -4 -2 0 2 Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 23 / 39 Concluding Remarks

Final Fields of Values for f Closest to 1 n=3 n=4 n=5 1.5 4 5 Crouzeix’s 4 3 1 Conjecture 3 2 2 0.5 Nonsmooth 1 1 Optimization of 0 0 0 the Crouzeix Ratio -1 Nonsmoothness of -1 -0.5 -2 the Crouzeix Ratio -2 -3 -1 BFGS -3 -4 Experiments -5 -4 -1.5 -1.5 -1 -0.5 0 0.5 -4 -2 0 2 -8 -6 -4 -2 0 Optimizing over A (order n ) and p (deg ≤ n − 1 ) n=6 n=7 n=8 Final Fields of 4 5 Values for Lowest 10 4 3 Computed f 3 Optimizing over 2 5 2 both p and A : Final 1 1 f ( p, A ) 0 0 0 Is the Ratio 0 . 5 -1 -1 Attained? -5 -2 -2 Final Fields of -3 Values for f Closest -10 -3 -4 to 1 -5 -4 Why is the Crouzeix -15 -10 -5 0 5 0 2 4 -6 -4 -2 0 2 Ratio One? Results for Larger Dimension n and Ice cream cone shape: Degree n − 1 exactly one eigenvalue at a vertex of the field of values Nonsmooth Analysis of the Crouzeix Ratio 23 / 39 Concluding Remarks

Why is the Crouzeix Ratio One? Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio Nonsmoothness of the Crouzeix Ratio BFGS Experiments Optimizing over A (order n ) and p (deg ≤ n − 1 ) Final Fields of Values for Lowest Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 24 / 39 Concluding Remarks

Why is the Crouzeix Ratio One? Because for this computed local minimizer, A is nearly unitarily similar to a block diagonal matrix Crouzeix’s Conjecture Nonsmooth diag( λ, B ) , λ ∈ R Optimization of the Crouzeix Ratio Nonsmoothness of so the Crouzeix Ratio W ( A ) ≈ conv( λ, W ( B )) BFGS Experiments Optimizing over A with λ active and the block B inactive , that is: (order n ) and p (deg ≤ n − 1 ) � p � W ( A ) is attained only at λ ■ Final Fields of Values for Lowest | p ( λ ) | > � p ( B ) � 2 ■ Computed f Optimizing over both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 24 / 39 Concluding Remarks

Why is the Crouzeix Ratio One? Because for this computed local minimizer, A is nearly unitarily similar to a block diagonal matrix Crouzeix’s Conjecture Nonsmooth diag( λ, B ) , λ ∈ R Optimization of the Crouzeix Ratio Nonsmoothness of so the Crouzeix Ratio W ( A ) ≈ conv( λ, W ( B )) BFGS Experiments Optimizing over A with λ active and the block B inactive , that is: (order n ) and p (deg ≤ n − 1 ) � p � W ( A ) is attained only at λ ■ Final Fields of Values for Lowest | p ( λ ) | > � p ( B ) � 2 ■ Computed f Optimizing over So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . both p and A : Final f ( p, A ) Is the Ratio 0 . 5 Attained? Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 24 / 39 Concluding Remarks

Why is the Crouzeix Ratio One? Because for this computed local minimizer, A is nearly unitarily similar to a block diagonal matrix Crouzeix’s Conjecture Nonsmooth diag( λ, B ) , λ ∈ R Optimization of the Crouzeix Ratio Nonsmoothness of so the Crouzeix Ratio W ( A ) ≈ conv( λ, W ( B )) BFGS Experiments Optimizing over A with λ active and the block B inactive , that is: (order n ) and p (deg ≤ n − 1 ) � p � W ( A ) is attained only at λ ■ Final Fields of Values for Lowest | p ( λ ) | > � p ( B ) � 2 ■ Computed f Optimizing over So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . both p and A : Final f ( p, A ) Furthermore, f is differentiable at this pair ( p, A ) , with zero gradient. Is the Ratio 0 . 5 Attained? Thus, such ( p, A ) is a smooth stationary point of f . Final Fields of Values for f Closest to 1 Why is the Crouzeix Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 24 / 39 Concluding Remarks

Why is the Crouzeix Ratio One? Because for this computed local minimizer, A is nearly unitarily similar to a block diagonal matrix Crouzeix’s Conjecture Nonsmooth diag( λ, B ) , λ ∈ R Optimization of the Crouzeix Ratio Nonsmoothness of so the Crouzeix Ratio W ( A ) ≈ conv( λ, W ( B )) BFGS Experiments Optimizing over A with λ active and the block B inactive , that is: (order n ) and p (deg ≤ n − 1 ) � p � W ( A ) is attained only at λ ■ Final Fields of Values for Lowest | p ( λ ) | > � p ( B ) � 2 ■ Computed f Optimizing over So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . both p and A : Final f ( p, A ) Furthermore, f is differentiable at this pair ( p, A ) , with zero gradient. Is the Ratio 0 . 5 Attained? Thus, such ( p, A ) is a smooth stationary point of f . Final Fields of Values for f Closest This doesn’t imply that it is a local minimizer, but the numerical to 1 Why is the Crouzeix results make this evident. Ratio One? Results for Larger Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 24 / 39 Concluding Remarks

Why is the Crouzeix Ratio One? Because for this computed local minimizer, A is nearly unitarily similar to a block diagonal matrix Crouzeix’s Conjecture Nonsmooth diag( λ, B ) , λ ∈ R Optimization of the Crouzeix Ratio Nonsmoothness of so the Crouzeix Ratio W ( A ) ≈ conv( λ, W ( B )) BFGS Experiments Optimizing over A with λ active and the block B inactive , that is: (order n ) and p (deg ≤ n − 1 ) � p � W ( A ) is attained only at λ ■ Final Fields of Values for Lowest | p ( λ ) | > � p ( B ) � 2 ■ Computed f Optimizing over So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . both p and A : Final f ( p, A ) Furthermore, f is differentiable at this pair ( p, A ) , with zero gradient. Is the Ratio 0 . 5 Attained? Thus, such ( p, A ) is a smooth stationary point of f . Final Fields of Values for f Closest This doesn’t imply that it is a local minimizer, but the numerical to 1 Why is the Crouzeix results make this evident. Ratio One? Results for Larger As n increases, ice cream cone stationary points become increasingly Dimension n and Degree n − 1 common and it becomes very difficult to reduce f below 1. Nonsmooth Analysis of the Crouzeix Ratio 24 / 39 Concluding Remarks

Results for Larger Dimension n and Degree n − 1 n=9 n=10 n=12 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of 0.7 0.7 0.7 the Crouzeix Ratio Nonsmoothness of 0.6 0.6 0.6 the Crouzeix Ratio 0.5 0.5 0.5 BFGS 0.4 0.4 0.4 0 100 200 300 400 500 0 100 200 300 400 0 2000 4000 6000 8000 Experiments Optimizing over A (order n ) and p n=14 n=15 n=16 (deg ≤ n − 1 ) 1.1 1.1 1.1 Final Fields of 1 1 1 Values for Lowest 0.9 0.9 0.9 Computed f Optimizing over 0.8 0.8 0.8 both p and A : Final 0.7 0.7 0.7 f ( p, A ) 0.6 0.6 0.6 Is the Ratio 0 . 5 0.5 0.5 0.5 Attained? Final Fields of 0.4 0.4 0.4 0 1000 2000 3000 0 1000 2000 3000 0 500 1000 1500 2000 Values for f Closest to 1 Why is the Crouzeix Sorted final values of the Crouzeix ratio f Ratio One? Results for Larger found starting from many randomly generated initial points. Dimension n and Degree n − 1 Nonsmooth Analysis of the Crouzeix Ratio 25 / 39 Concluding Remarks

Results for Larger Dimension n and Degree n − 1 n=9 n=10 n=12 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of 0.7 0.7 0.7 the Crouzeix Ratio Nonsmoothness of 0.6 0.6 0.6 the Crouzeix Ratio 0.5 0.5 0.5 BFGS 0.4 0.4 0.4 0 100 200 300 400 500 0 100 200 300 400 0 2000 4000 6000 8000 Experiments Optimizing over A (order n ) and p n=14 n=15 n=16 (deg ≤ n − 1 ) 1.1 1.1 1.1 Final Fields of 1 1 1 Values for Lowest 0.9 0.9 0.9 Computed f Optimizing over 0.8 0.8 0.8 both p and A : Final 0.7 0.7 0.7 f ( p, A ) 0.6 0.6 0.6 Is the Ratio 0 . 5 0.5 0.5 0.5 Attained? Final Fields of 0.4 0.4 0.4 0 1000 2000 3000 0 1000 2000 3000 0 500 1000 1500 2000 Values for f Closest to 1 Why is the Crouzeix Sorted final values of the Crouzeix ratio f Ratio One? Results for Larger found starting from many randomly generated initial points. Dimension n and Degree n − 1 There are other locally optimal values of f between 0.5 and 1 ! Nonsmooth Analysis of the Crouzeix Ratio 25 / 39 Concluding Remarks

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio The Clarke Subdifferential Nonsmooth Analysis of the Crouzeix Ratio The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 26 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis The Clarke subdifferential, or set of subgradients, of h at ¯ x is of the Crouzeix Ratio The Clarke Subdifferential � � The Gradient or ∂h (¯ x ) = conv lim x,x ∈ D ∇ h ( x ) . Subgradients of the x → ¯ Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis The Clarke subdifferential, or set of subgradients, of h at ¯ x is of the Crouzeix Ratio The Clarke Subdifferential � � The Gradient or ∂h (¯ x ) = conv lim x,x ∈ D ∇ h ( x ) . Subgradients of the x → ¯ Crouzeix Ratio Regularity Simplest Case where F.H. Clarke, 1973 (he used the name “generalized gradient”). Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis The Clarke subdifferential, or set of subgradients, of h at ¯ x is of the Crouzeix Ratio The Clarke Subdifferential � � The Gradient or ∂h (¯ x ) = conv lim x,x ∈ D ∇ h ( x ) . Subgradients of the x → ¯ Crouzeix Ratio Regularity Simplest Case where F.H. Clarke, 1973 (he used the name “generalized gradient”). Crouzeix Ratio is Nonsmooth If h is continuously differentiable at ¯ x , then ∂h (¯ x ) = {∇ h (¯ x ) } . c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis The Clarke subdifferential, or set of subgradients, of h at ¯ x is of the Crouzeix Ratio The Clarke Subdifferential � � The Gradient or ∂h (¯ x ) = conv lim x,x ∈ D ∇ h ( x ) . Subgradients of the x → ¯ Crouzeix Ratio Regularity Simplest Case where F.H. Clarke, 1973 (he used the name “generalized gradient”). Crouzeix Ratio is Nonsmooth If h is continuously differentiable at ¯ x , then ∂h (¯ x ) = {∇ h (¯ x ) } . c, ˆ (ˆ A ) is a Nonsmooth If h is convex, ∂h is the subdifferential of convex analysis. Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis The Clarke subdifferential, or set of subgradients, of h at ¯ x is of the Crouzeix Ratio The Clarke Subdifferential � � The Gradient or ∂h (¯ x ) = conv lim x,x ∈ D ∇ h ( x ) . Subgradients of the x → ¯ Crouzeix Ratio Regularity Simplest Case where F.H. Clarke, 1973 (he used the name “generalized gradient”). Crouzeix Ratio is Nonsmooth If h is continuously differentiable at ¯ x , then ∂h (¯ x ) = {∇ h (¯ x ) } . c, ˆ (ˆ A ) is a Nonsmooth If h is convex, ∂h is the subdifferential of convex analysis. Stationary Point of f ( · , · ) We say ¯ x is Clarke stationary for h if 0 ∈ ∂h (¯ x ) (a nonsmooth The General Case c, ˆ (ˆ A ) is a stationary point if ∈ ∂h (¯ x ) contains more than one vector) Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 27 / 39

The Clarke Subdifferential Assume h : R n → R is locally Lipschitz, and Crouzeix’s Conjecture let D = { x ∈ R n : h is differentiable at x } . Nonsmooth Optimization of Rademacher’s Theorem: R n \ D has measure zero. the Crouzeix Ratio Nonsmooth Analysis The Clarke subdifferential, or set of subgradients, of h at ¯ x is of the Crouzeix Ratio The Clarke Subdifferential � � The Gradient or ∂h (¯ x ) = conv lim x,x ∈ D ∇ h ( x ) . Subgradients of the x → ¯ Crouzeix Ratio Regularity Simplest Case where F.H. Clarke, 1973 (he used the name “generalized gradient”). Crouzeix Ratio is Nonsmooth If h is continuously differentiable at ¯ x , then ∂h (¯ x ) = {∇ h (¯ x ) } . c, ˆ (ˆ A ) is a Nonsmooth If h is convex, ∂h is the subdifferential of convex analysis. Stationary Point of f ( · , · ) We say ¯ x is Clarke stationary for h if 0 ∈ ∂h (¯ x ) (a nonsmooth The General Case c, ˆ (ˆ A ) is a stationary point if ∈ ∂h (¯ x ) contains more than one vector) Nonsmooth Stationary Point of f ( · , · ) Clarke stationarity is a necessary condition for local or global Is the Crouzeix Ratio optimality. Globally Clarke Regular? Concluding Remarks 27 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the Crouzeix Ratio Nonsmooth Analysis of the Crouzeix Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the gradient of p ( z θ ) w.r.t. z θ the Crouzeix Ratio ■ Nonsmooth Analysis of the Crouzeix Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the gradient of p ( z θ ) w.r.t. z θ the Crouzeix Ratio ■ Nonsmooth Analysis the gradient of z θ ( A ) = v ∗ θ Av θ w.r.t. A ■ of the Crouzeix Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the gradient of p ( z θ ) w.r.t. z θ the Crouzeix Ratio ■ Nonsmooth Analysis the gradient of z θ ( A ) = v ∗ θ Av θ w.r.t. A ■ of the Crouzeix Ratio If the max of | p ( z θ ) | is attained by a unique point ˆ θ , then all The Clarke Subdifferential these are evaluated at ˆ θ and combined with the gradient of | · | The Gradient or Subgradients of the to obtain the gradient of the numerator. Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the gradient of p ( z θ ) w.r.t. z θ the Crouzeix Ratio ■ Nonsmooth Analysis the gradient of z θ ( A ) = v ∗ θ Av θ w.r.t. A ■ of the Crouzeix Ratio If the max of | p ( z θ ) | is attained by a unique point ˆ θ , then all The Clarke Subdifferential these are evaluated at ˆ θ and combined with the gradient of | · | The Gradient or Subgradients of the to obtain the gradient of the numerator. Crouzeix Ratio Regularity Otherwise, need to take the convex hull of these gradients over Simplest Case where Crouzeix Ratio is all maximizing θ to get the subgradients of the numerator. Nonsmooth c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the gradient of p ( z θ ) w.r.t. z θ the Crouzeix Ratio ■ Nonsmooth Analysis the gradient of z θ ( A ) = v ∗ θ Av θ w.r.t. A ■ of the Crouzeix Ratio If the max of | p ( z θ ) | is attained by a unique point ˆ θ , then all The Clarke Subdifferential these are evaluated at ˆ θ and combined with the gradient of | · | The Gradient or Subgradients of the to obtain the gradient of the numerator. Crouzeix Ratio Regularity Otherwise, need to take the convex hull of these gradients over Simplest Case where Crouzeix Ratio is all maximizing θ to get the subgradients of the numerator. Nonsmooth c, ˆ (ˆ A ) is a For the denominator, combine: Nonsmooth Stationary Point of f ( · , · ) The General Case c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39

The Gradient or Subgradients of the Crouzeix Ratio For the numerator, we need the variational properties of z θ = v ∗ θ ∈ [0 , 2 π ] | p ( z θ ) | max where θ Av θ . Crouzeix’s Conjecture Nonsmooth the gradient of p ( z θ ) w.r.t. the coefficients of p ■ Optimization of the gradient of p ( z θ ) w.r.t. z θ the Crouzeix Ratio ■ Nonsmooth Analysis the gradient of z θ ( A ) = v ∗ θ Av θ w.r.t. A ■ of the Crouzeix Ratio If the max of | p ( z θ ) | is attained by a unique point ˆ θ , then all The Clarke Subdifferential these are evaluated at ˆ θ and combined with the gradient of | · | The Gradient or Subgradients of the to obtain the gradient of the numerator. Crouzeix Ratio Regularity Otherwise, need to take the convex hull of these gradients over Simplest Case where Crouzeix Ratio is all maximizing θ to get the subgradients of the numerator. Nonsmooth c, ˆ (ˆ A ) is a For the denominator, combine: Nonsmooth Stationary Point of the gradient or subgradients of the 2-norm (maximum singular ■ f ( · , · ) The General Case value) of a matrix (involves the singular vectors) c, ˆ (ˆ A ) is a Nonsmooth Stationary Point of f ( · , · ) Is the Crouzeix Ratio Globally Clarke Regular? Concluding Remarks 28 / 39