CSCI 1951-G – Optimization Methods in Finance Part 10: Conic Optimization April 6, 2018 1 / 34
This material is covered in the textbook, Chapters 9 and 10. Some of the materials are taken from it. Some of the figures are from S. Boyd, L. Vandenberge’s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/ . 2 / 34
Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 3 / 34
Cones A set C is a cone if for every x ∈ C and θ ≥ 0 , θx ∈ C Example: { ( x, | x | ) , x ∈ R } ⊂ R 2 Is this set convex? 4 / 34
Convex Cones A set C is a convex cone if, for every x 1 , x 2 ∈ C and θ 1 , θ 2 ≥ 0 , θ 1 x 1 + θ 2 x 2 ∈ C . Example: x 1 x 2 0 Figure 2.4 The pie slice shows all points of the form θ 1 x 1 + θ 2 x 2 , where θ 1 , θ 2 ≥ 0. The apex of the slice (which corresponds to θ 1 = θ 2 = 0) is at 0; its edges (which correspond to θ 1 = 0 or θ 2 = 0) pass through the points x 1 and x 2 . 5 / 34
Conic optimization Conic optimization problem in standard form: min c T x Ax = b x ∈ C where C is a convex cone in finite-dimensional vector space X . Note: linear objective function, linear constraints. If X = R n and C = R n + , then ...we get an LP! Conic optimization is a unifying framework for • linear programming, • second-order cone programming (SOCP), • semidefinite programming (SDP). 6 / 34
Norm cones Let � · � be any norm in R n − 1 . The norm cone associated to � · � is the set C = { x = ( x 1 , . . . , x n ) : x 1 ≥ � ( x 2 , . . . , x n ) �} It is a convex set. 7 / 34
Second-order cone in R 3 The second-order cone is the norm cone for the Euclidean norm � · � 2 . 1 0 . 5 t 0 1 1 0 0 − 1 − 1 x 2 x 1 2 ) 1 / 2 ≤ Figure 2.10 Boundary of second-order cone in R 3 , { ( x 1 , x 2 , t ) | ( x 2 1 + x 2 t } . What happens when we slice the second-order cone? I.e., when we take the intersection with a hyperplane ? We obtain ellipsoidal sets. 8 / 34
Rewriting constraints Let’s rewrite C = { x = ( x 1 , . . . , x n ) : x 1 ≥ � ( x 2 , . . . , x n ) � 2 } as x 2 1 − x 2 2 − · · · x 2 x 1 ≥ 0 , n ≥ 0 This is a combination of an linear and a quadratic constraints. Also: convex quadratic constraints can be expressed as second-order cone membership constraints. 9 / 34
Rewriting constraints Qadratic constraint: x T Px + 2 q T x + γ ≤ 0 Assume P w.l.o.g. positive definite, so the constraint is ...convex. Also assume, for technical reasons, that q T Pq − γ ≥ 0 . Goal: rewrite the above constraint as a combination of linear and second-order cone membership constraints. 10 / 34
Rewriting constraints Because P is positive definitive, it has a Cholesky decomposition : ∃ invertible R s.t. P = RR T . Rewrite the constraint as: ( R T x ) T ( R T x ) + 2 q T x + γ ≤ 0 Let y = ( y 1 , . . . , y n ) T = R T x + R − 1 q The above is a bijection between x and y . We are going to rewrite the constraint as a constraint on y . 11 / 34
Rewriting constraints The constraint: ( R T x ) T ( R T x ) + 2 q T x + γ ≤ 0 It holds y T y = ( R T x ) T ( R T x ) + 2 q T x + q T P − 1 q Since there is a bijection between y and x , the constraint can be satisfied if and only if ∃ y s.t. y = R T x + R − 1 q, y T y ≤ q T Pq − γ 12 / 34
Rewriting constraints The constraint is equivalent to: ∃ y s.t. y = R T x + R − 1 q, y T y ≤ q T Pq − γ Lets denote with y 0 the square root of the r.h.s. of the right inequality : � q T Pq − γ ∈ R + y 0 = Consider the vector ( y 0 , y 1 , . . . , y n ) . � �� � y The right inequality then is n � y 2 0 ≥ y T y = y 2 i i =1 Taking the square root on both sides: � � n � � y 2 y 0 ≥ i = � y � 2 � i =1 This is the membership constraint for the second-order cone in R n +1 . 13 / 34
Rewriting constraints We rewrite the convex quadratic constraint x T Px + 2 q T x + γ ≤ 0 as ( y 1 , . . . , y n ) T = R T x + R − 1 q � y 0 = q T Pq − γ ∈ R + ( y 0 , y 1 , . . . , y n ) ∈ C which is a combination of linear and second-order cone membership constraints. 14 / 34
Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 6. SDP and approximation algorithms 15 / 34
Benchmark-relative portfolio optimization Given a benchmark strategy x B (e.g., an index ) develop a portfolio x that tracks x B , but adds value by beating it. I.e., we want a portfolio x with positive expected excess return : µ T ( x − x B ) ≥ 0 and specifically want to maximize the expected excess return. Challenge: balance expected excess return with its variance . 16 / 34
Tracking error and volatility constraints The (predicted) tracking error of the portfolio x is � ( x − x B ) T Σ( x − x B ) TE ( x ) = It measure the variability of excess returns . In benchmark-relative portfolio optimization, we solve mean-variance optimization w.r.t. the expected excess return and tracking error: max µ T ( x − x B ) ( x − x B ) T Σ( x − x B ) ≤ T 2 Ax = b 17 / 34
Comparison with mean-variance optimization We have seen MVO as: min 1 2 x T Σ x max µ T x − δ 2Σ x µ T x ≥ R or Ax = b Ax = b How do they differ from max µ T ( x − x B ) ( x − x B ) T Σ( x − x B ) ≤ T 2 Ax = b The later is not a standard quadratic program: it has a nonlinear constraint. 18 / 34
max µ T ( x − x B ) ( x − x B ) T Σ( x − x B ) ≤ T 2 Ax = b The nonlinear constraint is ... convex quadratic We can rewrite it as a combination of linear and second-order cone membership, and solve the resulting convex conic problem. 19 / 34
Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 6. SDP and approximation algorithms 20 / 34
SemiDefinite Programming (SDP) The variables are the entries of a symmetric matrix in the cone of positive semidefinite matrices. 1 0 . 5 z 0 1 1 0 0 . 5 y − 1 0 x Figure 2.12 Boundary of positive semidefinite cone in S 2 . 21 / 34
Application: approximating covariance matrices Portfolio Optimization almost always requires covariance matrices . These are not directly available, but are estimated . Estimation of covariance matrices is a very challenging task , mathematically and computationally, because the matrices must satisfy various properties (e.g., symmetry, positive semidefiniteness). To be efficient , many estimation methods do not impose problem-dependent constraints. Typically, one is interested in finding the smallest distortion of the original estimate that satisfies the desired constraints; 22 / 34
Application: approximating covariance matrices • Let ˆ Σ ∈ S n be an estimate of a covariance matrix • ˆ Σ is symmetric ( ∈ S n ) but not positive semidefinite. Goal: find the positive semidefinite matrix that is closest to ˆ Σ w.r.t. the Frobenius norm : �� d F (Σ , ˆ (Σ ij − ˆ Σ ij ) 2 Σ) = i,j Formally: nearest covariance matrix problem: d F (Σ , ˆ min Σ) Σ Σ ∈ C n s where C n s is the cone of n × n symmetric and positive semidefinite matrices. 23 / 34
Application: approximating covariance matrices d F (Σ , ˆ min Σ) Σ Σ ∈ C n s Introduce a dummy variable t and rewrite the problem as min t d F (Σ , ˆ Σ) ≤ t Σ ∈ C n s The first constraint can be writen as a second-order cone constraint, so the problem is transformed into a conic optimization problem. 24 / 34
Application: approximating covariance matrices Variation of the problem with additional linear constraints : Let E ⊆ { ( i, j ) : 1 ≤ i ≤ n } Let ( ℓ ij , u ij ) , for ( i, j ) ∈ E be lower/upper bounds to impose to the entries. We want to solve: d F (Σ , ˆ min Σ) Σ ℓ ij < Σ ij < u ij , ∀ ( i, j ) ∈ E Σ ∈ C n s 25 / 34
Application: approximating covariance matrices For example, let ˆ Σ be an estimation of a correlation matrix. Correlation matrix have all diagonal entries equal to 1. We want to solve the nearest correlation matrix problem. We choose E = { ( i, i ) , 1 ≤ i ≤ n } , ℓ i = 1 = u i , 1 ≤ i ≤ n 26 / 34
Application: approximating covariance matrices Many other variants are possible: • Force some entries of ˆ Σ to remain the same in Σ ; • Weight the changes to different entries differently, because we trust some more than other; • Impose lower bounds to the minimum eigenvalue of Σ , to reduce instability; All of these can be easily solved with SDP sofware. 27 / 34
Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 6. SDP and approximation algorithms 28 / 34
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