Topologically relevant stationarity concepts Oliver Stein Institute - PowerPoint PPT Presentation

The unconstrained smooth case The constrained smooth case Mathematical programs with complementarity constraints Mathematical programs with vanishing constraints Topologically relevant stationarity concepts Oliver Stein Institute of Operations Research Karlsruhe Institute of Technology ICCP 2014 Humboldt-Universit¨ at zu Berlin August 4–8, 2014 1 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case The constrained smooth case Mathematical programs with complementarity constraints Mathematical programs with vanishing constraints Survey The unconstrained smooth case 1 The constrained smooth case 2 Mathematical programs with complementarity constraints 3 Mathematical programs with vanishing constraints 4 2 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case The constrained smooth case Mathematical programs with complementarity constraints Mathematical programs with vanishing constraints Four reasons to look at stationary points Candidates for local minimizers Design of homotopy methods Understanding the problem structure (Morse theory) Convergence results for KKT points 3 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case The constrained smooth case Mathematical programs with complementarity constraints Mathematical programs with vanishing constraints Four reasons to look at stationary points Candidates for local minimizers Design of homotopy methods Understanding the problem structure (Morse theory) Convergence results for KKT points 3 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = − 3 4 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = − 2 5 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = − 1 6 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = 0 7 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = 1 8 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = 2 9 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization f ( t , x ) = x 4 4 x 2 − tx 8 − 3 for t = 3 10 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization x 0 t 0 Unfolded set of global minimizers 11 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Example: parametric unconstrained smooth optimization x glob. min. loc. min. loc. max. 0 loc. min. glob. min. t 0 Unfolded set of critical points 12 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Nondegenerate critical points Necessary condition x ∈ R n local minimizer of f ¯ ⇒ ∇ f (¯ x ) = 0. Definitions x ∈ R n is called nondegenerate critical point of f ∈ C 2 ( R n , R ), ¯ x ) = 0, and D 2 f (¯ if ∇ f (¯ x ) is nonsingular. The number of negative eigenvalues of D 2 f (¯ x ) is called the Morse index or quadratic index of ¯ x , briefly QI (¯ x ). Theorem (Jongen/Jonker/Twilt, 1983) Generically, all critical points of f ∈ C 2 ( R n , R ) are nondegenerate. 13 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Nondegenerate critical points Necessary condition x ∈ R n local minimizer of f ¯ ⇒ ∇ f (¯ x ) = 0. Definitions x ∈ R n is called nondegenerate critical point of f ∈ C 2 ( R n , R ), ¯ x ) = 0, and D 2 f (¯ if ∇ f (¯ x ) is nonsingular. The number of negative eigenvalues of D 2 f (¯ x ) is called the Morse index or quadratic index of ¯ x , briefly QI (¯ x ). Theorem (Jongen/Jonker/Twilt, 1983) Generically, all critical points of f ∈ C 2 ( R n , R ) are nondegenerate. 13 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Nondegenerate critical points Necessary condition x ∈ R n local minimizer of f ¯ ⇒ ∇ f (¯ x ) = 0. Definitions x ∈ R n is called nondegenerate critical point of f ∈ C 2 ( R n , R ), ¯ x ) = 0, and D 2 f (¯ if ∇ f (¯ x ) is nonsingular. The number of negative eigenvalues of D 2 f (¯ x ) is called the Morse index or quadratic index of ¯ x , briefly QI (¯ x ). Theorem (Jongen/Jonker/Twilt, 1983) Generically, all critical points of f ∈ C 2 ( R n , R ) are nondegenerate. 13 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Nondegenerate critical points Characterization of local minimality For any nondegenerate critical point ¯ x of f we have ¯ x is a local minimizer of f ⇔ QI (¯ x ) = 0 . Theorem (Morse Lemma - local structure) x be a nondegenerate critical point of f ∈ C 2 ( R n , R ). Let ¯ Then, modulo a local C 1 diffeomorphism, locally around ¯ x we have f ( x ) = − x 2 1 − x 2 2 − . . . − x 2 x ) + x 2 x )+1 + . . . + x 2 n . QI (¯ QI (¯ 14 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Nondegenerate critical points Characterization of local minimality For any nondegenerate critical point ¯ x of f we have ¯ x is a local minimizer of f ⇔ QI (¯ x ) = 0 . Theorem (Morse Lemma - local structure) x be a nondegenerate critical point of f ∈ C 2 ( R n , R ). Let ¯ Then, modulo a local C 1 diffeomorphism, locally around ¯ x we have f ( x ) = − x 2 1 − x 2 2 − . . . − x 2 x ) + x 2 x )+1 + . . . + x 2 n . QI (¯ QI (¯ 14 / 45 Oliver Stein Topologically relevant stationarity concepts

The unconstrained smooth case Homotopy The constrained smooth case Necessary condition and nondegeneracy Mathematical programs with complementarity constraints Morse theory Mathematical programs with vanishing constraints Deformation and cell attachment - global structure lev α f gph f 15 / 45 Oliver Stein Topologically relevant stationarity concepts