On Second-Order Optimality Conditions for Conic Programming Héctor Ramírez C. 1 1 Departamento Ingeniería Matemática & Centro Modelamiento Matemático, Universidad de Chile, Santiago de Chile 6èmes Journées Franco-Chiliennes d’Optimisation May 19th, 2008 Université du Sud Toulon-Var Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 1 / 24
Outline Introduction and Motivation 1 Formulation of Our Problem Applications of SDP and SOCP Optimality Conditions Constraint Qualification Conditions Reduction Approach Main results 2 Duality Results for Conic Programming Strong Regularity Condition Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 2 / 24
Outline Introduction and Motivation 1 Formulation of Our Problem Applications of SDP and SOCP Optimality Conditions Constraint Qualification Conditions Reduction Approach Main results 2 Duality Results for Conic Programming Strong Regularity Condition Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 3 / 24
Conic Programming Consider the optimization problem over a closed convex cone K x ∈ X f ( x ) ; g ( x ) ∈ K ⊆ Y min (P) where X and Y are finite dimensional Hilbert spaces. For instance: − ⊂ R p × R m , K = { 0 } × R m (NLP) K = S m (SDP) − or K = Q m 1 + 1 × Q m 2 + 1 × ... × Q m J + 1 , (SOCP) y ⊤ ) ∈ R × R m j : y 0 ≥ � ¯ where Q m j + 1 = { y = ( y 0 , ¯ y �} Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 4 / 24
Conic Programming Consider the optimization problem over a closed convex cone K x ∈ X f ( x ) ; g ( x ) ∈ K ⊆ Y min (P) where X and Y are finite dimensional Hilbert spaces. For instance: − ⊂ R p × R m , K = { 0 } × R m (NLP) K = S m (SDP) − or K = Q m 1 + 1 × Q m 2 + 1 × ... × Q m J + 1 , (SOCP) y ⊤ ) ∈ R × R m j : y 0 ≥ � ¯ where Q m j + 1 = { y = ( y 0 , ¯ y �} Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 4 / 24
Conic Programming Consider the optimization problem over a closed convex cone K x ∈ X f ( x ) ; g ( x ) ∈ K ⊆ Y min (P) where X and Y are finite dimensional Hilbert spaces. For instance: − ⊂ R p × R m , K = { 0 } × R m (NLP) K = S m (SDP) − or K = Q m 1 + 1 × Q m 2 + 1 × ... × Q m J + 1 , (SOCP) y ⊤ ) ∈ R × R m j : y 0 ≥ � ¯ where Q m j + 1 = { y = ( y 0 , ¯ y �} Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 4 / 24
Conic Programming Consider the optimization problem over a closed convex cone K x ∈ X f ( x ) ; g ( x ) ∈ K ⊆ Y min (P) where X and Y are finite dimensional Hilbert spaces. For instance: − ⊂ R p × R m , K = { 0 } × R m (NLP) K = S m (SDP) − or K = Q m 1 + 1 × Q m 2 + 1 × ... × Q m J + 1 , (SOCP) y ⊤ ) ∈ R × R m j : y 0 ≥ � ¯ where Q m j + 1 = { y = ( y 0 , ¯ y �} Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 4 / 24
Applications Minimization of Maximum Eigenvalue of G ( x ) t ∈ R , x ∈ R n t ; G ( x ) − tI � 0 min Robust Linear Programming x ∈ R n { f ( x ) = c ⊤ x ; a ⊤ min i x ≤ b i ∀ a i ∈ E i , i = 1 , ..., m } (RLP) where P i = P ⊤ � 0, and E i := { ¯ a i + P i u : � u � ≤ 1 } i The robust linear constraint can be reformulated as follows max { a ⊤ a ⊤ i x : a i ∈ E i } = ¯ i x + � P i x � ≤ b i , � b i − ¯ a ⊤ � i x which is of the form: ∈ Q n + 1 P i x Then (RLP) can be casted as a (SOCP) problem Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 5 / 24
Applications Minimization of Maximum Eigenvalue of G ( x ) t ∈ R , x ∈ R n t ; G ( x ) − tI � 0 min Robust Linear Programming x ∈ R n { f ( x ) = c ⊤ x ; a ⊤ min i x ≤ b i ∀ a i ∈ E i , i = 1 , ..., m } (RLP) where P i = P ⊤ � 0, and E i := { ¯ a i + P i u : � u � ≤ 1 } i The robust linear constraint can be reformulated as follows max { a ⊤ a ⊤ i x : a i ∈ E i } = ¯ i x + � P i x � ≤ b i , � b i − ¯ a ⊤ � i x which is of the form: ∈ Q n + 1 P i x Then (RLP) can be casted as a (SOCP) problem Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 5 / 24
Applications Minimization of Maximum Eigenvalue of G ( x ) t ∈ R , x ∈ R n t ; G ( x ) − tI � 0 min Robust Linear Programming x ∈ R n { f ( x ) = c ⊤ x ; a ⊤ min i x ≤ b i ∀ a i ∈ E i , i = 1 , ..., m } (RLP) where P i = P ⊤ � 0, and E i := { ¯ a i + P i u : � u � ≤ 1 } i The robust linear constraint can be reformulated as follows max { a ⊤ a ⊤ i x : a i ∈ E i } = ¯ i x + � P i x � ≤ b i , � b i − ¯ a ⊤ � i x which is of the form: ∈ Q n + 1 P i x Then (RLP) can be casted as a (SOCP) problem Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 5 / 24
Applications Minimization of Maximum Eigenvalue of G ( x ) t ∈ R , x ∈ R n t ; G ( x ) − tI � 0 min Robust Linear Programming x ∈ R n { f ( x ) = c ⊤ x ; a ⊤ min i x ≤ b i ∀ a i ∈ E i , i = 1 , ..., m } (RLP) where P i = P ⊤ � 0, and E i := { ¯ a i + P i u : � u � ≤ 1 } i The robust linear constraint can be reformulated as follows max { a ⊤ a ⊤ i x : a i ∈ E i } = ¯ i x + � P i x � ≤ b i , � b i − ¯ a ⊤ � i x which is of the form: ∈ Q n + 1 P i x Then (RLP) can be casted as a (SOCP) problem Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 5 / 24
Applications Minimization of Maximum Eigenvalue of G ( x ) t ∈ R , x ∈ R n t ; G ( x ) − tI � 0 min Robust Linear Programming x ∈ R n { f ( x ) = c ⊤ x ; a ⊤ min i x ≤ b i ∀ a i ∈ E i , i = 1 , ..., m } (RLP) where P i = P ⊤ � 0, and E i := { ¯ a i + P i u : � u � ≤ 1 } i The robust linear constraint can be reformulated as follows max { a ⊤ a ⊤ i x : a i ∈ E i } = ¯ i x + � P i x � ≤ b i , � b i − ¯ a ⊤ � i x which is of the form: ∈ Q n + 1 P i x Then (RLP) can be casted as a (SOCP) problem Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 5 / 24
First Order Optimality Conditions Consider the optimization problem over a closed convex cone K min f ( x ) ; g ( x ) ∈ K (P) x Karush-Kuhn-Tucker Conditions We say that ( x ∗ , y ∗ ) is a KKT-point ( y ∗ ∈ Λ( x ∗ ) ) if it satisfies ∇ x L ( x ∗ , y ∗ ) = ∇ f ( x ∗ ) + Dg ( x ∗ ) ⊤ y ∗ = 0 , (KKT) y ∗ ∈ N K ( g ( x ∗ )) , where N K ( z ) is the normal cone to K at z ∈ K Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 6 / 24
First Order Optimality Conditions Consider the optimization problem over a closed convex cone K min f ( x ) ; g ( x ) ∈ K (P) x Karush-Kuhn-Tucker Conditions We say that ( x ∗ , y ∗ ) is a KKT-point ( y ∗ ∈ Λ( x ∗ ) ) if it satisfies ∇ x L ( x ∗ , y ∗ ) = ∇ f ( x ∗ ) + Dg ( x ∗ ) ⊤ y ∗ = 0 , � g ( x ∗ ) , y ∗ � = 0 , (KKT) g ( x ∗ ) ∈ K , y ∗ ∈ K − (= − K ) , where K − is the negative polar cone of K Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 6 / 24
Constraint Qualification Conditions Robinson’s Constraint Qualification Condition Let x ∗ be a feasible point of (P). Definition We say that x ∗ satisfies Robinson’s const. qualif. cond. if Dg ( x ∗ ) X + T K ( g ( x ∗ )) = Y (Rob) where T K ( g ( x ∗ )) is the tangent (or Bouligand) cone of K at g ( x ∗ ) NLP case: Mangasarian-Fromovitz condition: ∇ g i ( x ∗ ) , for all i ∈ { 1 , ..., p } , are l.i. and ∃ h ∈ R n such that: ∇ g i ( x ∗ ) ⊤ h = 0 , ∀ i ∈ { 1 , ..., p } , ∇ g i ( x ∗ ) ⊤ h < 0 , ∀ i ∈ { p + 1 , ..., p + m } s.t. g i ( x ∗ ) = 0 ∃ h ∈ R n such that E ⊤ Dg ( x ∗ ) hE ≺ 0 , SDP case: where the columns of E are an orthonormal basis of Ker g ( x ∗ ) Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 7 / 24
Constraint Qualification Conditions Robinson’s Constraint Qualification Condition Let x ∗ be a feasible point of (P). Definition We say that x ∗ satisfies Robinson’s const. qualif. cond. if 0 ∈ int { g ( x ∗ ) + Dg ( x ∗ ) X − K } (Rob) NLP case: Mangasarian-Fromovitz condition: ∇ g i ( x ∗ ) , for all i ∈ { 1 , ..., p } , are l.i. and ∃ h ∈ R n such that: ∇ g i ( x ∗ ) ⊤ h = 0 , ∀ i ∈ { 1 , ..., p } , ∇ g i ( x ∗ ) ⊤ h < 0 , ∀ i ∈ { p + 1 , ..., p + m } s.t. g i ( x ∗ ) = 0 ∃ h ∈ R n such that E ⊤ Dg ( x ∗ ) hE ≺ 0 , SDP case: where the columns of E are an orthonormal basis of Ker g ( x ∗ ) Héctor Ramírez C. (DIM & CMM, U. Chile) SOOC for Conic Programming Toulon 2008 7 / 24
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