On the local stability of semidefinite relaxations Diego Cifuentes Department of Mathematics Massachusetts Institute of Technology Joint work with Sameer Agarwal (Google), Pablo Parrilo (MIT), Rekha Thomas (U. Washington). arXiv:1710.04287 Real Algebraic Geometry and Optimization - ICERM - 2018 Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 1 / 21
Nearest point problems Given a variety X ⊂ R n , and a point θ ∈ R n , � x − θ � 2 min x s.t. x ∈ X A variety is the zero set of some polynomials X := { x ∈ R n : f 1 ( x ) = · · · = f m ( x ) = 0 } Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 2 / 21
Nearest point problems Given a variety X ⊂ R n , and a point θ ∈ R n , � x − θ � 2 min x s.t. x ∈ X A variety is the zero set of some polynomials X := { x ∈ R n : f 1 ( x ) = · · · = f m ( x ) = 0 } This problem is nonconvex , and computationally challenging. SDP relaxations have been successful in several applications. Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 2 / 21
Nearest point problems Given a variety X ⊂ R n , and a point θ ∈ R n , � x − θ � 2 min x s.t. x ∈ X A variety is the zero set of some polynomials X := { x ∈ R n : f 1 ( x ) = · · · = f m ( x ) = 0 } This problem is nonconvex , and computationally challenging. SDP relaxations have been successful in several applications. Goal Study the behavior of SDP relaxations in the low noise regime: when x is sufficiently close to X . Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 2 / 21
Nearest point problems Many different applications Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 3 / 21
Nearest point to the twisted cubic � x − θ � 2 , X := { ( x 1 , x 2 , x 3 ) : x 2 = x 2 min where 1 , x 3 = x 1 x 2 } x ∈ X The twisted cubic X can be parametrized as t �→ ( t , t 2 , t 3 ). Its Lagrangian dual is the following SDP: γ + � θ � 2 − θ 1 λ 1 − θ 2 λ 2 − θ 3 � 0 . − θ 1 1 − 2 λ 1 − λ 2 0 max γ, s.t. λ 1 − θ 2 − λ 2 1 0 γ,λ 1 ,λ 2 ∈ R λ 2 − θ 3 0 0 1 Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 4 / 21
Nearest point to the twisted cubic � x − θ � 2 , X := { ( x 1 , x 2 , x 3 ) : x 2 = x 2 min where 1 , x 3 = x 1 x 2 } x ∈ X Nearest point to the twisted cubic 3 1.0 zero duality gap 2 0.8 1 0.6 duality gap 0 3 0.4 3 = 3 1 1 0.2 2 3 0.0 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 1 Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 4 / 21
Nearest point problem to a quadratic variety Theorem If ¯ θ ∈ X is a regular point then there is zero-duality-gap for any θ ∈ R n that is sufficiently close to ¯ θ . Applications: Triangulation problem [Aholt-Agarwal-Thomas] Nearest (symmetric) rank one tensor Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 5 / 21
Parametrized QCQPs Consider a family of quadratically constrained programs (QCQPs): min g θ ( x ) x ∈ R N ( P θ ) h i θ ( x ) = 0 for i = 1 , . . . , m where g θ , h i θ are quadratic , and the dependence on θ is continuous . The Lagrangian dual is an SDP. Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 6 / 21
Parametrized QCQPs Consider a family of quadratically constrained programs (QCQPs): min g θ ( x ) x ∈ R N ( P θ ) h i θ ( x ) = 0 for i = 1 , . . . , m where g θ , h i θ are quadratic , and the dependence on θ is continuous . The Lagrangian dual is an SDP. Goal: Given ¯ θ for which the SDP relaxation is tight, analyze the behavior as θ → ¯ θ . Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 6 / 21
Parametrized QCQPs Consider a family of quadratically constrained programs (QCQPs): min g θ ( x ) x ∈ R N ( P θ ) h i θ ( x ) = 0 for i = 1 , . . . , m where g θ , h i θ are quadratic , and the dependence on θ is continuous . The Lagrangian dual is an SDP. Goal: Given ¯ θ for which the SDP relaxation is tight, analyze the behavior as θ → ¯ θ . Example: For a nearest point problem g θ ( x ) := � x − θ � 2 , h i ( x ) independent of θ The problem is trivial for any ¯ θ ∈ X . Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 6 / 21
SDP relaxation of a (homogeneous) QCQP Primal problem x T G θ x min x ∈ R N ( P θ ) x T H i θ x = b i i = 1 , . . . , m Dual problem max d ( λ ) := − � i λ i b i λ ∈ R m ( D θ ) Q θ ( λ ) � 0 where Q θ ( λ ) is the Hessian of the Lagrangian � λ i H i θ ∈ S N . Q θ ( λ ) := G θ + i Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 7 / 21
SDP relaxation of a (homogeneous) QCQP Primal problem x T G θ x min x ∈ R N ( P θ ) x T H i θ x = b i i = 1 , . . . , m Dual problem max d ( λ ) := − � i λ i b i λ ∈ R m ( D θ ) Q θ ( λ ) � 0 Problem statement θ ), i.e., ¯ Assume that val( P ¯ θ ) = val( D ¯ θ is a zero-duality-gap parameter. Find conditions under which val( P θ ) = val( D θ ) when θ is close to ¯ θ . Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 7 / 21
Characterization of zero-duality-gap Given x θ primal feasible, its Lagrange multipliers are: λ T ∇ h θ ( x θ ) = −∇ g θ ( x θ ) λ ∈ Λ θ ( x θ ) ⇐ ⇒ ⇐ ⇒ Q θ ( λ ) x θ = 0 . Lemma Let x θ ∈ R N , λ ∈ R m . Then x θ is optimal to ( P θ ) and λ is optimal to ( D θ ) with val( P θ ) = val( D θ ) iff: 1 h θ ( x θ ) = 0 (primal feasibility). 2 Q θ ( λ ) � 0 (dual feasibility). 3 λ ∈ Λ θ ( x θ ) (complementarity). Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 8 / 21
Characterization of zero-duality-gap Given x θ primal feasible, its Lagrange multipliers are: λ T ∇ h θ ( x θ ) = −∇ g θ ( x θ ) λ ∈ Λ θ ( x θ ) ⇐ ⇒ ⇐ ⇒ Q θ ( λ ) x θ = 0 . Lemma Let x θ ∈ R N , λ ∈ R m . Then x θ is optimal to ( P θ ) and λ is optimal to ( D θ ) with val( P θ ) = val( D θ ) iff: 1 h θ ( x θ ) = 0 (primal feasibility). 2 Q θ ( λ ) � 0 (dual feasibility). 3 λ ∈ Λ θ ( x θ ) (complementarity). Proof. If Q θ ( λ ) x θ = 0 and h θ ( x θ ) = 0, then 0 = x T θ Q θ ( λ ) x θ = x T � λ i x T θ G θ x θ + θ H i x θ = g θ ( x θ ) − d ( λ ) . i Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 8 / 21
Characterization of zero-duality-gap Lemma Let ¯ x , ¯ θ be a zero-duality-gap parameter with (¯ λ ) primal/dual optimal. Assume that θ (¯ 1 Q ¯ λ ) has corank-one (strict-complementarity) 2 ∃ x θ feasible for ( P θ ) , λ θ ∈ Λ θ ( x θ ) s.t. ( x θ , λ θ ) θ → ¯ θ x , ¯ − − − → (¯ λ ) . Then there is zero-duality-gap when θ is close to ¯ θ . Proof. Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 9 / 21
Characterization of zero-duality-gap Lemma Let ¯ x , ¯ θ be a zero-duality-gap parameter with (¯ λ ) primal/dual optimal. Assume that θ (¯ 1 Q ¯ λ ) has corank-one (strict-complementarity) 2 ∃ x θ feasible for ( P θ ) , λ θ ∈ Λ θ ( x θ ) s.t. ( x θ , λ θ ) θ → ¯ θ x , ¯ − − − → (¯ λ ) . Then there is zero-duality-gap when θ is close to ¯ θ . Proof. Q θ ( λ θ ) has a zero eigenvalue ( Q θ ( λ θ ) x θ = 0). Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 9 / 21
Characterization of zero-duality-gap Lemma Let ¯ x , ¯ θ be a zero-duality-gap parameter with (¯ λ ) primal/dual optimal. Assume that θ (¯ 1 Q ¯ λ ) has corank-one (strict-complementarity) 2 ∃ x θ feasible for ( P θ ) , λ θ ∈ Λ θ ( x θ ) s.t. ( x θ , λ θ ) θ → ¯ θ x , ¯ − − − → (¯ λ ) . Then there is zero-duality-gap when θ is close to ¯ θ . Proof. Q θ ( λ θ ) has a zero eigenvalue ( Q θ ( λ θ ) x θ = 0). θ (¯ Q θ ( λ θ ) → Q ¯ λ ) (the dependence on θ is continuous). Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 9 / 21
Characterization of zero-duality-gap Lemma Let ¯ x , ¯ θ be a zero-duality-gap parameter with (¯ λ ) primal/dual optimal. Assume that θ (¯ 1 Q ¯ λ ) has corank-one (strict-complementarity) 2 ∃ x θ feasible for ( P θ ) , λ θ ∈ Λ θ ( x θ ) s.t. ( x θ , λ θ ) θ → ¯ θ x , ¯ − − − → (¯ λ ) . Then there is zero-duality-gap when θ is close to ¯ θ . Proof. Q θ ( λ θ ) has a zero eigenvalue ( Q θ ( λ θ ) x θ = 0). θ (¯ Q θ ( λ θ ) → Q ¯ λ ) (the dependence on θ is continuous). θ (¯ Q ¯ λ ) has N − 1 positive eigenvalues. Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 9 / 21
Characterization of zero-duality-gap Lemma Let ¯ x , ¯ θ be a zero-duality-gap parameter with (¯ λ ) primal/dual optimal. Assume that θ (¯ 1 Q ¯ λ ) has corank-one (strict-complementarity) 2 ∃ x θ feasible for ( P θ ) , λ θ ∈ Λ θ ( x θ ) s.t. ( x θ , λ θ ) θ → ¯ θ x , ¯ − − − → (¯ λ ) . Then there is zero-duality-gap when θ is close to ¯ θ . Proof. Q θ ( λ θ ) has a zero eigenvalue ( Q θ ( λ θ ) x θ = 0). θ (¯ Q θ ( λ θ ) → Q ¯ λ ) (the dependence on θ is continuous). θ (¯ Q ¯ λ ) has N − 1 positive eigenvalues. Q θ ( λ θ ) also has N − 1 positive eigenvalues (continuity of eigenvalues). Cifuentes (MIT) Stability of semidefinite relaxations ICERM’18 9 / 21
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