Exact Algorithms for Semidefinite Programs with Degenerate Feasible Set Didier Henrion LAAS-CNRS, Technical University in Prague Czech Republic Simone Naldi Univ. Limoges, XLIM, CNRS Mohab Safey El Din Sorbonne Univ./Inria/CNRS 2018 0/11
Problem statement A 0 , . . . , A n symmetric matrices with entries in Q Input of size m × m . ℓ ∈ Q [ x 1 , . . . , x n ] of degree 1 Spectrahedron S ( A ) = { x ∈ R n | A 0 + x 1 A 1 + · · · + x n A n � 0 } . ℓ ⋆ = inf x ∈ S ( A ) ℓ ( x ) ❀ real algebraic number Assuming that ℓ ⋆ is reached Output v i ( t ) rational parametrization of a minimizer q ( t ) = 0 , x i = ∂ q /∂ t , 1 � i � n 1/11
State of the art Approximate solutions Computed numerically through interior point methods. ◮ under the assumption that S ( A ) has non-empty interior ◮ polynomial time at fixed precision ◮ super efficient solvers based on floating/double point arithmetics BUT not so unfrequent reliability issues. Exact solving of SDP The feasible set S ( A ) is a semi-algebraic set ◮ General algorithms for semi-algebraic sets ◮ Dedicated exact algorithms for solving LMI Henrion/Naldi/Safey El Din, SIOPT ◮ polynomial time when n or m is fixed ◮ regularity assumptions ❀ genericity of the A i ’s 2/11
Motivations ◮ Lyapunov stability for ˙ x = M x see e.g. Henrion/Garulli’05 P + (− M T P − P M ) ≻ 0 Find P such that ◮ SOS polynomials (sums of squares): f ( u ) = f 1 ( u ) 2 + · · · + f t ( u ) 2 , with u = ( u 1 , . . . , u n ) , is equivalent to a LMI of type f ( u ) = v ( u ) T · A · v ( u ) , A � 0 ◮ More generally: f ∗ = inf f ( u ) iff f ∗ = sup λ : f − λ � 0 SOS relaxation : f − λ = g 2 1 + · · · + g 2 t Irrational certificates : Scheiderer’s examples ❀ hardness of solving ! ◮ Many examples of feasible sets with empty interiors 3/11
Motivations ◮ Lyapunov stability for ˙ x = M x see e.g. Henrion/Garulli’05 P + (− M T P − P M ) ≻ 0 Find P such that ◮ SOS polynomials (sums of squares): f ( u ) = f 1 ( u ) 2 + · · · + f t ( u ) 2 , with u = ( u 1 , . . . , u n ) , is equivalent to a LMI of type f ( u ) = v ( u ) T · A · v ( u ) , A � 0 ◮ More generally: f ∗ = inf f ( u ) iff f ∗ = sup λ : f − λ � 0 SOS relaxation : f − λ = g 2 1 + · · · + g 2 t Irrational certificates : Scheiderer’s examples ❀ hardness of solving ! ◮ Many examples of feasible sets with empty interiors Need of exact algorithms for solving SDP when S ( A ) is degenerate 3/11
Main results Assumptions on the input ◮ ℓ ⋆ is reached ◮ genericity assumption on ℓ ∈ Q [ x 1 , . . . , x n ] ◮ Dedicated algorithm for computing a minimizer solution to the SDP rep- resented by a rational parametrization no assumption on S ( A ) � n + m ◮ arithmetic complexity polynomial in � n ◮ preliminary implementation for small sized problems Can handle degenerate cases Useful for just deciding the emptiness or grabbing sample points in the solution of a feasible set (possibly degnerate) 4/11
Overview of the ingredients Degenerate SDP Regularized situation A ( x ) + ǫ B � 0 ❀ non-empty interior Reduction to polynomial system solving Highly structured systems Solve using symbolic homotopy Take the limit ( ǫ → 0 ) 5/11
Projections of semi-algebraic sets First result Let R be a real closed field and S ⊂ R n be a closed semi-algebraic set. For generic ℓ ∈ Q [ x 1 , . . . , x n ] with deg ( ℓ ) = 1, ℓ ( S ) is closed generalizes a result in S./Schost ISSAC’03 We start with S ( A ) defined by A 0 + x 1 + · · · + x n A n � 0. ◮ For generic ℓ , ℓ ( S ( A )) is closed. Take B symmetric with B ≻ 0. ◮ S ( A ) ⊂ S ( A ( x ) + ǫ B ) ⊂ R n = R � ǫ � n ◮ ℓ ( S ( A ( x ) + ǫ B )) is closed. We shall use B and ǫ to regularize the problem. Let A ǫ ( x ) = A ( x ) + ǫ B 6/11
From semi-algebraic to algebraic formulation Define: For A ǫ ( x ) : D r = { x ∈ C � ǫ � n | rank ( A ǫ ( x )) � r } For A ǫ ( x ) , and S ( A ǫ ( x )) � = ∅ : r ( A ǫ ) = min { rank A ǫ ( x ) | x ∈ S ( A ǫ ( x )) } So one has nested sequences D 0 ⊂ · · · ⊂ D m − 1 D 0 ∩ R � ǫ � n ⊂ · · · ⊂ D m − 1 ∩ R � ǫ � n 7/11
From semi-algebraic to algebraic formulation Define: For A ǫ ( x ) : D r = { x ∈ C � ǫ � n | rank ( A ǫ ( x )) � r } For A ǫ ( x ) , and S ( A ǫ ( x )) � = ∅ : r ( A ǫ ) = min { rank A ǫ ( x ) | x ∈ S ( A ǫ ( x )) } So one has nested sequences D 0 ⊂ · · · ⊂ D m − 1 D 0 ∩ R � ǫ � n ⊂ · · · ⊂ D m − 1 ∩ R � ǫ � n Smallest Rank Property Henrion-Naldi-S. SIOPT 2015 Let C be a conn. comp. of D r ( A ǫ ) ∩ R n s.t. C ∩ S ( A ǫ ( x )) � = ∅ . Then C ⊂ S ( A ǫ ( x )) . In particular C ⊂ D r ( A ǫ ) \ D r ( A ǫ )− 1 . 7/11
Critical points and incidence varieties A ǫ ( x ) = A ( x ) + ǫ B 1st step Lifting of the determinantal variety : ( x 2 , y ) C y 1 , 1 . . . y 1 , m − r . . π 1 π 1 A ǫ ( x ) Y ( y ) = A ǫ ( x ) . . = 0 . . . y m , 1 . . . y m , m − r x 1 U Y ( y ) = I m − r If B and ℓ are generic, the lifted algebraic set V r is smooth and equidimensional 2nd step Compute critical points of the map ( x , y ) �→ ℓ ( x ) on V r : When ℓ is generic, there are finitely many critical points. 8/11
Symbolic homotopy We use Lagrange polynomial systems to encode critical points. A ǫ ( x ) Y ( y ) = 0 , UY ( y ) = Id → F ( x , y ) = 0 z is a vector of new variables F ( x , y ) = 0 , z . jac ( F , ℓ ) = 0 9/11
Symbolic homotopy We use Lagrange polynomial systems to encode critical points. A ǫ ( x ) Y ( y ) = 0 , UY ( y ) = Id → F ( x , y ) = 0 z is a vector of new variables F ( x , y ) = 0 , z . jac ( F , ℓ ) = 0 ◮ System defining incidence variety → bi-linear in ( x , y ) ◮ Lagrange system → globally tri-linear in ( x , y , z ) but all equations are bi-linear ◮ Multi-homogeneous structure not handled efficiently by the litterature except S. /Schost, JSC ’18 (Symbolic homotopy) → 0-dim case Complexity quadratic in the Multi-homogeneous B´ ezout bound 9/11
Complexity Using symbolic homotopy ◮ Here we have introduced one parameter ǫ to regularize the problem ◮ Handling infinitesimal parameters in real geometry Basu/Pollack/Roy, Rouillier/Roy/S. Efficient use of lifting techniques combined with degree bounds Schost 03 Classical strategy ◮ Instantiate ǫ to a randomly chosen value ◮ Solve the zero-dimensional system ◮ Lift ǫ and compute the limit. All steps run in time polynomial in the multi-homogeneous bound associated to the system � n + m � This is ! m 10/11
Conclusions / Perspectives ◮ First dedicated algorithm for solving SDP with degenerate feasible sets ◮ Reasonable overhead w.r.t. the regular situation ◮ Geometric results that may be used in further/more general algorithms ◮ What is missing: ◮ What happens when the linear form to optimize is not generic? ◮ Implementation of symbolic homotopy algorithms ◮ Potential impact on numerical algorithms? 11/11
Conclusions / Perspectives ◮ First dedicated algorithm for solving SDP with degenerate feasible sets ◮ Reasonable overhead w.r.t. the regular situation ◮ Geometric results that may be used in further/more general algorithms ◮ What is missing: ◮ What happens when the linear form to optimize is not generic? ◮ Implementation of symbolic homotopy algorithms ◮ Potential impact on numerical algorithms? Thank you ... 11/11
Conclusions / Perspectives ◮ First dedicated algorithm for solving SDP with degenerate feasible sets ◮ Reasonable overhead w.r.t. the regular situation ◮ Geometric results that may be used in further/more general algorithms ◮ What is missing: ◮ What happens when the linear form to optimize is not generic? ◮ Implementation of symbolic homotopy algorithms ◮ Potential impact on numerical algorithms? Thank you ... and many thanks to ´ Eric for giving this talk 11/11
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