Semidefinite representation of convex sets

Let be a semialgebraic set defined by multivariate polynomials g i ( x ). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S i ) be the boundary of S (resp. S i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMI...

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Published in:Mathematical programming Vol. 122; no. 1; pp. 21 - 64
Main Authors: Helton, J. William, Nie, Jiawang
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.03.2010
Springer
Springer Nature B.V
Subjects:
Applied mathematics
Applied sciences
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Control theory
Exact sciences and technology
Full Length Paper
Hypotheses
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization
Polynomials
Theorems
Theoretical
Poscurv-convex
65K05
Moments
Convex polynomials positive curvature
Schmüdgen and Putinar’s matrix Positivstellensatz
Semidefinite programming (SDP)
Extendable poscurv-convex
Convex sets
Sos-concave (sos-convex)
Sum of squares(SOS)
Modified Hessian
90C25
90C22
Semialgebraic geometry
Positive second fundamental form
Positive definite Lagrange Hessian (PDLH) condition
Linear matrix inequality (LMI)
47N10
46N10
Convex set
Semi definite programming
Convex programming
Optimization
Lagrange multiplier
Linear matrix inequality
Mathematical programming
Polynomial matrix
Minimization
Semi algebraic set
Applied mathematics
Positive definite matrix
S matrix
Matrix polynomial
Schmüdgen and Putinar's matrix Positivstellensatz
Curvature
ISSN:0025-5610, 1436-4646
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Summary:Let be a semialgebraic set defined by multivariate polynomials g i ( x ). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S i ) be the boundary of S (resp. S i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of , does there exist an LMI representable set Ŝ in some higher dimensional space whose projection down onto equals S . Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g i ( x ) are all concave on S . If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g i ( x ) is either sos-concave ( − ∇ 2 g i ( x ) =  W ( x ) T W ( x ) for some possibly nonsquare matrix polynomial W ( x )) or strictly quasi-concave on S , then S is SDP representable. (iii) If each S i is either sos-convex or poscurv-convex ( S i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇ g i ( x ) ≠ 0 on ∂ S i ∩ S ), then S is SDP representable. This also holds for S i for which ∂ S i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S . (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-008-0240-y