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Semidefinite representation of convex sets
, 2007
"... Let S = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is compact, convex and has nonempty interior. Let Si = {x ∈ R n: gi(x) ≥ 0} and ∂Si = {x ∈ R n: gi(x) = 0} be its boundary. This paper, as does the subject of semidefin ..."
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Cited by 48 (10 self)
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Let S = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is compact, convex and has nonempty interior. Let Si = {x ∈ R n: gi(x) ≥ 0} and ∂Si = {x ∈ R n: gi(x) = 0} be its boundary. 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 [6]. A question arising from [13], see [6, 14], is: given S ∈ R n, does there exist an LMI representable set ˆ S in some higher dimensional space R n+N whose projection down onto R n 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 gi(x) are all concave on S. If the positive definite Lagrange Hessian (PDLH) 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 gi(x) is either sosconcave (− ∇ 2 gi(x) = W(x) T W(x) for some matrix polynomial W(x)) or strictly quasiconcave on S, then S is SDP representable. (iii) If each Si is either sosconvex or poscurvconvex (Si is compact, convex and has smooth boundary with positive curvature), then S is SDP representable. This also holds for Si for which ∂Si ∩ S extends smoothly to the boundary of a poscurvconvex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)(iii).
G.: Exact SDP Relaxations for classes of nonlinear semidefinite programming problems
, 2012
"... An exact semidefinite linear programming (SDP) relaxation of a nonlinear semidefinite programming problem is a highly desirable feature because a semidefinite linear programming problem can efficiently be solved. This paper addresses the basic issue of which nonlinear semidefinite programming probl ..."
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Cited by 4 (3 self)
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An exact semidefinite linear programming (SDP) relaxation of a nonlinear semidefinite programming problem is a highly desirable feature because a semidefinite linear programming problem can efficiently be solved. This paper addresses the basic issue of which nonlinear semidefinite programming problems possess exact SDP relaxations under a constraint qualification. We do this by establishing exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOSconvex polynomials. These classes include SOSconvex semidefinite programming problems and fractional semidefinite programming problems with SOSconvex polynomials. The class of SOSconvex polynomials contains, in particular, convex quadratic functions and separable convex polynomials. Consequently, we also derive numerically checkable conditions, completely characterizing minimizers of these classes of semidefinite programming problems. We finally present a constraint qualification, which is, in a sense, the weakest condition guaranteeing these checkable optimality conditions. The SOSconvexity of polynomials is a sufficient condition for convexity and it can be checked by semidefinite programming whereas deciding convexity is generally NPhard. Key words. Polynomial optimization, convex semidefinite programming, SOSconvex polynomials, sum of squares polynomials, fractional programs. ∗Authors are grateful to the referee for the comments and suggestions which have contributed to the
Polynomial Matrix Inequality and Semidefinite Representation
, 908
"... Consider a convex set S = {x ∈ D: G(x) ≽ 0} where G(x) is an m × m symmetric matrix whose every entry is a polynomial or rational function, D ⊆ R n is a domain where G(x) is defined, and G(x) ≽ 0 means G(x) is positive semidefinite. The set S is called semidefinite programming (SDP) representable ..."
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Cited by 3 (1 self)
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Consider a convex set S = {x ∈ D: G(x) ≽ 0} where G(x) is an m × m symmetric matrix whose every entry is a polynomial or rational function, D ⊆ R n is a domain where G(x) is defined, and G(x) ≽ 0 means G(x) is positive semidefinite. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper studies sufficient conditions guaranteeing semidefinite representability of S. We prove that S is semidefinite representable in the following cases: (i) D = R n, G(x) is a matrix polynomial and matrix sosconcave; (ii) D is compact convex, G(x) is a matrix polynomial and strictly matrix concave on D; (iii) G(x) is a matrix rational function and qmodule matrix concave on D. Explicit constructions of SDP representations are given. Some examples are illustrated.
SUFFICIENT AND NECESSARY CONDITIONS FOR SEMIDEFINITE REPRESENTABILITY OF CONVEX HULLS AND SETS
"... Abstract. A set S ⊆ Rn is called to be semidefinite programming (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describa ..."
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Abstract. A set S ⊆ Rn is called to be semidefinite programming (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to construct SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets W1, · · · , Wm, we give an explicit construction of an SDP representation for conv(∪m k=1Wk). This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient: the boundary ∂S is nonsingular and positively curved, while necessary is: ∂S has nonnegative curvature at each nonsingular point. In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on ∂S and the curvature condition can be expressed as their strict versus nonstrict quasiconcavity of at those points on ∂S where they vanish. The gaps between them are ∂S having or not having singular points either of the gradient or of the curvature’s positivity. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sosconcavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is ∂cT, the maximum subset of ∂T contained in ∂conv(T). We prove sufficient for SDP representability: ∂cT is nonsingular and positively curved, and necessary is: ∂cT has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.
Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints
, 2011
"... Dedicated to Bill Helton on the occasion of his 65th birthday. Let V be a semialgebraic set parameterized as {(f1(x),..., fm(x)) : x ∈ T} for quadratic polynomials f0,..., fm and a subset T of R n. This paper studies semidefinite representation of the convex hull conv(V) or its closure, i.e., descr ..."
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Dedicated to Bill Helton on the occasion of his 65th birthday. Let V be a semialgebraic set parameterized as {(f1(x),..., fm(x)) : x ∈ T} for quadratic polynomials f0,..., fm and a subset T of R n. This paper studies semidefinite representation of the convex hull conv(V) or its closure, i.e., describing conv(V) by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that conv(V) is equal to the first order moment type semidefinite relaxation of V, up to taking closures. Similar results hold when every fi is a quadratic form and T is defined by two homogeneous (modulo constants) quadratic constraints, or when all fi are quadratic rational functions with a common denominator and T is defined by a single quadratic constraint, under some general conditions. 1
c©20xx INFORMS Polynomial Matrix Inequality and Semidefinite Representation
"... DOI xx.xxxx/xxxx.xxxx.xxxx ..."
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