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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 47 (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).
Convexity in semialgebraic geometry and polynomial optimization
 SIAM Journal on Optimization
"... Abstract. We review several (and provide new) results on the theory of moments, sums of squares and basic semialgebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite conver ..."
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Cited by 21 (4 self)
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Abstract. We review several (and provide new) results on the theory of moments, sums of squares and basic semialgebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semialgebraic set K is convex but its defining polynomials are not, we provide two algebraic certificate of convexity which can be checked numerically. The second is simpler and holds if a sufficient (and almost necessary) condition is satisfied, it also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie [6]. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen’s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures. 1.
Exposed faces of semidefinite representable sets
"... Abstract. A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images ..."
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Cited by 18 (5 self)
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Abstract. A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie.
Semidefinite representation of convex hulls of rational varieties
, 2009
"... Using elementary duality properties of positive semidefinite moment matrices and polynomial sumofsquares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section ..."
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Cited by 18 (0 self)
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Using elementary duality properties of positive semidefinite moment matrices and polynomial sumofsquares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension.
Detecting rigid convexity of bivariate polynomials
, 2009
"... Given a polynomial x ∈ R n ↦ → p(x) in n = 2 variables, a symbolicnumerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x: p(x) ≥ 0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LM ..."
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Cited by 11 (5 self)
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Given a polynomial x ∈ R n ↦ → p(x) in n = 2 variables, a symbolicnumerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x: p(x) ≥ 0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial p(x) is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety C = {x: p(x) = 0} is an algebraic curve of genus zero, a second algorithm based on Bézoutians is proposed to detect whether P has an LMI representation and to build such a representation from a rational parametrization of C. Finally, some extensions to positive genus curves and to the case n> 2 are mentioned.
LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future
, 2012
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Structured semidefinite representation of some convex sets
, 2008
"... Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it can also be useful to have “lifts” which are expressible as ..."
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Cited by 8 (6 self)
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Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it can also be useful to have “lifts” which are expressible as LMIs. We show here that this is a much less restrictive condition and give methods for actually constructing lifts and their LMI representation.
On semidefinite representations of plane quartics
, 2008
"... This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite represen ..."
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Cited by 8 (0 self)
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This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables. If the quartic is rationally parametrizable, an exact semidefinite representation with 2 lifting variables can be obtained. Various numerical examples illustrate the techniques and suggest further research directions. Keywords semidefinite programming; polynomials; algebraic plane curves 1
First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials
 SIAM Journal on Matrix Analysis and Applications
, 2009
"... A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex se ..."
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Cited by 8 (3 self)
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A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form SD(f) = {x ∈ D: f(x) ≥ 0}. Here D = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} is a convex domain defined by some “nice ” concave polynomials gi(x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over D, we prove that SD(f) has some explicit semidefinite representations under certain conditions called preordering concavity or qmodule concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria: f(u) + ∇f(u) T (x − u) − f(x) ≥ 0, ∀ x,u ∈ D. When f(x) is a polynomial or rational function having singularities on the boundary of SD(f), a perspective transformation is introduced to find some explicit semidefinite representations for SD(f) under certain conditions. In the particular case n = 2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that SD(f) always admits an explicitly constructible semidefinite representation. Key words: convex set, linear matrix inequality, perspective transformation, polynomial, Positivstellensatz, preordering convex/concave, qmodule convex/concave, rational function, singularity, semidefinite programming, sum of squares