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46
Sums of squares, moment matrices and optimization over polynomials
, 2008
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory ..."
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Cited by 151 (9 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zerodimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.
SUFFICIENT AND NECESSARY CONDITIONS FOR SEMIDEFINITE REPRESENTABILITY OF CONVEX HULLS AND SETS
, 2007
"... A set S ⊆ R n is called to be Semidefinite (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 ..."
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Cited by 32 (9 self)
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A set S ⊆ R n is called to be Semidefinite (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=1 Wk). 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 condition: the boundary ∂S is positively curved, and necessary condition: ∂S has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasiconcavity of defining polynomials on those points on ∂S where they vanish. The gaps between them are ∂S having positive versus nonnegative curvature and smooth versus nonsmooth points. 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 conditions for SDP representability: ∂cT is positively curved, and necessary conditions: ∂cT has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.
Convex sets with semidefinite representation. Optimization Online
, 2006
"... Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approxi ..."
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Cited by 29 (1 self)
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Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed ɛ> 0, there is a convex set Kɛ such that co(K) ⊆ Kɛ ⊆ co(K) + ɛB (where B is the unit ball of R n), and Kɛ has an explicit SDr in terms of the gj’s. For convex and compact basic semialgebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian Lf associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case. 1.
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.
Positive polynomials and projections of spectrahedra
, 2010
"... This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain th ..."
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Cited by 18 (1 self)
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This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of [17] on nonexposed faces. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.
A convex polynomial that is not sosconvex
 Mathematical Programming
"... A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvex ..."
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Cited by 18 (4 self)
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A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it has been recently speculated whether sosconvexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sosconvex. Interestingly, our example is found with software using sum of squares programming techniques and the duality theory of semidefinite optimization. As a byproduct of our numerical procedure, we obtain a simple method for searching over a restricted family of nonnegative polynomials that are not sums of squares. 1
NPhardness of deciding convexity of quartic polynomials and related problems
, 2010
"... We show that unless P=NP, there exists no polynomial time (or even pseudopolynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexi ..."
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Cited by 11 (2 self)
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We show that unless P=NP, there exists no polynomial time (or even pseudopolynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NPhard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.
Generic optimality conditions for semialgebraic convex programs
 Math. Oper. Res
"... We consider linear optimization over a nonempty convex semialgebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique “active ” manifold, around which F is “partly smooth”, and th ..."
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Cited by 11 (6 self)
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We consider linear optimization over a nonempty convex semialgebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique “active ” manifold, around which F is “partly smooth”, and the secondorder sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F, and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F. Key words: convex optimization, sensitivity analysis, partial smoothness, identifiable surface, active sets, generic, secondorder sufficient conditions, semialgebraic.