The geometry of SDP-exactness in quadratic optimization

Mathematical Programming, 2000

In this paper we study a class of quadratic maximization problems and their semide nite programming (SDP) relaxation. For a special subclass of the problems we show that the SDP relaxation provides an exact optimal solution. Another subclass, which is N P-hard, guarantees that the SDP relaxation yields an approximate solution with a worst-case performance ratio of 0:87856:::. This is a generalization of the well-known result of Goemans and Williamson for the maximum-cut problem. Finally, we discuss extensions of these results in the presence of a certain type of sign restrictions.

IEEE Signal Processing Magazine, 2010

arXiv (Cornell University), 2007

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x * Cx | x * A k x ≥ 1, x ∈ F n , k = 0, 1, ..., m}; and (2) max{x * Cx | x * A k x ≤ 1, x ∈ F n , k = 0, 1, ..., m}. If one of A k 's is indefinite while others and C are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2) when F is the real line R, and by O(m) when F is the complex plane C. This result is an extension of the recent work of Luo et al. [8]. For (2), we show that the same ratio is bounded from below by O(1/ log m) for both the real and complex case, whenever all but one of A k 's are positive semidefinite while C can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal et al. [2]. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.

2003

A polynomial SDP (semidefinite programs) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmet- ric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the variables to be in

2009

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic 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 semi-algebraic 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.

SIAM Journal on Optimization, 2008

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x * Cx | x * A k x ≥ 1, x ∈ F n , k = 0, 1, ..., m}; and (2) max{x * Cx | x * A k x ≤ 1, x ∈ F n , k = 0, 1, ..., m}. If one of A k 's is indefinite while others and C are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2) when F is the real line R, and by O(m) when F is the complex plane C. This result is an extension of the recent work of Luo et al. [8]. For (2), we show that the same ratio is bounded from below by O(1/ log m) for both the real and complex case, whenever all but one of A k 's are positive semidefinite while C can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal et al. [2]. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.

2003

In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is termed the extended trust region subproblem in this paper, to distinguish it from the ordinary trust region subproblem where the constraint is a single ellipsoid. The computational complexity of the extended trust region subproblem in general is still unknown. In this paper we consider several interesting cases related to this problem and show that for those cases the corresponding SDP relaxation admits no gap with the true optimal value, and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization. For the extended trust region subproblem itself, we introduce a parameterized problem and prove the existence of a trajectory which will lead to an optimal solution. Combining with a result obtained in the first part of the paper, we propose a polynomial-time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint.

Operations Research Letters, 2012

This paper addresses the issue of which nonlinear semidefinite linear programming problems possess exact semidefinite linear programming (SDP) relaxations under a constraint qualification. We establish exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOS-convex polynomials. These classes include SOS-convex semidefinite programming problems and fractional semidefinite programming problems with SOS-convex polynomials. The class of SOS-convex polynomials contains convex quadratic functions and separable convex polynomials. We also derive numerically checkable conditions, completely characterizing minimizers of these classes of problems.

Mathematical Programming, 2010

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

2011