GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality (LMI) relaxations of non-convex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sum-of-squares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations. 1

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 known as the extended trust region subproblem and the computational complexity of this problem is still unknown. 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.

Abstract. In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. S-lemma, S-procedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities

This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang [3]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [3] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of LMI. We also present an upper bound on the minimum rank among optimal solutions for a standard complex SDP problem, as a byproduct of the new rank-one decomposition result.

We study the problem of computing a (1+ɛ)-approximation to the minimum volume covering ellipsoid of a given set S of the convex hull of m full-dimensional ellipsoids in R n. We extend the first-order algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in R n, which, in turn, is a modification of Khachiyan’s algorithm. For fixed ɛ> 0, we establish a polynomial-time complexity, which is linear in the number of ellipsoids m. In particular, the iteration complexity of our algorithm is identical to that for a set of m points. The main ingredient in our analysis is the extension of polynomialtime complexity of certain subroutines in the algorithm from a set of points to a set of ellipsoids. As a byproduct, our algorithm returns a finite “core ” set X ⊆ S with the property that the minimum volume covering ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends only on the dimension n and ɛ, but not on the number of ellipsoids m. We also discuss the extent to which our algorithm can be used to compute the minimum volume covering ellipsoid of the convex hull of other sets in R n. We adopt the real number model of computation in our analysis.

Abstract. We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.

Abstract. In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a pre-specified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study when it can be represented by linear matrix inequalities. We also discuss the applications of the results in robust optimization, especially the robust quadratic matrix inequalities and the robust linear programming models. In the latter application the implementational errors of the solution is taken into account, and the problem is formulated as a semidefinite program. Key words. Linear matrix inequalities, convex cone, robust optimization, bi-quadratic functions AMS subject classifications. 15A48, 90C22

Given A: = {a 1,..., a m} ⊂ R n and ɛ> 0, we propose and analyze two algorithms for the problem of computing a (1 + ɛ)-approximation to the radius of the minimum enclosing ball of A. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem. We establish that this algorithm converges in O(1/ɛ) iterations with an overall complexity bound of O(mn/ɛ) arithmetic operations. In addition, the algorithm returns a “core set ” of size O(1/ɛ), which is independent of both m and n. The latter algorithm is obtained by incorporating “away ” steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one. While the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one, the latter algorithm has the potential to compute even smaller core sets in practice since, in contrast with the former one, it allows “dropping ” points from the working core set at each iteration. Our computational results indicate that the latter algorithm indeed returns smaller core sets in comparison with the first one. We also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of other input sets. In particular, we establish the existence of a core set of size O(1/ɛ) for a much wider class of input sets.

We present a general semidefinite relaxation scheme for general n-variate quartic polynomial optimization under homogeneous quadratic constraints. Unlike the existing sum-of-squares (SOS) approach which relaxes the quartic optimization problems to a sequence of (typically large) linear semidefinite programs (SDP), our relaxation scheme leads to a (possibly nonconvex) quadratic optimization problem with linear constraints over the semidefinite matrix cone in R n×n. It is shown that each α-factor approximate solution of the relaxed quadratic SDP can be used to generate in randomized polynomial time an O(α)-factor approximate solution for the original quartic optimization problem, where the constant in O(·) depends only on problem dimension. In the case where only one positive definite quadratic constraint is present in the quartic optimization problem, we present a polynomial time approximation algorithm which can provide a guaranteed relative approximation ratio of (1 − O(n −2)). 1

Abstract. This paper studies the so-called bi-quadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl