Cones of matrices and successive convex relaxations of nonconvex sets

1. Introduction. Consider a maximization problem with a linear objective function cT x: (1.1) maximize cT x subject to x ∈ F,
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where c denotes a constant vector in the n-dimensional Euclidean space Rn and F a subset of Rn . We can reduce a more general maximization problem with a nonlinear objective function f (x) to a maximization problem having a linear objective function represented by a new variable, xn+1 , if we replace f (x) by xn+1 and then add the inequality f (x) ≥ xn+1 to the constraint. Thus (1.1) covers such a general optimization problem. Throughout the paper we assume that F is compact. Then the problem (1.1) has a global maximizer whenever the feasible region F is nonempty. For any compact convex set C containing F , the maximization problem (1.2) maximize cT x subject to x ∈ C
¸ EL‡ MASAKAZU KOJIMA† AND LEVENT TUNC Abstract. Let F be a compact subset of the n-dimensional Euclidean space Rn represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets Ck (k = 1, 2, . . . ) of Rn such that (a) the convex hull of F ⊆ Ck+1 ⊆ Ck (monotonicity), (b) ∩∞ k=1 Ck = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lov´ asz–Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using this equivalence, we investigate some fundamental features of the two methods including (a) and (b) above. Key words. semidefinite programming, nonconvex quadratic optimization problem, linear matrix inequality, bilinear matrix inequality, semi-infinite programming, global optimization AMS subject classifications. 15A48, 52A47, 49M39, 90C05, 90C25, 90C26, 90C30, 90C34 PII. S1052623498336450
serves as a convex relaxation problem, which satisfies the properties that (i) the maximum objective value ζ of the problem (1.2) gives an upper bound for the maximum objective value ζ ∗ of the problem (1.1), i.e., ζ ≥ ζ ∗ , and
∗ Received by the editors March 31, 1998; accepted for publication (in revised form) July 19, 1999; published electronically March 21, 2000. /journals/siopt/10-3/33645.html † Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan (kojima@is.titech.ac.jp). ‡ Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (ltuncel@math.uwaterloo.ca). This work was completed while this author was visiting Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, on a sabbatical leave from University of Waterloo. The research of this author was supported in part by Tokyo Institute of Technology and by a research grant from NSERC of Canada.
SIAM J. OPTIM. Vol. 10, No. 3, pp. 750–778
c 2000 Society for Industrial and Applied Mathematics
CONES OF MATRICES AND SUCCESSIVE CONVEX RELAXATIONS OF NONCONVEX SETS∗
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SUCCESSIVE CONVEX RELAXATIONS OF NONCONVEX SETS
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ˆ ∈ C of (1.2) lies in F , it is a maximizer of (1.1). (ii) if a maximizer x Since the objective function of (1.1) is linear, we know that if we take the convex hull c.hull(F ) (defined as the intersection of all the convex sets containing F ) for C in (1.2), then (i)′ ζ = ζ ∗ , and (ii)′ the set of the maximizers of (1.2) forms a compact convex set whose extreme points are maximizers of (1.1). Therefore, if we solve the relaxation problem (1.2) with a convex feasible region C which closely approximates c.hull(F ), we can expect to get not only a good upper bound ζ for the maximum objective value ζ ∗ but also an approximate maximizer of the problem (1.1). We can further prove that for almost every c ∈ Rn (in the sense of measure), any maximizer x′ ∈ C = c.hull(F ) of (1.2) is an extreme point of c.hull(F ), which also lies in F ; hence x′ is a maximizer of (1.1). This follows from a result due to Ewald, Larman, and Rogers [5] for consequences of related results; see also [17]. ˆ ∈ C , we can Furthermore, for many representations of various convex sets C , given x ˆ. very efficiently find x∗ , an extreme point of C , such that cT x∗ ≥ cT x Indeed, the relaxation technique mentioned above has been playing an essential role in practical computational methods for solving various problems in the fields of combinatorial optimization and global optimization. It is often used in hybrid schemes with the branch-and-bound and branch-and-cut techniques in those fields. See, for instance, [2]. The aim of this paper is to present a basic idea on how we can approximate the convex hull of F . This is a quite difficult problem, and also too general. Before making further discussions, we at least need to provide an appropriate (algebraic) representation for the compact feasible region F of the problem (1.1) and the compact convex feasible region C of the relaxation problem (1.2). We employ quadratic inequalities for this purpose. n Let S n and S+ ⊂ S n denote the set of n × n symmetric matrices and the set of n × n symmetric positive semidefinite matrices, respectively. Given Q ∈ S n , q ∈ Rn , and γ ∈ R, we write a quadratic function on Rn with the quadratic term xT Qx, the linear term 2q T x, and the constant term γ as p(·; γ, q , Q): Then the set Q of quadratic functions on Rn and the set Q+ of convex quadratic functions are defined as Q ≡ {p(·; γ, q , Q) : Q ∈ S n , q ∈ Rn and γ ∈ R} and