Polynomial optimization problem with second-order cone complementarity constraints (SOCPOPCC) is a special case of mathematical program (SOCMPCC). In this paper, we consider how to apply Lasserre's type semidefinite relaxation method solve SOCPOPCC. To end, first reformulate SOCPOPCC equivalently as polynomial and then the reformulated method. For SOCPOPCC, present another reformulation optimiz...

Abstract The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as standard program, admits an exact convex conic formulation computationally intractable cone completely positive matrices. Replacing in this by larger but tractable doubly nonnegative matrices, i.e., semidefinite and componentwise one obtains so-called relaxation, whose optimal value yields lowe...

Recently, semidefinite programming has been used to bound the price of a single-asset European call option at a fixed time. Given the first n moments, a tight bound can be obtained by solving a single semidefinite programming problem of dimension n + 1. In this paper, we study the multi-asset case, which is generally more practical than the single-asset case. We construct a sequence of semidefi...

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimizati...

We study Lovász and Schrijver’s hieararchy of relaxations based on positive semidefiniteness constraints derived from the fractional stable set polytope. We show that there are graphsG for which a single application of the underlying operator, N+, to the fractional stable set polytope gives a nonpolyhedral convex relaxation of the stable set polytope. We also show that none of the current best ...

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on...

Maximum a posteriori (MAP) inference over discrete Markov random fields is a fundamental task spanning a wide spectrum of real-world applications, which is known to be NP-hard for general graphs. In this paper, we propose a novel semidefinite relaxation formulation (referred to as SDR) to estimate the MAP assignment. Algorithmically, we develop an accelerated variant of the alternating directio...

A semidefinite relaxation σ(Γ) for the problem of finding the maximum number κ(Γ) of edges in a complete bipartite subgraph of a bipartite graph Γ = (V1 ∪ V2, E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in [9]. It is shown that σ(Γ) is bounded from above by the Lovász theta function θ(LQ(Γ)) of the graph LQ(Γ) related to the line grap...

The rank function rank(·) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(·), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization p...

(ABSTRACT) Despite recent advances in convex optimization techniques, the areas of discrete and continuous nonconvex optimization remain formidable, particularly when globally optimal solutions are desired. Most solution techniques, such as branch-and-bound, are enumerative in nature, and the rate of their convergence is strongly dependent on the accuracy of the bounds provided, and therefore, ...