Etienne de Klerk

Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming

The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMIs). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

Erratum: On Semidefinite Programming Relaxations of the Traveling Salesman Problem

Provided are compositions which include 1-methyl-2-nitro-3-[(3-tetrahydrofuryl)methyl]guanidine and at least one compound of formula (I):wherein, R1 represents a halogen atom or a methyl group, R2 represents a halogen atom or a methyl group and R3 represents a hydrogen atom or a cyano group, as well as a method of controlling cockroaches.

Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube

We consider the problem of minimizing a polynomial on the hypercube

An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution

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On the Turing Model Complexity of Interior Point Methods for Semidefinite Programming.

and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmudgen [Math. Ann., 289 (1991), pp. 203-206]. The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the hypercube.

Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization

We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator

Comparison of Lasserre’s Measure-Based Bounds for Polynomial Optimization to Bounds Obtained by Simulated Annealing

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Special issue in honour of Professor Kees Roos’ 70th Birthday

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Relaxations of the Satisfiability Problem Using Semidefinite Programming

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