Global Optimization: Software, Test Problems, and Applications

We provide a concise review of the most prominent global optimization (GO) strategies currently available. This is followed by a discussion of GO software, test problems and several important types of applications, with additional pointers. The exposition is concentrated around topics related to continuous GO, although in certain aspects it is also pertinent to analogous … Read more

GloptiPoly – Global Optimization over Polynomials withMatlab and SeDuMi

GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality relaxations of the (generally non-convex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality or integer constraints. It generates a series of lower bounds monotonically converging to the global optimum. Numerical experiments show that for most of … Read more

The Sample Average Approximation Method for Stochastic Programs with Integer Recourse

This paper develops a solution strategy for two-stage stochastic programs with integer recourse. The proposed methodology relies on approximating the underlying stochastic program via sampling, and solving the approximate problem via a specialized optimization algorithm. We show that the proposed scheme will produce an optimal solution to the true problem with probability approaching one exponentially … Read more

A sequential convexification method (SCM) for continuous global optimization

A new method for continuous global minimization problems, acronymed SCM, is introduced. This method gives a simple transformation to convert the original objective function to an auxiliary function with gradually fewer local minimizers. All Local minimizers except a prefixed one of the auxiliary function is in the region where the function value of the original … Read more

Semidefinite programming vs LP relaxations for polynomial programming

We consider the global minimization of a multivariate polynomial on a semi-algebraic set \Omega defined with polynomial inequalities. We then compare two hierarchies of relaxations, namely, LP-relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams and recent SDP (semi definite programming) relaxations introduced by the … Read more

Products of positive forms, linear matrix inequalities, and Hilbert 17-th problem for ternary forms

A form p on R^n (homogeneous n-variate polynomial) is called positive semidefinite (psd) if it is nonnegative on R^n. In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert (later proven by Artin) is that a form p is psd if and only if … Read more

New Classes of Globally Convexized Filled Functions for Global Optimization

We propose new classes of globally convexized filled functions. Unlike the globally convexized filled functions previously proposed in literature, the ones proposed in this paper are continuously differentiable and, under suitable assumptions, their unconstrained minimization allows to escape from any local minima of the original objective function. Moreover we show that the properties of the … Read more

A New Second-Order Cone Programming Relaxation for MAX-CUT problems

We propose a new relaxation scheme for the MAX-CUT problem using second-order cone programming. We construct relaxation problems to reflect the structure of the original graph. Numerical experiments show that our relaxation approaches give better bounds than those based on the spectral decomposition proposed by Kim and Kojima, and that the efficiency of the branch-and-bound … Read more

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 … Read more

Adaptive Simulated Annealing (ASA)

Adaptive Simulated Annealing (ASA) is a C-language code developed to statistically find the best global fit of a nonlinear constrained non-convex cost-function over a D-dimensional space. Citation %A L. Ingber %R Global optimization C-code %I Caltech Alumni Association %C Pasadena, CA %T Adaptive Simulated Annealing (ASA) %D 1993 %K 200701 %L Ingber:1993:CODE-ASA %O URL … Read more