This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.

A Branch-and-Reduce Approach to Global Optimization

A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory

Optimization problems involving d.c. functions (differences of convex functions) and d.c. sets (differences of convex sets) occur quite frequently in operations research,economics, engineering design and other fields. We present a review of the theory, methods and algorithms for this class of global optimization problems which have been elaborated in recent years

D.C. Optimization: Theory, Methods and Algorithms

A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—II. Application of theory and test problems

New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems

A simple algorithm for globally minimizing a polynomial on a compact interval of the real line is proposed. The algorithm is based upon the idea of local polynomial approximation, is finitely convergent and reliable. Moreover, no derivatives of the polynomial are evaluated. Some examples of the numerical behavior of the algorithm are given.

Dallas R. Wingo

Papers

Globally minimizing polynomials without evaluating derivatives