Global optimization and its applications

Abstract

This special issue is devoted to selected invited papers presented at the Workshop on Global Optimization held at Imperial College London, 15–17 December 2007 (http://gow2007.ps. ic.ac.uk/). The Special Issue contains eight papers ranging from algorithms andmethodology to applications in engineering and finance. Methodological developments are considered by four papers. Floudas and Gounaris present a review of recent research in deterministic global optimization. It covers twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization with grey box/nonfactorable models, and bilevel nonlinear optimization. Further methodological issues are discussed by Lasserre in relation to moments and sums of squares for polynomial optimization and related problems. The duality between moment problems and sums of squares representations of positive polynomials is central. Subsequently, the paper discusses how such results are used to define convergent semidefinite programming relaxations in polynomial optimization as well as computing the convex envelope of a rational function and finding all zeros of a system of polynomial equations. Mitsos, Chachuat, Barton consider global bilevel dynamic optimization. A deterministic algorithm for bilevel programs with nonconvex functions is given, followed by a summary of deterministic algorithms for the global solution of optimization problems with nonlinear ordinary differential equations embedded. Parpas and Rustem present a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory information from the Lagrange multipliers is introduced into the SDE. The convergence analysis is provided. Two mixed-integer linear programming (MILP)-related papers are included. The first, by McAllister, DiMaggio, Floudas, presents a computational study for solving the distance-