Global optimization in the 21st century: Advances and challenges

This paper presents an overview of the research progress in global optimization during the last five years (1998-2003), and a brief account of our recent research contributions. The review part covers the areas of (a) twice continuously differentiable nonlinear optimization, (b) mixed-integer nonlinear optimization, (c) optimization with differential-algebraic models, (d) optimization with grey box/black box/nonfactorable models, and (e) bilevel nonlinear optimization. Our research contributions part focuses on (i) improved convex underestimation approaches that include convex envelope results for multilinear functions, convex relaxation results for trigonometric functions, and a piecewise quadratic convex underestimator for twice continuously differentiable functions, and (d) the recently proposed novel generalized alphaBB framework. Computational studies will illustrate the potential of these advances.