Deterministic Global Optimization:  Advances and Challenges

Christodoulos A. Floudas
Department of Chemical Engineering
Princeton University
Princteon, New Jersey

Global optimization addresses the computation and characterization of global optima (i.e., minima and maxima) of nonconvex functions constrained in a specified domain.  Given an objective function f that is to be minimized and a set of equality and inequality constraints S, the area of Deterministic Global Optimization focuses on the following important issues:

  1. Determine a global minimum of the objective function f (i.e., f has the lowest possible value in S) subject to the set of constraints S;
  2. Determine lower and upper bounds on the global minimum of the objective function f on S that are valid for the whole feasible region S;
  3. Enclose all solutions of the set of equality and inequality constraints S.

In the last decade, we have experienced an explosive growth in the area of Global Optimization.  This is complimented by the ubiquitous applications that include important problems such as the structure prediction in protein folding and peptide docking in computational chemistry and molecular biology, phase and chemical equilibrium in thermodynamics, pooling and blending in chemical refineries, robust stability analysis in control, parameter estimation and data reconciliation, energy recovery systems in process synthesis, and multiple steady states in reactors.

In this plenary talk, we will discuss recent advances in the area of deterministic global optimization.  We will focus on (i) constrained optimization problems that are twice-continuously differentiable, (ii) nonlinear and mixed integer optimization models, and (iii) systems with differential and algebraic constraints.  The difference of convex functions method, denoted as αBB which can address general continuous twice-differentiable problems that arise in a variety of engineering and applied science problems, will be presented.  The αBB (i) offers theoretical guarantee of attaining an 0-global optimum solution in a finite number of iterations, (ii) provides valid lower and upper bounds on the global solution, and (iii) identifies local optima close to the global minimum.  We will also discuss new important theoretical results on (a) the derivation of the convex envelope for products of functions, and (b) the generalization of the αBB.  Applications from a variety of disciplines will demonstrate the power of these recent advances.