Global optimization of mixed‐integer nonlinear problems

Abstract

Two no®el deterministic global optimization algorithms for noncon®ex mixed-integer ( ) problems MINLPs are proposed, using the ad®ances of the ␣ BB algorithm for noncon®ex NLPs of Adjiman et al. The special structure mixed-integer ␣ BB algorithm ( ) SMIN-␣ BB addresses problems with noncon®exities in the continuous ®ariables and linear and mixed-bilinear participation of the binary ®ariables. The general structure ( ) mixed-integer ␣ BB algorithm GMIN-␣ BB is applicable to a ®ery general class of problems for which the continuous relaxation is twice continuously differentiable. Both algorithms are de®eloped using the concepts of branch-and-bound, but they differ in their approach to each of the required steps. The SMIN-␣ BB algorithm is based on the con®ex underestimation of the continuous functions, while the GMIN-␣ BB algorithm is centered around the con®ex relaxation of the entire problem. Both algorithms rely on optimization or inter®al-based ®ariable-bound updates to enhance efficiency. A series of medium-size engineering applications demonstrates the performance of the algorithms. Finally, a comparison of the two algorithms on the same problems highlights the ®alue of algorithms that can handle binary or integer ®ariables without reformulation. IntroductionThe decision-making processes that take place during the design of new products or chemical plants can be made more rational and efficient thanks to the use of mathematical models within a global optimization framework. For instance, ap-Ž . proaches based on nonlinear programming NLP , such as Ž . those described in Horst and Tuy 1996 , have been used to determine optimum equipment sizes and operating condi-Ž . tions for a given process Floudas, 1995; Grossmann, 1996 . The economic benefits to be derived from identifying the global solution of the many nonconvex problems that arise in Ž chemical engineering has been amply illustrated Grossmann, . 1996 . The most significant contribution of mathematical approaches, however, comes from their ability to incorporate many alternative structures within a single problem. This is achieved through the introduction of integer variables, which leads to the formulation of a mixed-integer nonlinear prob-Ž .Ž lem MINLP Floudas, 1995;Floudas and Grossmann, 1995; . Grossmann, 1996 . Such an approach has already been used for a wide array of applications, including process synthesis.Correspondence concerning this article should be addressed to C. A. Floudas. Present address of: C. S. Adjiman, Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK; I. P. Androulakis, Exxon Research and Development, Annandale, NJ.The solution of many MINLPs relevant to chemical engineering is made challenging not only by the presence of integer variables but also by the nonconvexities in the models. As a result, the potential contributions of mixed-integer nonlinear optimization to general design problems have not yet been fully realized. The deterministic ap...

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results

“…[16,Def. 6]) and on solving the inner upper bounding and outer lower bounding problems, respectively (RIUB) and (LB), to global optimality, as required, using the αBB solver [1][2][3]. The outer upper bounding problem (UB), although nonconvex, does not need to be solved to global optimality.…”

“…In this section, we apply this theory to both branch-and-bound schemes, inner and outer, of the Branch-and-Sandwich algorithm introduced in [16] in order to set the foundations for its overall convergence, proved in Theorem 6, to an ε-optimal solution of problem (1). It is assumed throughout that an exhaustive partitioning scheme is used.…”

“…The Branch-and-Sandwich algorithm is ε-convergent, such that at termination we have F UB − F LB ≤ ε F , where F LB = min k∈L F (k) , and the incumbent solution (x UB , y UB ) is an ε-optimal solution of problem (1).…”

“…f (x, y) ≤ w(x), a restriction of problem (3), i.e., an upper bound on w(x), yields a relaxation of (2); consequently, a relaxation of (1). Similarly, a relaxation of (3) gives a restriction of (2) [19].…”

“…Problem (1) has been tackled extensively for specific classes of the participating functions. For instance, the linear and the nonlinear but convex bilevel programming problems have been addressed in [4,5,9,12,13,26,27].…”