Global Optimization with Nonfactorable Constraints

Abstract

This paper presents an approach for the global optimization of constrained nonlinear programming problems in which some of the constraints are non factorable, defined by a computational model for which no explicit analytical representation is available. A three-phase approach to the global optimization is considered. In the sampling phase, the nonfactorable functions and their gradients are evaluated and an interpolation function is constructed. In the global optimization phase, the interpolants are used as surrogates in a deterministic global optimization algorithm. In the final local optimization phase, the global optimum of the interpolation problem is used as a starting point for a local optimization of the original problem. The interpolants are designed in such a way as to allow valid over- and underestimation functions to be constructed to provide the global optimization algorithm with a guarantee of ε-global optimality for the surrogate problem.

Equation‐free optimal switching policies for bistable reacting systems

“…Through design of computational experiments, sampling and estimation, one can determine the form of surrogate models that are subsequently used to identify local or global (e.g. see References [20,21]) optima. In another approach, near optimal solutions are found for unconstrained optimization problems, under the assumption that absolute and relative error bounds are known for the computed objective function values [22].…”

A centroid‐based sampling strategy for kriging global modeling and optimization

“…closed-form models, they cannot be directly applied to the problem class addressed in this article because the formulation contains black-box models. A similar problem has been addressed by Meyer et al 6 for NLP containing nonanalytical constraints that are both known and differentiable. For this problem class, the solution is obtained using a global model whose form is that of a blending function from which overand under-estimators can be generated to provide e-guarantee of global optimality.…”

Global optimization in the 21st century: Advances and challenges

“…Byrne and Bogle (2000) studied the global optimization of modular flowsheeting systems, introduced an approach to modular based process simulation which is based on interval analysis and which can generate interval bounds, derivatives and their bounds for generic input-output modules, proposed a branch-and-bound global optimization algorithm, and applied it to an acyclic problem, and flowsheet with recycle. Meyer, Floudas, and Neumaier (2002) studied the global optimization of problems with nonfactorable constraints for which there does not exist an analytical form, proposed a sampling phase in which the nonfactorable functions and their gradients are sampled and a new blending function is constructed, presented a global optimization phase in which linear underestimators and overestimators are derived via interval analysis and the interpolants are used as surrogates in a branch-and-cut global optimization algorithm, discussed a local optimization stage where the global optimum solution of the interpolation problem becomes the starting point for optimizing locally the original problem, and illustrated their approach through a small benchmark problem, an oilshale pyrolysis problem, and a nonlinear continuous stirred tank reactor model. Theoretical and algorithmic advances outside of Chemical Engineering in this area include the work by Gutmann (2001), Jones (2001), Jones, Schonlau, and Welch (1998)and the recent book by Zabinsky (2003).…”