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This paper addresses the solution of bound-constrained optimization problems using algorithms that require only the availability of objective function values but no derivative information. We refer to these algorithms as derivative-free algorithms. Fueled by a growing number of applications in science and engineering, the development of derivativefree optimization algorithms has long been studied, and it has found renewed interest in recent time. Along with many derivative-free algorithms, many software implementations have also appeared. The paper presents a review of derivative-free algorithms, followed by a systematic comparison of 22 related implementations using a test set of 502 problems. The test bed includes convex and nonconvex problems, smooth as well as nonsmooth problems. The algorithms were tested under the same conditions and ranked under several criteria, including their ability to find near-global solutions for nonconvex problems, improve a given starting point, and refine a near-optimal solution. A total of 112,448 problem instances were solved. We find that the ability of all these solvers to obtain good solutions diminishes with increasing problem size. For the problems used in this study, TOMLAB/MULTI-MIN, TOMLAB/GLCCLUSTER, MCS and TOMLAB/LGO are better, on average, than other derivative-free solvers in terms of solution quality within 2,500 function evaluations. These global solvers outperform local solvers even for convex problems. Finally, TOMLAB/OQNLP, NEWUOA, and TOMLAB/MULTIMIN show superior performance in terms of refining a nearoptimal solution.

A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recourse-based stochastic programming, robust stochastic programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the state-of-the-art in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomial-time approximation schemes for multi-stage stochastic programs and the application of global optimization algorithms to two-stage and chance-constraint formulations.

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The Branch-And-Reduce Optimization Navigator (BARON) is a computational system for facilitating the solution of nonconvex optimization problems to global optimality. We provide a brief description of the algorithms used by the software, describe the types of problems that can be currently solved and summarize our recent computational experience. BARON is available by anonymousftpfromaristotle.me.uiuc.edu.

This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm.In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems.In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0 − 1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

This paper provides a critical review of the literature on heat-exchanger network synthesis, the most commonly studied problem in process synthesis. After a review of solution methods, we present the chronological milestones in the development of the field and we discuss separately each of 461 related works leading up to the turn of the century. Then, we present several classification schemes of this body of work based on a number of modeling and algorithmic criteria. Finally, we offer a critical assessment of the current status of research in this area and provide suggestions for future research.

This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.