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PROCESS DESIGN AND CONTROL Differential Evolution with Tabu List for Solving Nonlinear and Mixed-Integer Nonlinear Programming Problems Mekapati Srinivas and G. P. Rangaiah* Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, 4 Engineering DriVe 4, Singapore 117576
Differential evolution (DE), a population-based direct-search algorithm, has been gaining popularity in the recent past due to its simplicity and ability to handle nonlinear, nondifferentiable, and nonconvex functions. In this study, a method, namely, differential evolution with tabu list (DETL), is described and evaluated for solving constrained optimization problems encountered in chemical engineering. It incorporates the concept of tabu search (TS) (i.e., avoiding revisits during the search) in DE mainly to improve its computational efficiency. DETL is initially applied to many nonlinear programming problems (NLPs) involving 2-13 variables and up to 38 constraints. It is then tested on several mixed-integer nonlinear programming problems (MINLPs) encountered in chemical engineering practice. The performance results of DETL, DE, and modified differential evolution (MDE) (Babu, K. V.; Angira, R. Comput. Chem. Eng. 2006, 30, 989), for both NLPs and MINLPs, are presented, and the relative performance of the three methods is discussed. Introduction Many process design, synthesis, control, and scheduling problems in engineering involve formulating and solving nonlinear programming (NLP) or mixed-integer nonlinear programming (MINLP) problems with constraints. Examples from the chemical engineering area include heat exchanger networks,1 pooling problem,2 utility and refrigeration systems,3 and evaporation systems.4 In general, the problem can be stated as
Minimize f(x, y) subject to hi(x, y) ) 0, i ) 1, 2, ..., m1 gj(x, y) g 0, j ) 1, 2, ..., m2 xlk e xk e xuk, k ) 1, 2, ..., n ylk e yk e yuk, k ) 1, 2, ..., (p - n) where x and y are vectors representing continuous and discrete variables, respectively, and h and g are equality and inequality constraints, respectively. The numbers of equality constraints, inequality constraints, continuous variables, and discrete variables are, respectively, m1, m2, n, and (p - n). The challenging characteristic of NLP/MINLP problems is the existence of nonconvexities because of either the objective function and/or constraints. The use of traditional local optimization techniques for solving these problems leads to local solutions, and hence, the study of global optimization techniques for NLP/MINLP problems has been of immense interest in the recent past.5-7 Generally, global optimization techniques can be broadly divided into two types: deterministic and stochastic. Several * Author for correspondence. E-mail:
[email protected]. Fax: (65) 6779 1936. Phone: (65) 6516 2187.
deterministic algorithms have been proposed for the solution of NLP/MINLP problems in the literature.5,8-10 Kocis and Grossmann8 have solved MINLP problems using the outer approximation/equality relaxation (OA/ER) algorithm. OA/ER consists of two phases. In phase I, nonconvexities that cut off the global optimum are systematically identified with local and global tests. In phase II, a new master problem is solved to locate the global optimum that may have been overlooked in phase I. Floudas et al.9 proposed an approach to solve NLP and MINLP problems that involves the decomposition of the variable set into two sets: complicating and noncomplicating variables. The decomposition of the original problem induces special structure in the resulting subproblems, and a series of these subproblems are solved based on the generalized Benders decomposition for the global optimum. Ryoo and Sahinidis10 proposed a branch-and-bound-based method for MINLP problems. It is based on the solution of a sequence of convex underestimating subproblems generated by evolutionary subdivision of the search region. Adjiman et al.5 proposed two new global optimization techniques for MINLP problems involving functions that are twice-differentiable in continuous variables. Both the techniques are based on the R-branch-and-bound (RBB) global optimization algorithm11 for twice-differentiable NLP problems. Most of the above deterministic methods have a mathematical guarantee to provide the global optimum while exploiting the mathematical structure of the given problem such that the original problem is decomposed into a number of subproblems that can be solved easily. On the other hand, stochastic methods are problem-independent and converge to the global optimum with probability approaching 1 as their running time goes to infinity.12 Further, they are applicable to nonconvex and/or noncontinuous functions. Das et al.13 studied four different versions of simulated annealing (SA) for scheduling serial, multiproduct batch pro-
10.1021/ie070007q CCC: $37.00 © 2007 American Chemical Society Published on Web 10/03/2007
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cesses; of the four versions of SA studied, the Metropolis algorithm with the Aarts and van Laarhoven annealing schedule was found to give the best results. Salcedo14 proposed an adaptive random search method for NLP and MINLP problems. The results obtained reveal the adequacy of random search methods for nonconvex NLPs and MINLPs in chemical engineering. Cardoso et al.15 proposed an SA approach for the solution of MINLP problems. The method combines the original Metropolis algorithm16 with the nonlinear simplex method of Nelder and Mead.17 The proposed approach is shown to be reliable and efficient, especially for larger-scale and illconditioned problems. Jayaraman et al.18 applied an ant-colony framework for optimal design of batch plants and found it to be robust to locate the optimum with
