Abstract
Nonconvex mixed integer nonlinear programming problems arise quite frequently in engineering decision problems, in general, and in chemical process design synthesis and process scheduling applications, in particular. These problems are characterized by high dimensionality and multiple local optimal solutions. In this work, a novel approach is developed for determining the global optimum in nonlinear continuous and discrete domains. The mathematical foundations of the feature extraction algorithm are presented and the properties of the algorithm discussed in detail. The algorithm uses a partition and search strategy in which the problem domain is successively partitioned and a statistical approximation approach is used to characterize the objective function values and the constraint feasibility over a partition. Specifically, the general joint distribution function representing the objective function values is relaxed to a separable form and approximated using an expansion in terms of Bernstein functions. The coefficients of the expansion are determined by solving a small linear program. Feasibility is established by computing upper and lower bounds for the inequality constraint functions, while equality constraints are explicitly or numerically eliminated. Estimates of the volume averaged values of objective function and constraint feasibility are used to select efficient partitions for further investigation. These are refined successively so as to focus the search on the most promising decision regions. An alternative, constant resolution partitioning strategy is also developed using a suitably modified genetic search algorithm. Illustrative examples are used to demonstrate the key computational features of the method.
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References
M. Abramowitz and I.A. Stegun (eds.),Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington D.C., 1964).
G. Alefeld and J. Herzberger,Introduction to Interval Computations (Academic Press, New York, 1983).
I.P. Androulakis and V. Venkatasubramanian, A genetic algorithm framework for process design and optimization, Comp. Chem. Eng. 15(1991)217–228.
T.M. Apostol,Mathematical Analysis, 2nd ed. (Addison-Wesley, Reading, MA, 1974).
C.A. Floudas and V. Visweswaran, A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory, Comp. Chem. Eng. 14(1990)1397–1417.
C.A. Floudas, A. Aggarwal and A.R. Ciric, Global optimum search for nonconvex NLP and MINLP problems, Comp. Chem. Eng. 13(1989)1117–1132.
D.E. Goldberg,Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, MA, 1989).
J.H. Holland,Adaptation in Natural and Artificial Systems (The University of Michigan Press, Ann Arbor, 1975).
S. Karlin and W.J. Studden,Tchebycheff Systems with Applications in Analysis and Statistics (Interscience, New York, 1966).
R.S.H. Mah,Chemical Process Structures and Information Flows (Butterworths, Boston, 1990).
G. McCormick,Nonlinear Programming: Theory, Algorithms and Applications (Wiley, New York, 1983).
P.M. Pardalos and J.B. Rosen,Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268 (Springer, Berlin, New York, 1987).
S. Papageorgaki and G.V. Reklaitis, Optimal design of mutipurpose batch plants — 2. A decomposition solution strategy, Ind. Eng. Chem. Res. 29(1990)2062–2073.
A. Papoulis,Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).
H. Ratcheck, Inclusion functions and global optimization, Math. Progr. 33(1985)300–317.
H. Ratcheck and J. Rokne,Computer Methods for the Range of Functions (E. Horwood, Chichester, Halsted Press, New York, 1984).
H. Ratcheck and J. Rokne,New Computer Methods for Global Optimization (E. Horwood, Chichester, Halsted Press, New York, 1988).
G.V. Reklaitis, A. Ravindran and K.M. Ragsdell,Engineering Optimization: Methods and Applications (Wiley-Interscience, 1983).
A.G. Tsirukis, Scheduling of multipurpose batch chemical plants, Ph.D. Thesis, Purdue University (August, 1991).
A.G. Tsirukis, G.V. Reklaitis and M.F. Tenorio, Nonlinear optimization using generalized Hopfield networks, Neural Comp. 2(1989)511–521.
A.G. Tsirukis and G.V. Reklaitis, Application of generalized Hopfield networks to discrete nonlinear optimization problems, Comp. Chem. Eng. (1991).
J. Viswanathan and I.E. Grossmann, A combined penalty function and outer-approximation method for MINLP optimization, presented at theCOPS/TIMS/ORSA Meeting, Vancouver (1989) paper MD 18.5.
V. Visweswaran and C.A. Floudas, A global optimization algorithm (GOP) for certain classes of Nonconvex NLPs — II. Application of theory and test problems, Comp. Chem. Eng. 14(1990)1419–1434.
M.C. Wellons and G.V. Reklaitis, Scheduling of multipurpose batch plants — Part I. Formation of single-product production lines, Ind. Eng. Chem. Res. 30(1991)671–698.
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Tsirukis, A.G., Reklaitis, G.V. Feature extraction algorithms for constrained global optimization I. Mathematical foundation. Ann Oper Res 42, 229–273 (1993). https://doi.org/10.1007/BF02023177
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DOI: https://doi.org/10.1007/BF02023177