An extended cutting plane method is introduced. The extended method can be applied in the solution of a class of non-convex MINLP (mixed-integer non-linear ...
Pergamon
Computers chem. Engng Vol.22, No. 3, pp. 357-365, 1998 Copyright© 1998ElsevierScienceLtd.All rightsreserved Printedin GreatBritain PII: S0098-1354(97)00000-0 0098-1354/98 $19.00+0.00
An extended cutting plane method for a class of non-convex MINLP problems Tapio Westerlund ~, Hans Skrifvars l, Iiro Harjunkoski ~ and Ray PSrn 2 Process Design Laboratory,/~bo Akademi University, Biskopsgatan 8, FIN-20500/~BO, Finland. 2 Department of Mathematics,/~bo Akademi University, F~inriksgatan 3, FIN-20500/~BO, Finland.
(Received 26 March 1996; revised 16 June 1997) Abstract
An extended cutting plane method is introduced. The extended method can be applied in the solution of a class of non-convex MINLP (mixed-integer non-linear programming) problems, although the method was originally introduced for the solution of convex problems only. Global convergence properties of the method are given for pseudo-convex MINLP problems in the present paper and a numerical example from the paper-converting industry is finally provided to illustrate the numerical procedure. © 1998 Elsevier Science Ltd. All rights reserved Keywords: Optimization; Mixed-integer non-linear programming; Integer non-linear programming; Quasi-convex
functions; Pseudo-convex functions; Extended cutting plane method
1. Introduction
Several outer-approximation (OA) methods for MINLP (mixed-integer non-linear programming) problems have been introduced over the past few decades: the GBD (generalized benders decomposition) method (Geoffrion, 1972), the OA (outer-approximation) method (Duran and Grossmann, 1986), the GOA (generalized outer approximation) method (Yuan et al., 1989), the LP/NLP (linear programming/non-linear programming) based B&B (branch and bound) method (Quesada and Grossmann, 1992), the LOA (linear outer approximation) method (Fletcher and Leyffer, 1994) and the ECP (extended cutting plane) method (Westerlund and Pettersson, 1995), to name a few. The methods are based on different numerical approaches. The GBD, OA, GOA, LP/NLP based B&B and the LOA methods solve NLP subproblems while the ECP method does not solve any NLP subproblems. The methods have, however, a common feature: they all solve MILP subproblems. The size of the MILP problem increases with the number of iterations. Thus, efficient outer-approximation methods need efficient MILP solvers (Nernhauser and Wolsey, 1988). The MILP subproblems do not, however, need to be re-solved; only the tree must be updated. Improved efficiency in solving MILP subproblems using information from the subsequent iteration has been shown by Quesada and Grossmann (1992) and Fletcher and Leyffer (1995). However, no recursive MILP procedure
where re-solving of the MILP problems is completely avoided has, to the authors' knowledge, yet been given. In addition to linear outer-approximation based methods, methods based on tree enumeration (Dakin, 1965; Gupta and Ravindran, 1985), non-smooth outer approximation (Fletcher and Leyffer, 1994) as well as probabilistic type methods such as simulated annealing (Kirkpatrick et al., 1983) have been suggested for the solution of MINLP problems. Most of the MINLP methods have, however, another common property; global convergence is ensured for convex MINLP problems only. Different heuristic procedures for the solution of non-convex problems have been introduced (Viswanathan and Grossmann, 1990; Westerlund et al., 1994) for some of the outer-approximation methods. However, although these methods have been found to perform quite well in different applications, the convergence properties have not generally been extended from the convex case. Some interesting global optimization type algorithms that ensure convergence for certain classes of nonconvex MINLP problems have also been introduced recently (Androulakis et al., 1995; Visweswaran and Floudas, 1996; Ryoo and Sahinidis, 1995). In the global optimization algorithms, however, the function space is separated for the continuous and discrete variables, and the discrete variables can still only occur in the linear space. In a paper by Westerlund and Pettersson (1995), a
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convergence proof for the extended cutting plane (ECP) method was given for convex MINLP problems. The ECP method is an extension of Kelley's cutting plane (CP) method which was originally given for convex NLP problems (Kelley, 1960). In the present paper, the extended method is modified to cover a wider class of MINLP problems. A convergence proof for pseudoconvex problems is given below. The modified version of the method can be applied to non-convex MINLP problems, and global convergence is ensured for pseudo-convex problems. The procedure is an attractive alternative for the solution of different non-convex MINLP problems, especially pseudo-convex INLP problems for which the procedure converges to the global optimal solution in a finite number of iterations. The ECP method requires only the solution of MILP subproblems in each iteration. This makes the ECP algorithm easy to implement. The method has been fpund to be well suited for solving large MINLP, and especially for INLP, problems with a moderate degree of non-linearity.
2. Formulation of the minlp problem The MINLP problem corresponding to the method may be formulated as follows, min {c.~rx+ery} x,y
(1)
{x,y}e.NAL where N= {(x,y)lg(x,y)-0), with n=k~ and k=k/in (20) such that
e--0 and define Z(x,y)=erx+ eyy. Then there exists an index k0 for which we have, Z(xP,y p) - Z(xk,yk) < e
(25)
for all k>k0 . Now assume there exists a feasible point (x,y) to (P) with Z(x,y) - eh
(39)
The solution obtained with the procedure given by (28)(29) and (38) together with the termination criteria, (27) and (39), will then be feasible for g~(x,y)-
