global optimum search for nonconvex nlp and minlp ... - ScienceDirect

Compurers

them. Engng,

Vol. 13, No. 10, pp.

1117-1132, 1989

0098-1354189

GLOBAL

OPTIMUM

%3.00+ 0.00

Pergamon press plc

Printed in Great Britain

SEARCH FOR NONCONVEX MINLP PROBLEMS

NLP AND

C. A. FLOUDAS,? A. AGGARWAL and A. R. CIRIC Department of Chemical

Engineering,

Princeton

University,

Princeton,

NJ 08544-5263,

U.S.A.

(Received 10 February 1989; final revision received 8 May 1989; received for publication 8 June 1989) Abstract-Nonconvex nonlinear programming problems (NLPs) and mixed-integer nonlinear programming problems (MINLPs) abound in the synthesis, design and control of chemical processes. The presently available mathematical programming techniques very often do not lead to the global solution of these classes of problems. A new approach for global optimum search is presented in this paper which involves a decomposition of the variable set into two sets-complicating and noncomplicating variables. This results in a decomposition of the constraint set leading to two subproblems. The decomposition of the original problem induces special structure in the resulting subproblems and a senes of these subproblems are then solved to determine the optimal solution. Appendix A presents a systematic approach, based on graph theory, for determining the various possibilities of decomposmg the variable set. The key idea is to combine a judicious selection of the complicating variables with suitable transformations leading to subproblems which can attain their respective global solutions at each iteration. Mathematical properties of the proposed approach are presented and its effectiveness is illustrated through a number of nonconvex NLP and MINLP example problems.

INTRODLJCTXON A large number of process synthesis, design and control problems in chemical engineering can be formulated as nonlinear programming (NLP) models or mixed-integer nonlinear programming (MINLP) models that involve continuous variables and integer decisions. A common feature of this class of mathematical problems is the potential existence of nonconvexities due to the particular form of the objective function and/or the set of constraints. The use of current mathematical programming techniques very often leads to suboptimal solutions. Separation sequences (Floudas, 1987; Wehe and Westerberg, 1987; Floudas and Anastasiadis, 1988), heat exchanger network generation (Floudas et al., 1986; Floudas and Ciric, 1988), the “pooling” problem (Aggarwal and Floudas, 1988) and the various categories of quadratic programming problems (e.g. bilinear programming, definite and indefinite quadratic program1990) are Floudas, ming; Agganval and representative examples of nonconvex NLP problems. Examples of nonconvex MINLP problems in chemical engineering include nonsharp distillation sequences (Aggarwal and Floudas, 1989), simultaneous match and network optimization in heat exchanger networks (Floudas and Ciric, 1989) and optimization of reactor networks (Kokossis and Floudas, 1990). The currently available algorithms for NLP problems can result in suboptimal solutions in the presence of nonconvexities. These algorithms rely heavily on the starting point and in the case of a particularly tAuthor to whom all correspondence CACE

13,IG-c

should be addressed.

bad starting point may not provide any feasible solution even though such a solution might exist. Most of the recent work on the solution of nonconvex MINLP problems has been based on projection on the integer variables and then application of either the Outer Approximation (Duran and Grossmann, 1986) or the Generalized Benders’ Decomposition technique (Geoffrion, 1972). Both these approaches involve the solution of a series of NLP and MILP subproblems. The NLP is the original problem solved for a fixed set of values of the integer variables and the MILP (refered to as the master problem) is solved to provide new values of the integer variables. The two approaches differ in the formulation of the master problem, that is, the first one is based on linearization of the constraints and the objective function while the second one is based upon the dualization of the original problem. Kocis and Grossmann (1987) presented an algorithm using Outer Approximation with Equality Relaxation that can handle explicitly equality constraints. Paules and Floudas (1989) proposed an algorithmic development procedure, APROS, for the automatic solution of mathematical programming problems that involve some form of decomposition and require extensive communication of data between a set of subproblems whose size and structure may vary during the solution procedure. APROS can be applied to MINLP problems using the two previously-mentioned approaches. It should be noted though, that in the above approaches, a nonconvex NLP still has to be solved at each iteration and therefore the search for the global solution is not performed. Kocis and Grossmann (1988) proposed a twophase strategy for solving nonconvex MINLP proh1117

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A.

FLOUDAS et al

lems through the Outer Approximation/Equality Relaxation algorithm. In phase I, the OA/ER algorithm 1s applied and then nonconvexities are identified through local and global tests. In phase II, the invalid outer-approximations to the nonconvex functions are systematlcally relaxed to yield a modified master problem that attempts to locate an improved solution from the one obtained in the first phase. This approach can sometimes fail to identify the nonconvexities in phase I. In this paper an approach is presented for the search of the global optimal solution of nonconvex NLP and MINLP problems. This approach is based on the decomposition of the variable set leading to a decomposition of the original nonconvex problem into two subproblems. A series of these subproblems is then solved based on the Generalized Benders’ Decomposition Method in search of the global optimum. The crucial element of this procedure is that the decomposition is performed in such a manner that both the resulting subproblems can be solved for their respective global solutions at each iteration. As will be shown. this approach can be applied to both nonconvex NLP and MINLP problems. BACKGROUND

J. F. Benders (1962) proposed a method for solving mathematical programming problems based upon the partitioning of the original problem into two subproblems, the primal and the master. and then iterating between these subproblems to arrive at a final solution. Geoffrion (1972) generalized this approach for a mathematical programming problem of the form (I) shown below:

subject to

g(.K, y) < 0,

(1)

where .Xand y are column vectors of variables, X and Y are feasible sets of these variables and g(x, _v) is a vector of inequality constraint functions. The Generalized Benders’ Decomposition technique has been used mainly for mixed-integer problems by partitioning the problem so that the integer variables y may be solved for independently of the continuous variables x. For this purpose, a master problem is formulated. This master problem contams a number of constraints which must be evaluated prior to the solution of this problem. To overcome this difficulty, Geoffrion (1972) proposed an approach that solves the master problem as a series of relaxed subproblems. The derivation of the master problem begins with a partitioning of problem (I) into an equivalent formulation featuring an inner and outer optimization problem: min V(Y).

subject to

,VEYrb v, where I/ z \‘~a: g(x, y) < 0

for some

XEX}.

(2a)

The inner optimization problem over zceX is simply the original problem (1) solved for some fixed value of y:

subject to

g(.u. .r 1 < 0.

(2b)

The solution of this problrm for every possible value of y provides the function L.(,I.). The outer optimization problem seeks to minim& I.(Y) over every value of y in &.. defined as the set of all _Vthat provide a feasible solution to the constraint set g(s. r)> and arc in Y. the origmal feasible set of )‘. The difficulty with (2b) is that the function L.(,v) and the set V are known only implicitly-. This difTiculty can be overcome by invoking the dual representation of L’ formulation. and V to provide the following min v(J,). it-i’ subject to

inf (j. ‘g(.y. _I*)]< 0 ,r.Y

all

j.EA.

(3)

Problem (3) is the master problem. This problem can be difficult to solve in its above form since it still involves inner optimization problems. The master problem may be solved by relaxing the formulation to contain constraints and not inner optimization problems. If the solution to this relaxed master problem satisfies the inner optimization problems, then the solution to the original problem has been achieved. If. however, the solution does not satisfy the inner optimization problems, further constraints This must he added to the relaxed master problem. updated formulation can be solved again, creating an iterative procedure that forms the basis of the Generalized Benders’ Decomposition Algorithm. Testing the solution of the relaxed problem requires solving problem (2b). If this problem 1s feasible. then a new constrain1 for the relaxed master may be generated from the Lagrange multipliers of g(x, 1’). If this problem is infeasible, then a minimizatlon of infeasibilities subproblem must be solved to generate appropriate values of the Lagrange multipliers ;_. As the relaxed master problem contains fewer constraints than (?a) [which is an equivalent formula-

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Global optimum search for nonconvex problems tion of the original problem (I)], its optimal value must be lower than or equal to the optimal value of (1). Thus the relaxed master problem provides a lower bound on the final solution. Conversely, as the Y variables are fixed in (2b), it contains more constraints than (2a). Thus, the optimal value of (2b) provides an upper bound on the final solution. The stopping criteria is met when the lower bound provided by the master problem meets the upper bound provided by subproblem (2b). To obtain the Lagrange functions for the master problem in explicit form, Geoffrion stated a property which allows the infimum over X to be taken essentially independently of Y. The usual situation, but by no means the only possible one, is that xs is an optimal solution of the primal problem where ri is an optimal multiplier vector. Under such conditions the master problem in the algorithm can be stated as follows: min pa YEY !4

subject to na af(xk,Y) Qk)‘g(x”,

+

(uk)rg(xk,

y)

k = 1,

Y) < 0

where x” is the optimal solution of the corresponding subproblem Lagrange multipliers. Motivating

Consider

Example

k = 1, . . . , Krcar,

. . , K”‘-,

in the kth iteration with the associated

f(x,.x*)=

the following

-x,

+x,x,-xx,

and f lin is the linearization (X:? x$) and is given by

of f around

the point

The points at which the tests are applied can be determined by perturbing the NLP problem and trying to identify points which lead to a reduction in the objective function. In this case, the local optimum of - 1.0052 is a strong local optimum and such a search is not successful. So, two points are arbitrarily selected in the neighbourhood of the local solution (0.916, 1.062). Consider the two points (1,1.125) and (0.833,1). At both these points f(x,, x2) is equal to -1. S’ln(x,,x2) is - 1.0053 for the first point and - 1.00513 for the second point. So the inequality (i) is satisfied for both these points. Therefore, the solution found by MINOS 5.2 might be taken to be the global optimum. But the global solution of this problem occurs at x, = 1,167, x1 = 0.5 and y = 0 with the value of the objective function being equal to - 1.0833. The Generalized Benders’ Decomposition method was then applied to this problem with the same starting point of - 1.0052. This algorithm stopped after one iteration with the starting point as the optimal solution. Motivating

1

Example

2

This example represents the design of multiproduct batch plants and is taken from Grossmann and Sargent (1979). The plant consists of M processing stages manufacturing fixed amount Qi of N products. For each stage j, the number of parallel units N, and their sizes V, have to be determined along with the batch sizes B, and cycle times TL, for each product i. The data for this problem is given in Table 1. The MINLP formulation for this problem is given below:

problem:

mm-x,+x,xr-xX1+Y, X,.” st.

where

- 6x, + 8x, < 3, 3x, - x2 < 3, O