On the Generalized Benders Decomposition

Abstract

Generalized Benders Decomposition is a procedure to solve certain types of NLP and MINLP problems. The use of this procedure has been recently suggested as a tool for solving process design problems. This paper analyzes the solution of nonconvex problems through different implementations of the Generalized Benders Decomposition. It is demonstrated that in certain cases only local minima may be found, whereas in other cases not even convergence to local optima can be achieved. A criterion to identify whether the converged value is a candidate for being a local minimum is provided. It is also shown that in the presence of a dual gap, a particular implementation of the Generalized Benders Decomposition may provide upper and lower bounds on the global optimum. The conceptual steps undertaken in establishing the various implementations of GBD are summarized as follows: • -Convexity of X and of F ( x, y) and G ( x, y) in x, implies that (1) is equivalent to (4). When these conditions do not apply a gap between v( y) and its dual may exist (Remark 2.1). • -Problem (4) is replaced by a sequence of relaxed master problems (Remark 2.3). • -Convexity of X and of F ( x, y) and G ( x, y) in x (as well as satisfaction of certain other conditions) guarantees ϵ-convergence of the GBD iterations (Geoffrion, 1972, Theorem 2.5). It is implicitly understood, that for these results to hold, both the primal and the relaxed master problem must be solved globally. • -Global solution of the primal stems readily from convexity of X and of F ( x, y) and G ( x, y) in x. Global solution of the relaxed master may come either through the use of special algorithms or by the establishment of convexity of L *( y, u) and L * ( y, λ) in y. One case for which convexity of these functions can be established is when F ( x, y) and G ( x, y) are jointly convex in x and y (Geoffrion, 1972, Section 4.2). • -Prior to the solution of the relaxed master problem, the evaluation of L*( y, u*) and L * ( y, λ*) is first required through (5) and (6). • -One case for which L*( y, u*) and L * ( y, λ*) can be obtained in explicit form is when the global minimum (over x) of (5) and (6) can be obtained independently of y [ Property ( P)]. In this case one can construct the Benders iterations by selecting x to be the solution of the primal and u (or λ) the corresponding Lagrange multiplier. • -When the Benders iterations are built based on the primal solution, i.e. assuming that the solution of the minimization problem (5) or (6) is equal to the solution of the primal x*, [irrespective of the satisfaction of Property ( P)], the NP-GBD algorithm is obtained. When x* is indeed the solution of the minimization problem, it is said that Property ( P′) holds. • -When the Benders iterations are built through the introduction of additional constraints as in (11), and through global solutions of (5) and (6) (problems R and S), then the PL-GBD is obtained.