Reversed geometric programming: A branch-and-boundmethod involving linear subproblems F. COLE, W. GOCHET, F. VAN ASSCHE Department of Applied Economics, Catholic University of Louvain, Belgium J. ECKER Department of Mathematical Sciences, RensselaerPolytechnic Institute and Y. SMEERS Center for Operations Research, Catholic University o f Louvain, Belgium Received November 1978 Revised Match 1979 The paper proposes a branch-and-bound method to find the global solution of general polynomial programs. The problem ~is first transformed into a reversed posynomial program. The procedure, which is a combination of a previ- ously developed branch-and-bound method and of a well- known cutting plane algorithm, only requires the solution of linear subproblems. 1. Introduction A signomial geometric program is a nonconvex optimization program that can be defined as the minimization (maximization) of a 'general' poly- nomial over a set defined by (in)equalities on general- ized polynomials. A generalized polynomial is a poly- nomial in several variables, i.e. a function of the type iEl ] with ci, i E I and ai/, i E L / E J, arbitrary constants. This paper proposed a branch- © North-Holland Publishing Company European Journal of Operational-Research 5 (1980) 26-35 and-bound method to find the global solution of such programs. It uses the result that every signomial pro- gram can be transformed into a 'reversed' geometric program [4] i.e. a minimization program with all coef- ficients c i positive and both 'less than' and 'larger than' type constraints appear in the constraint set. The subproblems to be solved at every iteration are linear programming problems. Previous papers using branch-and-bound methods to solve signomial geo- metric programs include [8,9,1 2 and 15 ]. In this paper we modify the algorithm proposed in [10] to avoid solving nonlinear subproblems. A well- known cutting plane method, see e.g. [ 1,6,9], is combined with the branch-and-bound scheme of [10] to generate a method involving only linear subprob- lems. Convergence of the method is proved under the usual assumption of cutting plane methods (bounded feasible region). Although the algorithm presented here assumes that all cuts are kept in subsequent iterations, the reader can easily see from the type of argument used in the convergence proof that condi- tions for dropping cuts [5] could be adopted for this method. In [8,15] the general algorithms of [7] for sep- arable nonconvex programming is applied to sig- nomial programming. Therefore, each signomial function is first transformed into a linear combina- tion of exponential functions of one variable. The negative exponential[ terms are approximated in an interval by their convex hull and these approxima- tions are improved in a branch-and-bound scheme. Except for an illustrative example in [8], no numeri- cal experience has been reported with these methods. Other methods do not use the separability property of geometric programs. In [ 12, 13], a convex relaxa- tion of a reversed constraint with k terms (a reversed constraint is defined in Section 2) is constructed by introducing k one term posynomial constraints. In [10] a reversed constraint is approximated by just one one-term posynomial constraint. Results for the same illustrative example, the well-known gravel box problem, are given in these three papers. It appears 26