An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints

We extend a previous algorithm in order to solve mathematical programming problems of the form: Find x = (x₁, ..., x_n) to minimize \sum \varphi _i0(x_i) subject to x \in G, l \leqq x \leqq L and \sum \varphi _ij(x_i) \leqq 0, j = 1, ..., m. Each \varphi _ij is assumed to be lower semicontinuous, possibly nonconvex, and G is assumed to be closed. The algorithm is of the branch and bound type and solves a sequence of problems in each of which the objective function is convex. In case G is convex each problem in the sequence is a convex programming problem. The problems correspond to successive partitions of the set C = { x | l \leqq x \leqq L}. Two different rules for refining the partitions are considered; these lead to convergence of the algorithm under different requirements on the problem functions. An example is given, and computational considerations are discussed.