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 ∑φ_i0(x_i) subject to x ∈ G, l ≦ x ≦ L and ∑φ_ij(x_i) ≦ 0, j = 1, …, m. Each φ_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 ≦ x ≦ 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.