Branch-and-reduce algorithm for convex programs with additional multiplicative constraints

This article presents a branch-and-reduce algorithm for globally solving for the first time a convex minimization problem (P) with p⩾1p⩾1 additional multiplicative constraints. In each of these p additional constraints, the product of two convex functions is constrained to be less than or equal to a positive number. The algorithm works by globally solving a 2p2p-dimensional master problem (MP) equivalent to problem (P). During a typical stage k of the algorithm, a point is found that minimizes the objective function of problem (MP) over a nonconvex set FkFk that contains the portion of the boundary of the feasible region of the problem where a global optimal solution lies. If this point is feasible in problem (MP), the algorithm terminates. Otherwise, the algorithm continues by branching and creating a new, reduced nonconvex set Fk+1Fk+1 that is a strict subset of FkFk. To implement the algorithm, all that is required is the ability to solve standard convex programming problems and to implement simple algebraic steps. Convergence properties of the algorithm are given, and results of some computational experiments are reported.