Sign reversion approach to concave minimization problems

Abstract

We consider a problem of minimization of a concave function subject to affine constraints. By using sign reversion techniques we show that the initial problem can be transformed into a family of concave maximization problems. This property enables us to suggest certain algorithms based on the parametric dual optimization problem.

One algorithm for branch and bound method for solving concave optimization problem

“…Thus, the branch and bound method is only necessary in order to build this equivalent auxiliary convex programming problem. This theoretical result for the concave programming problem with linear constraints has been proposed and substantiated in [5,6].…”

“…So, for * 0 () II  x conditions (2)-( 3) and ( 6)-( 7) are the same. It is obvious that any local minimum point of problem (1) is a solution some problem of form (5). Conversely, if the point * x solves the problem (5) and * D  x , then * x is the stationary point of the problem (1).…”

“…Consequently, the solution of the problem (1) can be found as a point of local minimum obtained as the solution of the problem (5) for some index set, with a minimum value of the objective function. Therefore, it can be found by trying all possible subsets of indices and solution for their auxiliary problem (5).…”

“…1. An easy way to solve the auxiliary problem (5) in the node tree. The dual problem to (5) also has the form of quadratic programming problem and there is an explicit formula for obtaining the solution of the direct problem (5) ' x with solution of dual problem…”