Extended Reverse Convex Programming: An Active-Set Approach to Global Optimization

Reverse convex programming (RCP) represents an important class of global optimization problems consisting of concave cost and inequality constraint functions. While useful in many practical scenarios due to the frequent appearance of concave models, a more powerful, though somewhat abstractly recognized, characteristic of the RCP problem is its ability to approximate a very general class of nonconvex nonlinear programming (NLP) problems to arbitrary precision. The goal of the present work is to make this abstract idea concrete by formalizing an extended RCP framework with a nearly algorithmic procedure to approximate the general NLP problem by an RCP one. Furthermore, an active-set RCP algorithm, which may be seen as an improved and modernized version of Ueing’s method [39], is proposed and described in detail. Some preliminary results are presented for several NLP problems to demonstrate the potential of the proposed framework together with its shortcomings.