On global optimality conditions for D.C. minimization problems with D.C. constraints

Abstract

. The paper addresses the nonconvex nonsmooth optimization problem with the cost function, and equality and inequality constraints given by d.c. functions, i.e. represented as a difference of convex functions. The original problem is reduced to a problem without constraints with the help of the exact penalization theory. After that, the penalized problem is represented as a d.c. minimization problem without constraints, for which the new mathematical tools under the form of global optimality conditions (GOCs) are developed. The GOCs reduce the nonconvex problem in question to a family of convex (linearized with respect to the basic nonconvexities) problems. In addition, the GOCs are related to some nonsmooth form of the KKT-theorem for the original problem. Besides, the GOCs possess the constructive (algorithmic) property, which, when the GOCs are broken down, implies the producing of a feasible point that is better (in the original problem) than the one in question. The effectiveness of the GOCs is demonstrated by examples.