Application to Global Optimization: Lagrange and Penalty Functions

Abstract

One of the main approaches to the solution of convex optimization problems involves the exploitation of properties of the traditional linear Lagrange function (Lagrangian) and the penalty function. In particular, the zero duality gap property between the primal convex optimization problem and its Lagrange (penalty) dual problem has enabled important algorithms to be proposed and developed, see for example [21, 57, 113, 136] and references therein. The zero duality gap property guarantees that there exists a sequence of unconstrained problems, such that their solutions tend to a solution of the initial constrained optimization problem. These problems are formed by means of a special choice of Lagrange multipliers (penalty parameters). The situation becomes much simpler if there exists the exact Lagrange multiplier (penalty parameter). In this case it is possible to build an unconstrained problem, which is equivalent to the initial problem. The exact penalty parameter can be found in many instances, however sometimes this parameter is very large, so the unconstrained problem discussed above becomes ill-conditioned.