Generalized logarithmic penalty function method for solving smooth nonlinear programming involving invex functions

Abstract

In this paper, we have reviewed some penalty function methods for solving constrained optimization problems in the literature and proposed a continuously differentiable logarithmic penalty function which consists of the proposed logarithmic penalty function and modified Courant-Beltrami penalty function for equality and inequality constraints, respectively. Furthermore, we hybridized the two and came up with the general form of both (equality and inequality) constraints. However, in the first part, the equivalence between the sets of optimal solutions in the original optimization problem and its associated penalized logarithmic optimization problem constituted by invex functions with equality and inequality constraints has been established. In the second part, we have validated the general form of the logarithmic penalty function and compared the results with absolute value penalty function results by solving nine small problems from Hock-Schittkowski collections of test problems with different classifications. The experiments were carried out via quasi-newton algorithm using a fminunc routine function in matlab2018a. The general form yields a better objective value compared to absolute value penalty function.