SADDLE POINT CRITERIA AND THE EXACT MINIMAX PENALTY FUNCTION METHOD IN NONCONVEX PROGRAMMING

• Published in: Taiwanese journal of mathematics Vol. 17; no. 2; pp. 559 - 581
• Main Author: Antczak, Tadeusz
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.04.2013
• Subjects: Constrained optimization; Lagrange multipliers; Mathematical functions; Minimax; Penalty function; Saddle points
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm.17.2013.1823

A new characterization of the exact minimax penalty function method is presented. The exactness of the penalization for the exact minimax penalty function method is analyzed in the context of saddle point criteria of the Lagrange function in the nonconvex differentiable optimization problem with both inequality and equality constraints. Thus, new conditions for the exactness of the exact minimax penalty function method are established under assumption that the functions constituting considered constrained optimization problem are invex with respect to the same functionη(exception with those equality constraints for which the associated Lagrange multipliers are negative - these functions should be assumed to be incave with respect to the same functionη). The threshold of the penalty parameter is given such that, for all penalty parameters exceeding this treshold, the equivalence holds between a saddle point of the Lagrange function in the considered constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function. 2010Mathematics Subject Classification: 49M30, 90C26, 90C30. Key words and phrases: Exact minimax penalty function method, Minimax penalized optimization problem, Exactness of the exact minimax penalty function, Saddle point, Invex function.