Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

Abstract

ABSTRACTSecond-order optimality conditions for the vector nonlinear programming problems with inequality constraints and continuously differentiable data are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce notions of a second-order pseudoconvex vector function and KKT-pseudoconvex vector problem with inequality constraints. We derive necessary and sufficient conditions for efficiency in the vector problem with inequality constraints. Three examples are presented.