Optimality conditions of strict minimality in optimization problems under inclusion constraints

In this paper, we extend the concepts of linearizing cone, regularity assumption and Lagrange multiplier rule due to Maurer and Zowe [H. Maurer, J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program. 16 (1979) 98–110] to an optimization problem under inclusion constraints. By virtue of the Robinson–Ursescu open mapping theorem, we obtain a Kuhn–Tucker necessary optimality condition. Moreover, we propose a Lagrangian by using the support function for set-valued maps, and establish some second-order sufficient and necessary optimality conditions for a strict local minimizer of order 2 based on the second-order derivative of the Lagrangian. As applications, we also investigate some second-order optimality conditions for the strict local minimizer of order 2 of a smooth scalar optimization problem with equality and inequality constraints.