THE P-th ORDER OPTIMALITY CONDITIONS FOR DEGENERATE INEQUALITY CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, we present necessary and sufficient optimality conditions for optimization problems with inequality constraints in the finite dimensional spaces. We focus on the degenerate (nonregular) case when the linear independence constraint qualification (LICQ) and Mangasarian-Fromovitz constraint qualification (MFCQ) are not satisfied at the solution of the optimization problem. For the problems satisfying the p-regularity constraint qualification or p-regularity conditions, we present necessary and sufficient conditions that resemble the structure of the classical conditions and give new and nontrivial conditions for degenerate inequality constrained problems. We also present second-order necessary conditions and corresponding sufficient conditions. The optimality conditions can be applied to discretizations of calculus of variations and optimal control problems. In addition, we prove that the 2-regularity condition is weaker than the MFCQ.