Classical Optimality Conditions under Weaker Assumptions
Simon. Di · SIAM Journal on Optimization · 1996
In this article, an optimization problem that incorporates equality constraints, inequality constraints, and an abstract constraint is considered. We prove the classical optimality conditions (first-order and second-order, necessary, and sufficient) under weaker assumptions: functions involved are continuous around and differentiable at the optimal point instead of continuously differentiable or strictly differentiable and/or twice differentiable. The proof is based on the exact expressions for the contingent cone and the second-order contingent hull to the feasible set established in this article. It is interesting to note that our second-order Lagrange multiplier rule for minimization problems incorporating a more generalized abstract constraint has an improved appearance. Direct applications to “max-type” minimization problems and Pareto optimality are mentioned.