Maximal monotonicity and convex programming
- Tiba, Dan
2010 Mathematics Subject Classification
We introduce an explicit constraint qualification condition which is necessary and sufficient for the nondegenerate Lagrange multipliers rule to hold. We compare it with metric regularity conditions and we show that it is strictly weaker than the Slater assumption. Under certain weak smoothness hypotheses, our condition, the Slater condition and the existence of nondegenerate Lagrange multipliers are equivalent. The basic ingredient in the proof of the main result is the theory of maximal monotone operators (Minty's theorem). Another approach using a direct exact penalization argument yields a modified nondegenerate Lagrange multipliers rule involving the positive part of the constraint mapping. Examples and applications to abstract optimal control problems are also indicated.