In this paper, by assuming that a non-Lipschitz penalty function is exact, new conditions for the existence of Lagrange multipliers are established for an inequality and equality-constrained continuously differentiable optimization problem. This is done by virtue of a first-order necessary optimality condition of the penalty problem, which is obtained by estimating Dini upper-directional derivatives of the penalty function in terms of Taylor expansions, and a Farkas lemma. Relations among the obtained results and some well-known constraint qualifications are discussed.
- Dini directional derivative
- Generalized calmness condition
- Lagrange multiplier
- non-Lipschitz penalty function
ASJC Scopus subject areas
- Applied Mathematics
- Management Science and Operations Research