Degenerate Equality Constrained Optimization Problems and P-Regularity Theory

Abstract

We consider necessary optimality conditions for optimization problems with equality constraints given in the operator form as $F(x)=0$, where F is an operator between Banach spaces. The paper addresses the case when the Lagrange multiplier $\lambda _0$ associated with the objective function might be equal to zero. If the equality constraints are not regular at some point $x^*$ in the sense that the Fréchet derivative of F at $x^*$ is not onto, then the point $z^*=(x^*, \lambda ^*_0, \lambda ^*)$ is a degenerate solution of the classical Lagrange system of optimality conditions ${\mathcal {L}}(x, \lambda _0, \lambda )=0$, where $x^*$ is a solution of the optimization problem and $(\lambda ^*_0, \lambda ^*)$ is a corresponding generalized Lagrange multiplier. We derive new conditions that guarantee that $z^*$ is a locally unique solution of the Lagrange system. We also introduce a modified Lagrange system and prove that $z^*$ is its regular locally unique solution. The modified Lagrange system introduced in the paper can be used as a basis for constructing numerical methods for solving degenerate optimization problems. Our results are based on the construction of p–regularity and are illustrated by examples.

This work was supported in part by the Russian Foundation for Basic Research, project No. 21-71-30005.