Degenerate Equality Constrained Optimization Problems and P-Regularity Theory

Abstract

AbstractWe 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.KeywordsEquality constrained optimization problemsDegeneracyGeneralized necessary conditions