Come Back to Lagrange. The p-Factor Analysis of Optimality Conditions

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 article addresses the case when the Lagrange multiplier λ0 associated with the objective function might be equal to zero. If the equality constraints are not regular at some point in the sense that the Fréchet derivative of F at is not onto, then the point is a degenerate solution of the classical Lagrange system of optimality conditions ℒ(x, λ0, λ) = 0, where is a solution of the optimization problem and is a corresponding generalized Lagrange multiplier. We derive new conditions that guarantee that is a locally unique solution of the Lagrange system. We also introduce a modified Lagrange system and prove that is its regular locally unique solution. In addition, we propose new conditions that guarantee that the point is an isolated local minimizer of the optimization problem. The modified Lagrange system introduced in this article 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.