Stable sequential Kuhn-Tucker theorem in iterative form or a regularized Uzawa algorithm in a regular nonlinear programming problem

Abstract

A parametric nonlinear programming problem in a metric space with an operator equality constraint in a Hilbert space is studied assuming that its lower semicontinuous value function at a cho sen individual parameter value has certain subdifferentiability properties in the sense of nonlinear (nonsmooth) analysis. Such subdifferentiability can be understood as the existence of a proximal sub gradient or a Fréchet subdifferential. In other words, an individual problem has a corresponding gen eralized Kuhn-Tucker vector. Under this assumption, a stable sequential Kuhn-Tucker theorem in nondifferential iterative form is proved and discussed in terms of minimizing sequences on the basis of the dual regularization method. This theorem provides necessary and sufficient conditions for the sta ble construction of a minimizing approximate solution in the sense of Warga in the considered prob lem, whose initial data can be approximately specified. A substantial difference of the proved theorem from its classical same named analogue is that the former takes into account the possible instability of the problem in the case of perturbed initial data and, as a consequence, allows for the inherited insta bility of classical optimality conditions. This theorem can be treated as a regularized generalization of the classical Uzawa algorithm to nonlinear programming problems. Finally, the theorem is applied to the "simplest" nonlinear optimal control problem, namely, to a time optimal control problem. SUMIN Example 0.1. Consider the minimization of a strongly convex quadratic function of two variables on a set defined by an affine equality constraint:The exact solution of the problem is x* = (1/2, 1/2). Since problem (0.1) is rather simple, it can be directly shown that λ α = (λ 1, α , λ 2, α ) = (-1, α) ∀α ∈ ‫ޒ‬ 1 is its Kuhn-Tucker vector. Therefore, by the classical Kuhn-Tucker theorem (see, e.g., [1, 2]), any pair (х*, λ α ) of the indicated form, but no other one, consists of solutions to the original problem (0.1) and to its dual. Consider the following perturbation of problem (0.1) with δ > 0:Obviously, its unique feasible point x δ = ( , ) = (1 -δ, δ) is its solution. It was shown in the analysis of this example in [3, Example 0.1] that this perturbed problem has the (unique) Kuhn-Tucker vector λ δ = ( , ) = . In other words, by the classical Kuhn-Tucker theorem, (х δ , λ δ ) is a unique pair consisting of solutions to the perturbed problem and its dual. Summarizing, on the one (for mal) hand, the point x δ is a candidate for an approximation to the solution of the original problem (0.1).On the other hand, it does not converge to its unique exact solution х* = (1/2, 1/2) as δ 0. Moreover, the δ dependent quantities in the problem have a jump discontinuity at δ = 0. Note that, since the two dimensional system specifying the equality constraint in problem (0.1) is degenerate, in any of its "neigh borhoods," there obviously exist perturbed problems with an empty feasible set and, hence, with +∞ value.Naturally, there are "absolut...