On nonclassical optimality conditions or on regularization of the Lagrange multiplier rule for stable approximation of solutions to convex constrained optimization problems
Optimization Eruditorum, Volume 4, Issue 1, April 2027, Pages 62–87
Mikhail I. Sumin
Department of Mathematics, Derzhavin Tambov State University, 33 Internatsionalnaya St., Tambov 392000, Russian Federation
Abstract
This paper deals with a convex optimization problem with a strongly convex objective functional, an operator equality-constraint (i.e., defined by an operator with an infinite-dimensional image) in a Hilbert space, and a finite number of functional inequality-constraints. In relation to this problem, using the combination of methods of constrained optimization, convex analysis, and the theory of ill-posed problems, we discuss necessary and sufficient nonclassical optimality conditions (NCOCs), which are inextricably linked to the classical Lagrange multiplier rule (LMR). The necessary (sufficient) NCOCs discussed are necessary (sufficient) conditions for ordinary optimality, but they are written in a nonclassical form. Unlike the LMR, the necessary NCOCs are satisfied under only one condition of the solvability of the constrained optimization problem. Simultaneously, NCOCs represent regularizing algorithms for the problem under consideration, providing stable to input data errors approximation of its solution by extremals of a regular Lagrange function while the dual variable is selected in accordance with Tikhonov's regularization procedure. Thus, the NCOCs, particularly in the form of the so-called regularized LMR, overcome the fundamental difficulties associated with the ill-posedness of classical analogs, which shows itself in the possible unsatisfiability (impracticability) and instability of the latter. The paper concludes by showing how a classical LMR, in the case of a convex (possibly not strongly convex) objective functional, can be obtained through a limit transition using NCOC relations in the form of a regularized LMR.
Cite this Article as
Mikhail I. Sumin, On nonclassical optimality conditions or on regularization of the Lagrange multiplier rule for stable approximation of solutions to convex constrained optimization problems, Optimization Eruditorum, 4(1), 62–87, 2027