The Perturbation Method and Regularization of the Lagrange Multiplier Rule in Convex Constrained Optimization Problems

Pub Date : 2024-08-20 DOI:10.1134/s0081543824030155
M. I. Sumin
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Abstract

We consider a regularization of the Lagrange multiplier rule (LMR) in the nondifferential form in a convex constrained optimization problem with an operator equality constraint in a Hilbert space and a finite number of functional inequality constraints. The objective functional of the problem is assumed to be strongly convex, and the convex closed set of its admissible elements also belongs to a Hilbert space. The constraints of the problem contain additively included parameters, which enables using the so-called perturbation method to study it. The main purpose of the regularized LMR is the stable generation of generalized minimizing sequences (GMSs), which approximate the exact solution of the problem using extremals of the regular Lagrange functional. The regularized LMR itself can be interpreted as a GMS-generating (regularizing) operator, which assigns to each set of input data of the constrained optimization problem the extremal of its corresponding regular Lagrange functional, in which the dual variable is generated following one or another procedure for stabilizing the dual problem. The main attention is paid to (1) studying the connection between the dual regularization procedure and the subdifferential properties of the value function of the original problem; (2) proving the convergence of this procedure in the case of solvability of the dual problem; (3) an appropriate update of the regularized LMR; (4) obtaining the classical LMR as a limiting version of its regularized analog.

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凸约束优化问题中的扰动法和拉格朗日乘数规则的正规化
我们考虑在一个具有希尔伯特空间中的算子相等约束和有限数量的函数不等式约束的凸约束优化问题中,以非微分形式对拉格朗日乘法法则(LMR)进行正则化。假设问题的目标函数为强凸函数,其可接受元素的凸闭集也属于希尔伯特空间。问题的约束条件包含可加参数,因此可以使用所谓的扰动法进行研究。正则化 LMR 的主要目的是稳定生成广义最小化序列 (GMS),利用正则拉格朗日函数的极值逼近问题的精确解。正则化 LMR 本身可以解释为一个 GMS 生成(正则化)算子,它为约束优化问题的每一组输入数据分配相应正则拉格朗日函数的极值,其中对偶变量是按照稳定对偶问题的一种或另一种程序生成的。主要关注点在于:(1) 研究对偶正则化程序与原始问题值函数的次微分性质之间的联系;(2) 在对偶问题可解的情况下证明该程序的收敛性;(3) 正则化拉格朗日函数的适当更新;(4) 获得经典拉格朗日函数作为其正则化相似函数的极限版本。
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