On the regularization method for conflicting problems of convex programming
We consider the problem \[ \min\{f_0(x): x\in X\},\tag{1} \] where \(X= \{x: f(x)\leq 0\}\), \(f(x)= [f_1(x),\dots, f_m(x)]^T\), and the \(f_i(x)\) are convex differentiable functions defined on \(\mathbb{R}^n\) \((i=0,1,\dots, m)\). For this problem, we construct a Lagrange function regularized with respect to the direct and dual variables \[ L_\sigma(x,\lambda)= f_0(x)+ (\lambda,f(x))+ \alpha|x|^2- \beta|\lambda|^2, \] where \([x,\lambda]\in \mathbb{R}^n\times \mathbb{R}_+^m\), \(\sigma= [\alpha,\beta]>0\), \(|\cdot|\) stands for the Euclidean norm in a corresponding space. The function \(L_\sigma (x,\lambda)\) for any \(\sigma>0\) has a unique saddle point \([x^\sigma, \lambda^\sigma]\) in the domain \(\mathbb{R}^n\times \mathbb{R}_+^m\). This advantageous property distinguishes our case from the customary Lagrange function \(L_0(x,\lambda)= f_0(x)+ (\lambda,f(x))\) for problem (1). For instance, the latter function has no saddle points if the system of constraints \(f_i(x)\leq 0\), \(i=1,\dots, m\), is an inconsistent one. On the other hand, the problem of convex programming (CP) with conflicting constraints is an example of an improper optimization problem, which often occurs in practice of mathematical simulation. Therefore, the question of construction of effective schemes for correction of such problems seems to be actual. In particular, we consider this correction in terms of the function \(L_\sigma(x,\lambda)\). In earlier articles we had previously established estimates, which characterize convergence of the points \(x^\sigma\) and \(\lambda^\sigma\) for a resolvable regular problem of CP. The connection between \(\lambda\) behavior of \([x^\sigma, \lambda^\sigma]\) and the solution of certain approximate problem for (1) was studied for some classes of improper CP problems in recent papers of the author. On the whole, these articles deal with an analysis of convergence of the \(x\)-components of the saddle points as \(\sigma\to 0\), and the question of the behavior of \(\lambda^\sigma\) under variation of the parameter \(\sigma\) thus remains open. The present article is dealing mainly with the properties of the sequence of dual variables \(\lambda^\sigma\).
scientific article; zbMATH DE number 1122695