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Convergence analysis of an extended auxiliary problem principle with various stopping criteria. (English)
Optim. Methods Softw. 26, No. 1, 127-154 (2011).
The paper deals with the following problem. Let $K \subset {\mathbb R}^n$ be a closed convex set, ${\mathcal F} : K \rightarrow {\mathbb R}^n$ be a continuous operator with certain monotonicity properties, and ${\mathcal Q} : K \rightarrow 2^{{\mathbb R}^n}$ be a maximal monotone and multi-valued operator. The author studies the variational inequality: find $x^{\ast}\in K$ and $q^{\ast}\in {\mathcal Q}(x^{\ast})$ such that $$\langle {\mathcal F}(x^{\ast}) + q^{\ast}, x - x^{\ast}\rangle \geq 0,\;\;\; \forall x\in K, \tag1$$ where $\langle \cdot, \cdot \rangle$ means the canonical inner product in ${\mathbb R}^n$. An application of the extended proximal auxiliary problem principle scheme with some weakened assumptions on ${\mathcal Q}$ is proposed. The convergence analysis of the inexact method using the stopping criterion of fixed-relative-error type is performed.
Reviewer: Sergei V. Rogosin (Minsk)