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<p>Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but broadly applicable problem class from composite optimization in finite dimensions. While prioritizing accessibility over mathematical details, we introduce subgradients of arbitrary functions and the resulting optimality conditions, describe approximations and the need for going beyond pointwise and uniform convergence, and summarize proximal methods. We derive dual problems from parametrization of the actual problem and the resulting relaxations. The paper ends with an introduction to second-order theory and its role in stability analysis of optimization problems.</p>

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Variational Analysis of a Nonconvex and Nonsmooth Optimization Problem: An Introduction

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月15日

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  • Johannes O. Royset

摘要

Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but broadly applicable problem class from composite optimization in finite dimensions. While prioritizing accessibility over mathematical details, we introduce subgradients of arbitrary functions and the resulting optimality conditions, describe approximations and the need for going beyond pointwise and uniform convergence, and summarize proximal methods. We derive dual problems from parametrization of the actual problem and the resulting relaxations. The paper ends with an introduction to second-order theory and its role in stability analysis of optimization problems.