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<p>In this paper, we address two main topics. First, we study the problem of minimizing the sum of a smooth function and the composition of a weakly convex function with a linear operator on a closed vector subspace. For this problem, we propose a projected variable smoothing algorithm and establish a complexity bound of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}(\epsilon ^{-3})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>ϵ</mi> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to achieve an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-approximate solution. Second, we investigate the Moreau envelope and the proximity operator of functions defined as the supremum of weakly convex functions, and we compute the proximity operator in two important cases. In addition, we apply the proposed algorithm for solving a distributionally robust optimization problem, the LASSO with linear constraints, and the max dispersion problem. We illustrate numerical results for the max dispersion problem.</p>

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A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions

  • 期刊论文

  • 发表日期：

    2026年2月20日

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  • Sergio López-Rivera,
  • Pedro Pérez-Aros,
  • Emilio Vilches

摘要

In this paper, we address two main topics. First, we study the problem of minimizing the sum of a smooth function and the composition of a weakly convex function with a linear operator on a closed vector subspace. For this problem, we propose a projected variable smoothing algorithm and establish a complexity bound of \(\mathcal {O}(\epsilon ^{-3})\) O ( ϵ - 3 ) to achieve an \(\epsilon \) ϵ -approximate solution. Second, we investigate the Moreau envelope and the proximity operator of functions defined as the supremum of weakly convex functions, and we compute the proximity operator in two important cases. In addition, we apply the proposed algorithm for solving a distributionally robust optimization problem, the LASSO with linear constraints, and the max dispersion problem. We illustrate numerical results for the max dispersion problem.