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<p>Many optimization problems in machine learning can be studied through the lens of Riemannian nonsmooth optimization, where a nonsmooth objective is originally constrained to lie on a Riemannian manifold. In addition to this geometric view, the multiobjective approach has expansive applications in such problems, as one may aim to optimize more than one objective simultaneously, which makes evaluating trade-offs between objectives essential. In this paper, we study a retraction-based trust-region algorithm for solving nonsmooth multiobjective optimization problems defined on Riemannian manifolds. Considering general Riemannian manifolds, we analyze the convergence behavior of the proposed method. This general approach enables one to apply the proposed algorithm to a wider variety of cases, from unconstrained problems on Euclidean spaces to problems with manifold constraints. Furthermore, we study the application of the method in matrix machine learning problems such as fair sparse PCA problems, and multiobjective <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-regularized least squares problems. We demonstrate the efficiency of the algorithm by reporting the numerical experiments on the mentioned applications.</p>

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A nonsmooth Riemannian trust-region method for multiobjective optimization with applications in machine learning problems

  • 期刊论文

  • 发表日期：

    2026年8月1日

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  • Nima Eslami,
  • Amir Ardestani-Jaafari,
  • Javad Tavakoli

摘要

Many optimization problems in machine learning can be studied through the lens of Riemannian nonsmooth optimization, where a nonsmooth objective is originally constrained to lie on a Riemannian manifold. In addition to this geometric view, the multiobjective approach has expansive applications in such problems, as one may aim to optimize more than one objective simultaneously, which makes evaluating trade-offs between objectives essential. In this paper, we study a retraction-based trust-region algorithm for solving nonsmooth multiobjective optimization problems defined on Riemannian manifolds. Considering general Riemannian manifolds, we analyze the convergence behavior of the proposed method. This general approach enables one to apply the proposed algorithm to a wider variety of cases, from unconstrained problems on Euclidean spaces to problems with manifold constraints. Furthermore, we study the application of the method in matrix machine learning problems such as fair sparse PCA problems, and multiobjective \(\ell _1\) ℓ 1 -regularized least squares problems. We demonstrate the efficiency of the algorithm by reporting the numerical experiments on the mentioned applications.