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<p>We propose a new algorithm for minimizing a nonsmooth convex function <i>f</i>, applying one step of the fast gradient algorithm by Nesterov to the Moreau-Yosida regularization of the so-called Elzinga-Moore-Ouorou function, as first termed by Antonio Frangioni. In contrast with our previous fast proximal algorithms for nonsmooth convex optimization based on the Moreau-Yosida regularization of <i>f</i>, the proposed algorithm uses Chebychev centers as trial solutions, and has a descent property. Moreover, the generated approximate solutions are shown to converge to an optimal solution. The complexity bound has the accelerated convergence rate of the fast gradient algorithm, plus an error term that depends on the accuracy achieved at all subproblems solved in past iterations. From the theoretical development, we derive some practical choices for the proximity parameter. For numerical illustration, we consider the Held and Karp dual of a traveling salesman problem with datasets from TSPLIB, and some small-size academic test problems. The numerical experiments, although limited, indicate the relevance of the approach on large-scale problems in comparison with our previous proximal algorithm which is also based on Chebychev centers.</p>

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Fast proximal algorithms based on Chebychev centers for nonsmooth optimization

  • 期刊论文

  • 发表日期：

    2026年1月8日

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  • Adam Ouorou

摘要

We propose a new algorithm for minimizing a nonsmooth convex function f, applying one step of the fast gradient algorithm by Nesterov to the Moreau-Yosida regularization of the so-called Elzinga-Moore-Ouorou function, as first termed by Antonio Frangioni. In contrast with our previous fast proximal algorithms for nonsmooth convex optimization based on the Moreau-Yosida regularization of f, the proposed algorithm uses Chebychev centers as trial solutions, and has a descent property. Moreover, the generated approximate solutions are shown to converge to an optimal solution. The complexity bound has the accelerated convergence rate of the fast gradient algorithm, plus an error term that depends on the accuracy achieved at all subproblems solved in past iterations. From the theoretical development, we derive some practical choices for the proximity parameter. For numerical illustration, we consider the Held and Karp dual of a traveling salesman problem with datasets from TSPLIB, and some small-size academic test problems. The numerical experiments, although limited, indicate the relevance of the approach on large-scale problems in comparison with our previous proximal algorithm which is also based on Chebychev centers.