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<p>In this paper, we introduce an adaptive primal-dual algorithm (PDAc-A) for solving the structured convex-concave saddle point problems with a generic smooth non-bilinear coupling term. The proposed method incorporates a convex combination and an adaptive step size selection, eliminating the need for additional computations, such as linesearch. We establish the global pointwise convergence and an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}(1/N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> ergodic sublinear convergence rate for the proposed algorithm, where <i>N</i> is the iteration counter. Furthermore, we extend PDAc-A to solve non-linear compositional convex optimization problems. We also develop an accelerated algorithm (aPDAc-A) for the strongly convex case, which achieves an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}(1/N^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> ergodic convergence rate. Numerical results on quadratically constrained quadratic programming problems demonstrate the superiority of PDAc-A and aPDAc-A over existing methods.</p>

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Adaptive Primal-Dual Algorithms for Generic Saddle Point Problems

  • 期刊论文

  • 发表日期：

    2026年5月30日

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  • Siyuan Huang,
  • Cuijie Zhang,
  • Qiao-Li Dong,
  • D. R. Sahu

摘要

In this paper, we introduce an adaptive primal-dual algorithm (PDAc-A) for solving the structured convex-concave saddle point problems with a generic smooth non-bilinear coupling term. The proposed method incorporates a convex combination and an adaptive step size selection, eliminating the need for additional computations, such as linesearch. We establish the global pointwise convergence and an \(\mathcal {O}(1/N)\) O ( 1 / N ) ergodic sublinear convergence rate for the proposed algorithm, where N is the iteration counter. Furthermore, we extend PDAc-A to solve non-linear compositional convex optimization problems. We also develop an accelerated algorithm (aPDAc-A) for the strongly convex case, which achieves an \(\mathcal {O}(1/N^2)\) O ( 1 / N 2 ) ergodic convergence rate. Numerical results on quadratically constrained quadratic programming problems demonstrate the superiority of PDAc-A and aPDAc-A over existing methods.