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<p>This paper presents a primal-dual prediction-correction (PD-PC) method for solving linearly constrained time-varying convex optimization problems, which frequently arise in control, signal processing, and online learning applications. The proposed method establishes a novel integration of primal-dual gradient dynamics with a discrete-time prediction-correction structure, specifically designed for problems with time-dependent linear constraints. A tunable memory parameter is introduced in the prediction phase to perform linear extrapolation using past iterates, enabling a flexible trade-off between the amount of historical information stored and the computational cost of correction. In the correction phase, primal and dual variables are updated via gradient descent-ascent iterations, thus maintaining the computational efficiency of a first-order method without requiring Hessian or high-order derivative computations. Theoretical analysis shows that the method achieves <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\cal O}({h}^{2})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">O</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi>h</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> asymptotic tracking accuracy for both primal and dual variables, matching the state-of-the-art performance among first-order methods even in unconstrained settings. Numerical experiments on problems with both time-invariant and time-varying constraints validate the theoretical findings and demonstrate the method’s effectiveness.</p>

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Primal-Dual Prediction-Correction Method with Tunable Memory for Linearly Constrained Time-Varying Convex Optimization

  • 期刊论文

  • 发表日期：

    2026年4月28日

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  • Zhonghao Lin,
  • Xianlin Zeng,
  • Jie Hou,
  • Jian Sun,
  • Jie Chen

摘要

This paper presents a primal-dual prediction-correction (PD-PC) method for solving linearly constrained time-varying convex optimization problems, which frequently arise in control, signal processing, and online learning applications. The proposed method establishes a novel integration of primal-dual gradient dynamics with a discrete-time prediction-correction structure, specifically designed for problems with time-dependent linear constraints. A tunable memory parameter is introduced in the prediction phase to perform linear extrapolation using past iterates, enabling a flexible trade-off between the amount of historical information stored and the computational cost of correction. In the correction phase, primal and dual variables are updated via gradient descent-ascent iterations, thus maintaining the computational efficiency of a first-order method without requiring Hessian or high-order derivative computations. Theoretical analysis shows that the method achieves \({\cal O}({h}^{2})\) O ( h 2 ) asymptotic tracking accuracy for both primal and dual variables, matching the state-of-the-art performance among first-order methods even in unconstrained settings. Numerical experiments on problems with both time-invariant and time-varying constraints validate the theoretical findings and demonstrate the method’s effectiveness.