<p>In recent years, the Peaceman-Rachford splitting method (PRSM) has garnered significant attention due to the various demands of machine learning and big data relevant optimization. This paper focuses on solving a family of separable convex optimization problems with linear equality constraints, where the objective function is the sum of a convex but possibly nonsmooth function and an average of many smooth convex component functions. To handle this kind of problems, we develop an inexact accelerated stochastic PRSM with convex combination proximal centers (IAS-PRSM-ccpc). The involved smooth subproblem in IAS-PRSM-ccpc is addressed by using a linearization technique and an accelerated stochastic gradient method that incorporates the variance reduction technique, while the nonsmooth subproblem is solved inexactly under a relative error criterion to avoid the potential unavailability of the proximal operator. In addition, the convex combination technique is introduced into the proximal center of each subproblem simultaneously. Moreover, we extend the range of the convex combination parameters from [0.618, 1) to [0, 1), while still guaranteeing convergence. By an appropriate choice for the sample number of stochastic iterations in the inner loop, we prove the ergodic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {O}(((1-\textrm{sgn}(|1-\varrho |))\log K +1)/K^{\min \{1, \varrho \}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mtext>sgn</mtext> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mn>1</mn> <mo>-</mo> <mi>ϱ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>log</mo> <mi>K</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> </mrow> <msup> <mi>K</mi> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>ϱ</mi> <mo stretchy="false">}</mo> </mrow> </msup> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> sublinear convergence rates of IAS-PRSM-ccpc in the sense of expectation measured by the function value residual and constraints violation, where <i>K</i> represents the number of outer iterations and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varrho >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϱ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is used to determine the sample number of stochastic iterations in the inner loop. Under the strong convexity, we further establish the convergence of ergodic iterates in the sense of expectation. Finally, numerical experiments demonstrate that the proposed method is effective for solving separable convex optimization problems.</p>

An Inexact Accelerated Stochastic PRSM with Convex Combination Proximal Centers for Separable Convex Optimization

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发表日期：

2026年2月9日

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Jinbao Jian,
Xianke Tang,
Jianghua Yin,
Wenrui Chen,
Guodong Ma

摘要

In recent years, the Peaceman-Rachford splitting method (PRSM) has garnered significant attention due to the various demands of machine learning and big data relevant optimization. This paper focuses on solving a family of separable convex optimization problems with linear equality constraints, where the objective function is the sum of a convex but possibly nonsmooth function and an average of many smooth convex component functions. To handle this kind of problems, we develop an inexact accelerated stochastic PRSM with convex combination proximal centers (IAS-PRSM-ccpc). The involved smooth subproblem in IAS-PRSM-ccpc is addressed by using a linearization technique and an accelerated stochastic gradient method that incorporates the variance reduction technique, while the nonsmooth subproblem is solved inexactly under a relative error criterion to avoid the potential unavailability of the proximal operator. In addition, the convex combination technique is introduced into the proximal center of each subproblem simultaneously. Moreover, we extend the range of the convex combination parameters from [0.618, 1) to [0, 1), while still guaranteeing convergence. By an appropriate choice for the sample number of stochastic iterations in the inner loop, we prove the ergodic $\mathcal {O}(((1-\textrm{sgn}(|1-\varrho |))\log K +1)/K^{\min \{1, \varrho \}})$ $O (((1 - sgn (| 1 - ϱ |)) log K + 1) / K^{min {1, ϱ}})$ sublinear convergence rates of IAS-PRSM-ccpc in the sense of expectation measured by the function value residual and constraints violation, where K represents the number of outer iterations and $\varrho >0$ $ϱ > 0$ is used to determine the sample number of stochastic iterations in the inner loop. Under the strong convexity, we further establish the convergence of ergodic iterates in the sense of expectation. Finally, numerical experiments demonstrate that the proposed method is effective for solving separable convex optimization problems.