<p>In this paper, we investigate a class of two-stage stochastic mixed variational inequalities (SMVI), which can be viewed as a generalization of the two-stage stochastic variational inequality. First of all, we present some key properties of the model under strong monotonicity, strict monotonicity and co-coercivity. After that, we propose an inexact stochastic forward-backward (ISFB) algorithm to solve the two-stage SMVI. For the strong monotonicity case, we verify that it achieves the convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(q^K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mi>K</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <i>K</i> being the iteration number. For the strict monotonicity case, we show that the iteration sequence converges almost surely. For the co-coercivity case, under which the second stage problem may have multiple solutions, we give a quantitative relationship between the natural residual solution and the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-strong solution. It helps to avoid the accuracy loss, and then to establish the convergence rate of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$O(1/\sqrt{K})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msqrt> <mi>K</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. It should be highlighted that our approach does not need variance reduction techniques, which might be computationally expensive. Finally, numerical experiments demonstrate the effectiveness and the efficiency of the ISFB algorithm.</p>

Inexact Stochastic Forward-Backward Method for Two-stage Stochastic Mixed Variational Inequalities

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发表日期：

2026年7月1日

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Bin Zhou,
Xiaoying Zhu,
Hailin Sun,
Jie Jiang

摘要

In this paper, we investigate a class of two-stage stochastic mixed variational inequalities (SMVI), which can be viewed as a generalization of the two-stage stochastic variational inequality. First of all, we present some key properties of the model under strong monotonicity, strict monotonicity and co-coercivity. After that, we propose an inexact stochastic forward-backward (ISFB) algorithm to solve the two-stage SMVI. For the strong monotonicity case, we verify that it achieves the convergence rate of $O(q^K)$ $O (q^{K})$ for some $q\in (0,1)$ $q \in (0, 1)$ and K being the iteration number. For the strict monotonicity case, we show that the iteration sequence converges almost surely. For the co-coercivity case, under which the second stage problem may have multiple solutions, we give a quantitative relationship between the natural residual solution and the $\epsilon $ $ϵ$ -strong solution. It helps to avoid the accuracy loss, and then to establish the convergence rate of $O(1/\sqrt{K})$ $O (1 / \sqrt{K})$ . It should be highlighted that our approach does not need variance reduction techniques, which might be computationally expensive. Finally, numerical experiments demonstrate the effectiveness and the efficiency of the ISFB algorithm.