A single-projection proximal algorithm for stochastic mixed variational inequalities with applications to breast cancer screening

Mohammad Dilshad; Ibrahim Al-Dayel; Francis O. Nwawuru; Praveen Agarwal; Mohammad Dilshad; Ibrahim Al-Dayel; Francis O. Nwawuru; Praveen Agarwal

doi:10.3934/math.2026434

AIMS Mathematics

2026, Volume 11, Issue 4: 10533-10565. doi: 10.3934/math.2026434

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Research article

A single-projection proximal algorithm for stochastic mixed variational inequalities with applications to breast cancer screening

1.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk-71491, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box-65892, Riyadh 11566, Saudi Arabia
3.
Analysis, Control System and Optimization Research Group (ACoSORG), Department of Mathematics, Faculty of science, Chukwuemeka Odumegwu Ojukwu University, Anambra State, Nigeria
4.
Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India
5.
Nonlinear Dynamics Research Centre (NDRC), Ajman University, Ajman, UAE

Received: 11 January 2026 Revised: 04 March 2026 Accepted: 18 March 2026 Published: 17 April 2026
MSC : 47H25, 54E70, 65K15, 90C33, 90C15

We studied a stochastic mixed variational inequality problem (SMVIP) that encompasses stochastic optimization, stochastic variational inequality problems, and a composite convex minimization problem as special cases. To solve this problem, we proposed a single-projection proximal algorithm (SiPPA) that combined golden ratio dynamics with an adaptive stepsize strategy. In contrast to classical stochastic extragradient and subgradient extragradient methods, the proposed algorithm required only one projection and one averaged stochastic oracle call per iteration, resulting in reduced computational cost. Under mild assumptions on the stochastic oracle and monotonicity of the expected operator, we established almost sure convergence of the generated sequence. Moreover, when the operator was strongly monotone, we proved that the algorithm converges at an $ R- $linear rate. Numerical experiments on benchmark problems and real-world learning tasks on breast cancer screening, illustrate the effectiveness and efficiency of the proposed approach relative to existing stochastic methods.
- projection,
- golden ratio,
- variational inequality,
- stochastic optimization,
- almost sure convergence,
- $ R- $linear convergence
Citation: Mohammad Dilshad, Ibrahim Al-Dayel, Francis O. Nwawuru, Praveen Agarwal. A single-projection proximal algorithm for stochastic mixed variational inequalities with applications to breast cancer screening[J]. AIMS Mathematics, 2026, 11(4): 10533-10565. doi: 10.3934/math.2026434

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Abstract

We studied a stochastic mixed variational inequality problem (SMVIP) that encompasses stochastic optimization, stochastic variational inequality problems, and a composite convex minimization problem as special cases. To solve this problem, we proposed a single-projection proximal algorithm (SiPPA) that combined golden ratio dynamics with an adaptive stepsize strategy. In contrast to classical stochastic extragradient and subgradient extragradient methods, the proposed algorithm required only one projection and one averaged stochastic oracle call per iteration, resulting in reduced computational cost. Under mild assumptions on the stochastic oracle and monotonicity of the expected operator, we established almost sure convergence of the generated sequence. Moreover, when the operator was strongly monotone, we proved that the algorithm converges at an $ R- $linear rate. Numerical experiments on benchmark problems and real-world learning tasks on breast cancer screening, illustrate the effectiveness and efficiency of the proposed approach relative to existing stochastic methods.

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