Differential Stochastic Variational Inequalities with Parametric Optimization

TL;DR

This paper introduces a new framework called DSVI-O, which models complex stochastic systems involving variational inequalities and parametric optimization, and provides convergence analysis and practical applications.

Contribution

It develops a weak solution theory for DSVI-O, proposes a convergent discrete approximation scheme, and demonstrates applications in elderly health care using synthetic data.

Findings

Proved existence of weak solutions for DSVI-O.

Established convergence of the discrete scheme.

Applied the framework to elderly health care with synthetic data.

Abstract

The differential stochastic variational inequality with parametric convex optimization (DSVI-O) is an ordinary differential equation whose right-hand side involves a stochastic variational inequality and solutions of several dynamic and random parametric convex optimization problems. We consider that the distribution of the random variable is time-dependent and assume that the involved functions are continuous and the expectation is well-defined. We show that the DSVI-O has a weak solution with integrable and measurable solutions of the parametric optimization problems. Moreover, we propose a discrete scheme of DSVI-O by using a time-stepping approximation and the sample average approximation and prove the convergence of the discrete scheme. We illustrate our theoretical results of DSVI-O with applications in an embodied intelligence system for the elderly health by synthetic health care data generated by Multimodal Large Language Models.