标题： 带有参数优化的微分随机变分不等式 显示英文标题

标题： Differential Stochastic Variational Inequalities with Parametric Optimization

摘要： 带有参数凸优化的微分随机变分不等式（DSVI-O）是一个常微分方程，其右边涉及一个随机变分不等式以及几个动态和随机参数凸优化问题的解。 我们考虑随机变量的分布是时间相关的，并假设涉及的函数是连续的，期望是有定义的。 我们证明DSVI-O具有可积且可测的参数优化问题的弱解。 此外，我们通过使用时间步进近似和样本平均近似提出DSVI-O的离散方案，并证明该离散方案的收敛性。 我们通过使用由多模态大语言模型生成的合成医疗数据，在老年人健康的身体智能系统中的应用来说明DSVI-O的理论结果。 显示英文摘要

摘要： 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.