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Differential Stochastic Variational Inequalities and Optimization The variational inequality (VI) provides a broad unifying setting for the study of equilibrium problems and optimization. The differential variational inequality and stochastic variational inequality are generalizations of the VI and promising modeling paradigms when dynamics and uncertainties are involved in data analysis and decision making. We first discuss the mathematical formulations and recent results of the differential VI and stochastic VI. Next we introduce the differential stochastic VI with stochastic optimization (DSVIO), an ordinary differential equation whose right-hand side is defined by a stochastic VI and solutions of several stochastic optimization problems. The DSVIO provides a powerful modeling framework for various applications in data science and artificial intelligence. We provide sufficient conditions for the existence of a solution pair for the DSVIO in the space of absolutely continuous functions and the space of measurable functions. We discuss a convergent sample average approximation and time-stepping discretization scheme for numerical solutions of the DSVIO. Finally, we show applications of the DSVIO for optimizing multimodal large model-based embodied intelligence systems for the elderly mobility.Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices
