Convergence Rate Analysis of a Class of Uncertain Variational Inequalities

Abstract

Variational inequalities theory is a potent tool that can be employed to tackle diverse optimization problems, with applications spanning physics, economics, finance and beyond. Solving the uncertain variational inequalities (UVI) has emerged as a prominent research topic in academia. However, conventional regularization methods for solving UVI are subject to certain limitations on the convergence rate of the algorithm. To overcome this challenge, we opt a class of regularized gap functions which is lower semi-continuous and $\phi $ -bounded true convex properties. We establish an expected value equivalent optimization model and demonstrate that the exponential convergence of the regularized gap function. Furthermore, we investigate the subdifferential of the lower semi-continuous and $\phi $ -bounded true convex functions (LSTCOS) and develop a deviation between the LSTCOS and its closed convex hull. By utilizing their deviations, we transform the convergence rate of the UVI solution to the uniform exponential convergence of LSTCOS. Finally, we prove that the UVI solution uniformly converges at an exponential rate based on the generalized equation.

Keywords

• Uncertain variational inequality
• ɸ-bounded true convex function%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">ɸ</italic>-bounded true convex function
• exponential convergence