Deep solution operators for variational inequalities via proximal neural networks

Abstract

Following Bauschke and Combettes (Convex analysis and monotone operator theory in Hilbert spaces, Springer, Cham, 2017), we introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in real, separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g., in valuation of American-style contracts in Black–Scholes financial market models. In the finite-dimensional setting, the VIs reduce to matrix VIs in Euclidean space, and ProxNets emulate classical projected matrix iterations, such as projected Jacobi and projected SOR methods.