A global linear and local superlinear/quadratic inexact non-interior continuation method for variational inequalities

We use the concept of barrier-based smoothing approximations introduced in [ C. B. Chua and Z. Li, A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones, SIOPT 23(2), 2010] to extend the non-interior continuation method proposed in [B. Chen and N. Xiu, A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIOPT 9(3), 1999] to an inexact non-interior continuation method for variational inequalities over general closed convex sets. Newton equations involved in the method are solved inexactly to deal with high dimension problems. The method is proved to have global linear and local superlinear/quadratic convergence under suitable assumptions. We apply the method to non-negative orthants, positive semidefinite cones, polyhedral sets, epigraphs of matrix operator norm cone and epigraphs of matrix nuclear norm cone

Approximate Program Smoothing Using Mean-Variance Statistics, with Application to Procedural Shader Bandlimiting

This paper introduces a general method to approximate the convolution of an arbitrary program with a Gaussian kernel. This process has the effect of smoothing out a program. Our compiler framework models intermediate values in the program as random variables, by using mean and variance statistics. Our approach breaks the input program into parts and relates the statistics of the different parts, under the smoothing process. We give several approximations that can be used for the different parts of the program. These include the approximation of Dorn et al., a novel adaptive Gaussian approximation, Monte Carlo sampling, and compactly supported kernels. Our adaptive Gaussian approximation is accurate up to the second order in the standard deviation of the smoothing kernel, and mathematically smooth. We show how to construct a compiler that applies chosen approximations to given parts of the input program. Because each expression can have multiple approximation choices, we use a genetic search to automatically select the best approximations. We apply this framework to the problem of automatically bandlimiting procedural shader programs. We evaluate our method on a variety of complex shaders, including shaders with parallax mapping, animation, and spatially varying statistics. The resulting smoothed shader programs outperform previous approaches both numerically, and aesthetically, due to the smoothing properties of our approximations.Comment: 13 pages, 6 figure