Smoothing approximation for solving convex inequalities

Solving large-scale systems of nonlinear equations/inequalities is a fundamental problem in computing and optimization. In this paper, we propose a smoothing approximate framework for this problem. We first transform the convex inequalities into a mini-max problem which is non-differentiable, and then we show that an approximate solution of the convex inequalities problem can be obtained by a smoothing approximation technique. We approach a Newton-type algorithm to solve the approximation problem. Some properties of the novel approximate function are presented and the global convergence of the algorithm is proved under some mild assumptions. Numerical results are conducted to illustrate the efficiency of the proposed method.