A smooth approximation p(x; ) to the plus function: maxfx; 0g, is obtained by integrating the sigmoid function 1=(1 + e ? x ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for su ciently large. In the special case when a Slater constraint quali cation is satis ed, an exact solution can be obtained for nite . Speedup over MINOS 5.4 was as high as 1142 times for linear inequalities of size 2000 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCP's with as many as 10,000 variables, the proposed approach was as much as 63 times faster than Lemke's method.
In §4 we described a particular choice of smoothing the 1 penalty function using integrals of the sigmoid function for neural networks proposed by Chen and Mangasarian REF in the context of handling convex inequalities and linear complementarity problems.
Smoothing methods for convex inequalities and linear complementarity problems
Math. Program.
Mathematics Computer Science