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<TITLE>Smoothing Methods for Convex Inequalities and Linear Complementarity Problsms </TITLE>
<H2>Smoothing Methods for Convex Inequalities and Linear Complementarity Problsms </H2>
 Chunhui Chen and  Olvi L. Mangasarian<BR>
CS-TR-93-1191<BR>
November 1993
<p>
 A smooth approximation <i>\bold{p(x, \alpha)}</i> to the plus function: <i>\bold{\max \{ x, 0 \}}</i>, is obtained by integrating the sigmoid function <i>\bold{ 1/(1+e^{- \alpha x}) }</i>, 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 <i>\bold{\alpha}</i> sufficien tly large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite <i>\bold{\alpha}</i>. Speed up over MINOS 5.4 was as high as 515 times for linear inequalities of size <i>\bold{ 1000 \times 1000}</i>, and 580 times for convex inequalities with 400 vari ables. 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 400 variables, the proposed approach was as much as 85 times faster than Lemke's method.<P>
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