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Smoothing Methods for Convex Inequalities and Linear Complementarity Problems
 Mathematical Programming
, 1993
"... A smooth approximation p(x; ff) to the plus function: maxfx; 0g, is obtained by integrating the sigmoid function 1=(1 + e \Gammaffx ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization probl ..."
Abstract

Cited by 62 (6 self)
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A smooth approximation p(x; ff) to the plus function: maxfx; 0g, is obtained by integrating the sigmoid function 1=(1 + e \Gammaffx ), 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 ff sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite ff. Speedup over MINOS 5.4 was as high as 515 times for linear inequalities of size 1000 \Theta 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 400 variables, the proposed approach was as much as 85 times faster than Lemke's method. Key Words: Smo...
An Analysis of Zero Set and Global Error Bound Properties of a Piecewise Affine Function Via Its Recession Function
, 1996
"... For a piecewise affine function f : R n ! R m , the recession function is defined by f 1 (x) := lim !1 f(x) : In this paper, we study the zero set and error bound properties of f via f 1 . We show, for example, that f has a zero when f 1 has a unique zero (at the origin) with a nonvanis ..."
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Cited by 7 (3 self)
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For a piecewise affine function f : R n ! R m , the recession function is defined by f 1 (x) := lim !1 f(x) : In this paper, we study the zero set and error bound properties of f via f 1 . We show, for example, that f has a zero when f 1 has a unique zero (at the origin) with a nonvanishing index. We also characterize the global error bound property of a piecewise affine function in terms of the recession cones of the zero sets of the function and its recession function.
Smoothing Methods in Mathematical Programming
"... sity function. 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 sufficiently small positive value of the smoot ..."
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Cited by 1 (0 self)
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sity function. 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 sufficiently small positive value of the smoothing param eter fl. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite fl. Speedup over the linear/nonlinear programming package MINOS 5.4 was as high as 1142 times for linear inequali ties of size 2000 x 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems(LCPs) were treated by converting them into a system of smooth nonlinear equations and are solved by a quadratically con vergent Newton method. For monotone LCPs with as many as 10,000 variables, the proposed approach was as much as 63 times faster than Lemke's method. Our smooth approach can also be used to solve nonlinear and mixed comple menrarity problems (NCPs and MCPs) by converting them to classes of smooth parametric nonlinear equations. For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equation as well as the NCP or MCP, is established for sufficiently large value of a smoothing param eter c = l. An efficient smooth algorithm, based on the NewtonArmijo approach with an adjusted smoothing parameter, is also given and its global and local quadratic convergence is established. For NCPs, exact solutions of our smooth nonlinear equation for various values of the parameter c, generate an interior path, which is different from the central path for the interior point method. Computational results for 52 test problems compare favorably with those for another Ne...