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A class of smoothing functions for nonlinear and mixed complementarity problems

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Abstract

We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)+=max{0, x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter α. Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter α, generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].

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This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR-9322479.

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Chen, C., Mangasarian, O.L. A class of smoothing functions for nonlinear and mixed complementarity problems. Comput Optim Applic 5, 97–138 (1996). https://doi.org/10.1007/BF00249052

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