Minimum-support solutions of polyhedral concave programs *

Motivated by the successful application of mathematical programming techniques to difficult machine learning problems, we seek solutions of concave minimization problems over polyhedral sets with minimum number of nonzero components. We that if such problems have a solution, they have a vertex solution with a minimal number of zeros. This includes linear programs and general linear complementarity problems. A smooth concave exponential approximation to a step function solves the minimumsupport problem exactly for a finite value of the smoothing parameter. A fast finite linear-programming-based iterative method terminates at a stationary point, which for many important real world problems provides very useful answers. Utilizing the complementarity property of linear programs and linear complementarity problems, an upper bound on the number of nonzeros can be obtained by solving a single convex minimization problem on a polyhedral set

[1]  Jon W. Tolle,et al.  Exact penalty functions in nonlinear programming , 1973, Math. Program..

[2]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[3]  O. Mangasarian.  Characterization of linear complementarity problems as linear programs , 1978 .

[4]  O. Mangasarian.  Simple computable bounds for solutions of linear complementarity problems and linear programs , 1985 .

[5]  O. Mangasarian.  Some Applications of Penalty Functions in Mathematical Programming. , 1986 .

[6]  O. Mangasarian.  A simple characterization of solution sets of convex programs , 1988 .

[7]  S. J. Chung.  NP-Completeness of the linear complementarity problem , 1989 .

[8]  David H. Wolpert,et al.  On the Connection between In-sample Testing and Generalization Error , 1992, Complex Syst..

[9]  MinimizationO. L. Mangasarian.  Misclassiication Minimization , 1994 .

[10]  David H. Wolpert,et al.  The Mathematics of Generalization: The Proceedings of the SFI/CNLS Workshop on Formal Approaches to Supervised Learning , 1994 .

[11]  Vladimir Vapnik,et al.  The Nature of Statistical Learning , 1995 .

[12]  Christopher J. C. Burges,et al.  Simplified Support Vector Decision Rules , 1996, ICML.

[13]  Olvi L. Mangasarian,et al.  Machine Learning via Polyhedral Concave Minimization , 1996 .

[14]  Olvi L. Mangasarian,et al.  Exact Penalty Functions for Mathematical Programs with Linear Complementarity Constraints , 1996 .

[15]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[16]  Alexander J. Smola,et al.  Support Vector Regression Machines , 1996, NIPS.

[17]  J. Pang,et al.  Exact penalty for mathematical programs with linear complementarity constraints , 1997 .

[18]  O. L. Mangasariany.  Solution of General Linear Complementarity Problems via Nondiierentiable Concave Minimization Dedicated to Professor Hoang Tuy on the Occasion of His Seventieth Birthday , 1997 .

[19]  Paul S. Bradley,et al.  Parsimonious Least Norm Approximation , 1998, Comput. Optim. Appl..

[20]  Paul S. Bradley,et al.  Feature Selection via Mathematical Programming , 1997, INFORMS J. Comput..

[21]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.