Subgradient-based  neural network for nonconvex optimization problems in support vector machines with indefinite kernels

Fengqiu Liu; Xiaoping Xue

doi:10.3934/jimo.2016.12.285

Article Contents

2016, Volume 12, Issue 1: 285-301. Doi: 10.3934/jimo.2016.12.285

This issue Previous Article Quadratic optimization over a polyhedral cone Next Article Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback

Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels

Fengqiu Liu^1, and
Xiaoping Xue^2,

1.
Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080
2.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001

Received: May 31, 2012

Revised: December 31, 2014

Published: December 2015

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Abstract

Support vector machines (SVMs) with positive semidefinite kernels yield convex quadratic programming problems. SVMs with indefinite kernels yield nonconvex quadratic programming problems. Most optimization methods for SVMs rely on the convexity of objective functions and are not efficient for solving such nonconvex problems. In this paper, we propose a subgradient-based neural network (SGNN) for the problems cast by SVMs with indefinite kernels. It is shown that the state of the proposed neural network has finite length, and as a consequence it converges toward a singleton. The coincidence between the solution and the slow solution of SGNN is also proved starting from the initial value of SGNN. Moreover, we employ the Łojasiewicz inequality to exploit the convergence rate of trajectory of SGNN. The obtained results show that each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points through a quantitative evaluation of the Łojasiewicz exponent at the equilibrium points. This method is easy to implement without adding any new parameters. Three benchmark data sets from the University of California, Irvine machine learning repository are used in the numerical tests. Experimental results show the efficiency of the proposed neural network.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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