Subgradient-based Neural Network for Nonconvex Optimization Problems in Support Vector Machines with Indefinite Kernels

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 subgradientbased 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 Lojasiewicz 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 Lojasiewicz 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.