One-Dimensional Analysis of Exponential Convergence Condition for Dual Neural Network

Abstract

In view of its fundamental role arising in numerous fields of science and engineering, the problem of online solving quadratic programs (QP) has been investigated extensively for the past decades. One of the state-of-the-art recurrent neural network (RNN) solvers is dual neural network (DNN). The dual neural network is of simple piecewise-linear dynamics and has global convergence to optimal solutions. Its exponential-convergence property relies on a so-called exponential convergence condition. Such a condition often exists in practice but seems difficult to be proved. In this paper, we investigate the proof complexity of such a condition by analyzing its one-dimensional case. The analysis shows that in general the exponential convergence condition often exists for dual neural networks, and always exists at least for the one-dimensional case. In addition, the analysis is very complex.KeywordsQuadratic programmingRedundant systemsDual neural networkOnline solutionExponential convergenceProof complexity