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Stability analysis of gradient-based neural networks for optimization problems

Stability analysis of gradient-based neural networks for optimization problems
Stability analysis of gradient-based neural networks for optimization problems
The paper introduces a new approach to analyze the stability of neural network models without using any Lyapunov function. With the new approach, we investigate the stability properties of the general gradient-based neural network model for optimization problems. Our discussion includes both isolated equilibrium points and connected equilibrium sets which could be unbounded. For a general optimization problem, if the objective function is bounded below and its gradient is Lipschitz continuous, we prove that (a) any trajectory of the gradient-based neural network converges to an equilibrium point, and (b) the Lyapunov stability is equivalent to the asymptotical stability in the gradient-based neural networks. For a convex optimization problem, under the same assumptions, we show that any trajectory of gradient-based neural networks will converge to an asymptotically stable equilibrium point of the neural networks. For a general nonlinear objective function, we propose a refined gradient-based neural network, whose trajectory with any arbitrary initial point will converge to an equilibrium point, which satisfies the second order necessary optimality conditions for optimization problems. Promising simulation results of a refined gradient-based neural network on some problems are also reported.
gradient-based neural network, equilibrium point, equilibrium set, asymptotic stability, exponential stability
0925-5001
363-381
Han, Qiaoming
a795292c-fd69-4f98-8f3c-b18a2d0bd677
Liao, Li-Zhi
c79b29f2-ea1e-4e0e-badc-756956646ee1
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Qi, Liqun
69936be7-f1aa-4c1f-b403-5bd5f3ba7d4c
Han, Qiaoming
a795292c-fd69-4f98-8f3c-b18a2d0bd677
Liao, Li-Zhi
c79b29f2-ea1e-4e0e-badc-756956646ee1
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Qi, Liqun
69936be7-f1aa-4c1f-b403-5bd5f3ba7d4c

Han, Qiaoming, Liao, Li-Zhi, Qi, Houduo and Qi, Liqun (2001) Stability analysis of gradient-based neural networks for optimization problems. Journal of Global Optimization, 19 (4), 363-381. (doi:10.1023/A:1011245911067).

Record type: Article

Abstract

The paper introduces a new approach to analyze the stability of neural network models without using any Lyapunov function. With the new approach, we investigate the stability properties of the general gradient-based neural network model for optimization problems. Our discussion includes both isolated equilibrium points and connected equilibrium sets which could be unbounded. For a general optimization problem, if the objective function is bounded below and its gradient is Lipschitz continuous, we prove that (a) any trajectory of the gradient-based neural network converges to an equilibrium point, and (b) the Lyapunov stability is equivalent to the asymptotical stability in the gradient-based neural networks. For a convex optimization problem, under the same assumptions, we show that any trajectory of gradient-based neural networks will converge to an asymptotically stable equilibrium point of the neural networks. For a general nonlinear objective function, we propose a refined gradient-based neural network, whose trajectory with any arbitrary initial point will converge to an equilibrium point, which satisfies the second order necessary optimality conditions for optimization problems. Promising simulation results of a refined gradient-based neural network on some problems are also reported.

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More information

Published date: 2001
Keywords: gradient-based neural network, equilibrium point, equilibrium set, asymptotic stability, exponential stability
Organisations: Operational Research

Identifiers

Local EPrints ID: 29636
URI: http://eprints.soton.ac.uk/id/eprint/29636
ISSN: 0925-5001
PURE UUID: 234aac86-3669-4664-ad37-1d3d582ebb57
ORCID for Houduo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 11 May 2006
Last modified: 16 Mar 2024 03:41

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Contributors

Author: Qiaoming Han
Author: Li-Zhi Liao
Author: Houduo Qi ORCID iD
Author: Liqun Qi

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