Researches On Solving Methods Of Variational Inequalities Based On Neural Computation

Variational inequality (?) is a significant research direction of optimization field, which can be viewed as a natural framework for unifying the treatment of var-ious optimization problems and equilibrium problems. Many important optimization problems, such as complementary problems, quadratic programming problems and nonlinear programming problems, etc., can be converted into ? problemd. Nowa-days, VI problems enjoy extensive application in elaborating and researching the optimal model and balance model in the field of engineering and science, such as operational research, automatic control, signal and image processing and system identification. Therefore, the research on the solving methods of ? problems is of great significance and practical value. In the past two decades, solving VI and its related problems by neural networks based on circuit implementation, namely neu-ral computation, have been extensively investigated and gained rich research results in domestic and overseas, due to its advantages of massively parallel operations and rapid convergence compared with traditional numerical iteration algorithm. How-ever, the theory of solving optimization problems with neural networks is not perfect yet, there still exist some problems worth to be further studied and discussed, for ex-ample:(1)most existing neural networks for solving ? did not consider time-delay effect in real applications. Although the time-delay may influence the network's sta-bility, if you properly introduce the time-delay into neural networks, it can improve the solving performance of the neural networks by changing its topological structure: (2)due to the limitation of the analysis methods, the stability conditions of some of the existing (delayed/no delayed) neural networks for solving ? are conservative to some extent (especially the requirement of ?'s monotonicity), which restrict the networks' solving range and practicability; (3)in these years, various extensions of ? have been presented, such as general ? and inverse ?, etc., but there are still not many neural networks developed for solving these emerging problems. In this dissertation, the above issues have been intensively explored and studied from the the dynamic system perspective, and the main research of this dissertation can be brieflv described as follows:1. Study the stability problem of a class of no-delayed projection neural network for solving linear variational inequality (LVI). Present new global exponen-tial stability criteria for the neural network, through the theory of functional differential equation, the Lyapunov stability theory and LMI technique, re-spectively. Based on the obtained results, the feasible set can be any closed convex set and the monotone assumption of LVI is not necessary anymore. In other words, the neural network can solve a class of non-monotone LVI and related non-convex optimization problems, which implies a obvious extension of its application scope.2. Present a projection neural network with time-varying delay for solving LVI. Prove the existence and uniqueness of the network's equilibrium. Put for-ward several global exponential stability criteria for the neural network via the theory of functional differential equation, the Lyapunov stability theory and LMI technique, respectively. The obtained results are easy to be verified and possess low conservatism, they do not require the monotone assumption of LVI and the non-singularity of matrix I-?M, that is, the presented network can solve a class of non-monotone L? and related non-convex optimization problem. Moreover, the obtained results can deal with both of slow and fast time-varying delays, which implies its broad scope of application.3. Study the stability problem of a class of Neutral-type delayed projection neural network for solving LVI. Analyze the implicit conservatism in existing results. By constructing a proper Lyapunov functional and employing the technique of LMI and free-weighted matrix, new global exponential stability criteria for the neural network are obtained. Compared with the existing literature, the obtained results fully consider the neurons' excitatory and inhibitory effect, and possess lower conservatism. Moreover, the new obtained results do not require monotone assumption of LVI, which is required in existing results. That is to say, the neural network can solve a class of non-monotone LVI, which implies a obvious extension of its application scope.4. A projection neural network with discrete and distributed delays, ie. mixed delays, is presented for solving LVI. Prove the existence and uniqueness of the network's equilibrium. Employing the LMI and free weighted matrix tech-nique, put forward several global exponential stability criteria for the neural network via the theory of functional differential equation and the Lyapunov stability theory, respectively. Compared with the network with single discrete delay:in terms of network structure, the network with mixed delays is more general; in terms of functional, the network with mixed delays not only can solve a class of non-monotone LVI, but also can resolve the sensitivity problem of the network with single discrete delay to the delay values.5. Present a novel neural computation model for solving generalized linear varia-tional inequality (GLVI). Prove the existence and uniqueness of the network's equilibrium. Employing the LMI and free weighted matrix technique, put for-ward several global exponential stability criteria for the neural network via the theory of functional differential equation and the Lyapunov stability the-ory, respectively. Compared with the existing networks for solving GLVI, the proposed one dose not require the monotonicity of the GLVI, that is to say, it can solve a class of no-monotone GLVI and related non-convex optimization problems, which shows its larger solving scope.6. A novel neural computation model is proposed for solving nonlinear inverse variational inequality (IVI). Prove the existence and uniqueness of the net-work's equilibrium. Employing the LMI and free weighted matrix technique, several global exponential stability criteria for the neural network are obtained based on the Lyapunov stability theory. Compared with the existing networks for solving nonlinear IVI, the proposed network dose not require the mono-tonicity and the smoothness of the IVI, that is to say, it can solve a class of non-monotone and non-smooth IVI problems, which implies its larger solving scope.