Recent Developments in General Quasi Variational Inequalities

Muhammad Aslam Noor, Khalida Inayat Noor and Michael Th. Rassias
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Muhammad Aslam Noor: COMSATS University Islamabad
Khalida Inayat Noor: COMSATS University Islamabad
Michael Th. Rassias: Hellenic Military Academy

A chapter in Geometry and Non-Convex Optimization, 2025, pp 315-359 from Springer

Abstract: Abstract We introduce and consider a new class of quasi variational inequalities involving two arbitrary operators, which is called general quasi variational inequality. Several interesting new and known classes of variational inequalities, complementarity problems, and nonlinear optimization are discussed. We have established the equivalence between the general quasi variational inequalities and fixed point problems applying the projection technique. This equivalent formulation is considered to discuss the existence of the solution. We use the projection methods, Wiener-Hopf equations, dynamical systems, and nonexpansive mappings. Convergence analysis of these methods is investigated under suitable conditions. It is shown that the general quasi variational inequalities are equivalent to the extended general variational inequalities for a particular case of the convex-valued sets. Our results present a significant improvement of previously known methods for solving quasi variational inequalities, system of absolute value equations, Lax-Milgram lemma, representation theorems, and related optimization problems. Since the general quasi variational inequalities include variational inequalities, complementarity problems, and system of absolute value equations as special cases. Results obtained in this chapter continue to hold for these problems. The implementation of these new algorithms and comparison with other methods need further efforts.

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-031-87057-6_13

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DOI: 10.1007/978-3-031-87057-6_13

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