Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.677538
Title: Variational inequalities and optimization problems
Author: Liu, Yina
ISNI:       0000 0004 5369 0447
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2015
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Abstract:
The main purpose of this thesis is to study weakly sharp solutions of a variational inequality and its dual problem. Based on these, we present finite convergence algorithms for solving a variational inequality problem and its dual problem. We also construct the connection between variational inequalities and engineering problems. We consider a variational inequality problem on a nonempty closed convex subset of R^n. In order to solve this variational inequality problem, we construct the equivalence between the solution set of a variational inequality and optimization problems by using two gap functions, one is the primal gap function and the other is the dual gap function. We give properties of these two gap functions. We discuss su�cient conditions for the subdifferentiability of the primal gap function of a variational inequality problem. Moreover, we characterize relations between the G^ateaux differentiabilities of primal and dual gap functions. We also build some results for the Lipschitz and locally Lipschitz properties of primal and dual gap functions as well. Afterwards, several su�cient conditions for the relevant mapping to be constant on the solution set of a variational inequality has been obtained, including the relations between solution sets of a variational inequality and its dual problem as well as the optimal solution sets to gap functions. Based on these, we characterize weak sharpness of the solution set of a variational inequality by its primal gap function g and its dual gap function G. In particular, we apply error bounds of g, G and g + G on C. We also construct finite convergence of algorithms for solving a variational inequality by considering the convergence of a local projection. We carry out these results in terms of the weak sharpness of solution sets of a variational inequality as well as the error bounds of gap functions of a variational inequality problem.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.677538  DOI: Not available
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