This work aims to investigate some applications of the
conjugate duality for scalar and vector optimization problems to
the construction of gap functions for variational inequalities and
equilibrium problems. The basic idea of the approach is to
reformulate variational inequalities and equilibrium problems into
optimization problems depending on a fixed variable, which allows
us to apply duality results from optimization problems.
Based on some perturbations, first we consider the conjugate
duality for scalar optimization. As applications, duality
investigations for the convex partially separable optimization
problem are discussed.
Afterwards, we concentrate our attention on some applications of
conjugate duality for convex optimization problems in finite and
infinite-dimensional spaces to the construction of a gap function
for variational inequalities and equilibrium problems. To verify
the properties in the definition of a gap function weak and strong
duality are used.
The remainder of this thesis deals with the extension of this
approach to vector variational inequalities and vector equilibrium
problems. By using the perturbation functions in analogy to the
scalar case, different dual problems for vector optimization and
duality assertions for these problems are derived. This study
allows us to propose some set-valued gap functions for the vector
variational inequality. Finally, by applying the Fenchel duality
on the basis of weak orderings, some variational principles for
vector equilibrium problems are investigated.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:18561 |
| Date | 25 July 2006 |
| Creators | Lkhamsuren, Altangerel |
| Contributors | Wanka, Gert, Weidner, Petra, Enkhbat, Rentsen, Technische Universität Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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