Ph.D. (Doctor of Philosophy)
Department of Mathematical Sciences
Multi-objective optimization problems and game theory problems have a wide array of
applications and because of this there are different types of solutions available. This dissertation
explores two areas of optimization and a solution type for each. First, substantial
efficiency (SE) as a type of solution to multi-objective optimization problems that extends
proper efficiency. Secondly, strong Nash equilibria (SNE) as a type of solution to game
theoretic problems that extends Nash equilibria. Substantial efficiency is demonstrated to
be a superior solution to the more rudimentary notion of proper efficiency in solving some
multi-objective financial market and economic problems. Using this as motivation, a careful
examination is given for SE solutions in linear multi-objective optimization problems. Examples
are given showing that SE solutions are non-trivial and may not always exist. A
method for testing for substantial efficiency is provided. Existence of a SE solution can also
be guaranteed in a specific context. Both a topological inspection and recession analysis are
provided for the set of SE solutions. A situation where a SNE can be guaranteed is provided.
An algorithm is also provided that when convergent will be to a SNE.
Harris, Glenn Matthew, "Providing Better Choices: An Exploration of Solutions in Multi-Objective Optimization and Game Theory Using Variational Analysis" (2020). Graduate Research Theses & Dissertations. 7096.
Northern Illinois University
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