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Topics in multiple criteria optimization Warburton, Arthur Raymond


Several aspects of multiple criteria optimization are investigated. First, sufficient conditions are obtained for the upper semi-continuity of the set of maximal alternatives for a nonscalar parametric optimization problem in which the set of alternatives, the objective functions, and a partial order defined on the set of outcomes may all vary. One of the conditions involves continuity of a function f on a product space and the relationship between various continuity conditions on f that have appeared on the literature are investigated. Second, we study the set of Pareto optimal and weakly Pareto optimal solutions to a vector maximization problem defined by continuous vector valued quasiconcave criterion functions and a closed convex set of decisions, S. If S is compact, it is shown that the set of weakly Pareto optimal decisions is connected, but that the set of Pareto optimal decisions is not necessarily connected. However, the set of Pareto optima is shown to be connected for some important sub-classes of quasiconcave criteria. Under reasonable conditions the compactness assumption on S may be relaxed and connectedness maintained. Connectivity may fail if preferences are given by cones other than the Pareto cone. Finally, bicriterion mathematical programs of the form P: max {u(f₁(x), f₂(x))|xεS} are considered, where S is a bounded polyhedral set, f₁ and f₂ are linear fractional functions, and u is a real valued function, non-decreasing in each argument. It is shown that the solution of P may be essentially reduced to a one parameter linear program. Simple, computationally effective, finite algorithms are obtained for the cases where u is a weighted sum of f₁ and f₂, and where u is the Chebychev function min [f₁(x), f₂(x)]. It is also shown how a simple sequence of quasiconcave problems may be constructed whose solutions converge to the solution of max {min[f₁(x),...,f[sub p](x)]|xεS) if f₁,…f[sub p] are quasiconcave and S is convex.

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