On Generalizations of the Frank-Wolfe Theorem to Convex and Quasi-Convex Programmes |
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Authors: | Wiesława T Obuchowska |
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Institution: | (1) Department of Mathematics, Chowan College, Murfreesboro, NC, 27855 |
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Abstract: | In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained optimization
problem.
We present necessary and sufficient conditions for boundedness of either a faithfully convex or a quasi-convex polynomial
function over the feasible set defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial
inequalities, where the faithfully convex functions satisfy some mild assumption. The conditions are provided in the form
of an algorithm, terminating after a finite number of iterations, the implementation of which requires the identification
of implicit equality constraints in a homogeneous linear system. We prove that the optimal solution set of the considered
problem is nonempty, this way extending the attainability result well known as the so-called Frank-Wolfe theorem. Finally
we show that our extension of the Frank-Wolfe theorem immediately implies continuity of the solution set defined by the considered
system of (quasi)convex inequalities. |
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Keywords: | convex and quasi-convex constrained programs existence of optimal solutions cone of recession |
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