On generalizations of the Frank–Wolfe theorem to convex and quasi-convex programs

On generalizations of the Frank–Wolfe theorem to convex and quasi-convex programs

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Article ID: iaor20062906
Country: Netherlands
Volume: 33
Issue: 2/3
Start Page Number: 349
End Page Number: 364
Publication Date: Mar 2006
Journal: Computational Optimization and Applications
Authors:
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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