Abstract
In 1956, Frank and Wolfe extended the fundamental existence theorem of linear programming by proving that an arbitrary quadratic function f attains its minimum over a nonempty convex polyhedral set X provided f is bounded from below over X. We show that a similar statement holds if f is a convex polynomial and X is the solution set of a system of convex polynomial inequalities. In fact, this result was published by the first author already in a 1977 book, but seems to have been unnoticed until now. Further, we discuss the behavior of convex polynomial sets under linear transformations and derive some consequences of the Frank–Wolfe type theorem for perturbed problems.
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Belousov, E.G., Klatte, D. A Frank–Wolfe Type Theorem for Convex Polynomial Programs. Computational Optimization and Applications 22, 37–48 (2002). https://doi.org/10.1023/A:1014813701864
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DOI: https://doi.org/10.1023/A:1014813701864