Journal
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
Volume 22, Issue 1, Pages 37-48Publisher
KLUWER ACADEMIC PUBL
DOI: 10.1023/A:1014813701864
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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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