Topology of Global Optimization
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pendently of each other, establish the celebrated Hahn-Banach linear extension theorem; by means of an obvious reformulation it shows itself to be a ST. Here too the purpose is to have lemmas for proving other theorems ~ in functional analysis and geometry. Over several years TA and ST have been carried on as disjoint theories. Recently, thanks to the great development of optimization and to the increasing use of TA and ST in the theory of optimization, it has been recognized that TA and ST are different 'languages' for expressing the same 'structural' property (this does not imply that one of them should be deleted; on the contrary, different languages let us achieve more properties) and, overall, that they are not only tools for proving theorems; indeed, they have been raised to the basis for the theory of constrained extrema. After a short review of some TA, their application to prove fundamental theorems of optimization will be shown. Then, we will briefly describe the recent approach to the theory of constrained extrema which is based on TA and ST. Matrices and vectors will be real-valued. F a r k a s L e m m a . [7]. Let A be a matrix of the order m × n, a be a row n-vector, and x a column n-vector. Ax > 0 implies ax >_ 0 if and only if there exists a nonnegative row m-vector z such that zA - a. This lemma receives a useful vector interpretation. The rows of A can be seen as vectors of Rn; call C the (convex) cone generated by them, and set C* "- {x E R n" Ax > 0}. Since the elements of C are the only vectors which have a nonnega-
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