New PDF release: A Course in Combinatorial Optimization

By Schrijver A.

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By this observation it seems reasonable to consider convex sets which are the convex hull of a smaller set and identify which property on the smaller set S one needs to guarantee that the convex set is closed. Looking at the following counterexample it is not sufficient to impose that the set S is closed and this implies that we need a stronger property on S. 3 If convex hull given by clearly not closed. then S is closed and its is In the above counterexample the closed set S is unbounded and this prevents to be closed.

Functions this property is preserved. c. c.. c. c.. c. c.. c. function can be seen as a pointwise limit of an increasing sequence of real valued continuous functions. c.. 2 There exists an increasing sequence of continuous functions satisfying for every 3 with nonempty. Proof. We only give a proof of 1 2 since the other implications are obvious. c. c. c. function where and apply the first part. c. functions is the smallest set of functions, which are closed under taking the sup operation to any collection of functions belonging to this set and which contain the set of continuous real valued functions.

Without proof we now list the following result. For related results, valid in infinite dimensional topological vector spaces, the reader should consult [38]. 6 For any nonempty convex cone it follows that This concludes our section on sets. In the next section we will consider functions studied within convex and quasiconvex analysis. 3. 1 the different classes of functions studied within convex and quasiconvex analysis and derive their algebraic properties. 1. 1 we know how to apply hull operations to sets and using this it is also possible to construct so-called hull functions.

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A Course in Combinatorial Optimization by Schrijver A.

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