Borwein,Lewis's Convex Analysis and Non Linear Optimization Theory and PDF

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Carath´ eodory’s theorem [48]) Suppose {ai | i ∈ I} is a finite set of points in E. For any subset J of I, define the cone µi ai | 0 ≤ µi ∈ R, (i ∈ J) . CJ = i∈J (a) Prove the cone CI is the union of those cones CJ for which the set {ai | i ∈ J} is linearly independent. Furthermore, prove directly that any such cone CJ is closed. (b) Deduce that any finitely generated cone is closed. (c) If the point x lies in conv {ai | i ∈ I}, prove that in fact there is a subset J ⊂ I of size at most 1 + dim E such that x lies in conv {ai | i ∈ J}.

Suppose the k × k matrix A has each entry aij nonnegative. We say A has doubly stochastic pattern if there is a doubly stochastic matrix with exactly the same zero entries as A. Define a set Z = {(i, j)|aij > 0}, and let RZ denote the set of vectors with components indexed by Z and RZ+ denote those vectors in RZ with all nonnegative components. Consider the problem ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ inf subject to (i,j)∈Z (p(xij ) − xij log aij ) i:(i,j)∈Z xij = 1, for j = 1, 2, . . , k, j:(i,j)∈Z xij = 1, for i = 1, 2, .

Proof. Suppose some point y in E satisfies h(y) = −∞. Since yˆ lies in core (dom h) there is a real t > 0 with yˆ + t(ˆ y − y) in dom (h), and hence a real r with (ˆ y + t(ˆ y − y), r) in epi (h). Now for any real s, (y, s) lies in epi (h), so we know yˆ, t r + ts 1 (ˆ y + t(ˆ y − y), r) + (y, s) ∈ epi (h), = 1+t 1+t 1+t Letting s → −∞ gives a contradiction. 3 we saw that the Karush-Kuhn-Tucker conditions needed a regularity condition. 7) There exists xˆ in dom (f ) with gi (ˆ x) < 0 for i = 1, 2, .

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Convex Analysis and Non Linear Optimization Theory and Examples by Borwein,Lewis


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