By Borwein,Lewis

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**Example text**

M − 1, c, y > 0, y ∈ Y, or equivalently PY ai , y ≤ 0 for i = 1, 2, . . , m − 1, PY c, y > 0, y ∈ Y, has no solution. By the induction hypothesis applied to the subspace Y, there are nonm−1 negative reals µ1 , µ2, . . , µm−1 satisfying i=1 µi PY ai = PY c, so the vector m−1 i c − 1 µia is orthogonal to the subspace Y = (span (am ))⊥ . 10) m−1 µiai . 10) to obtain then we can substitute am = −λ−1 m 1 a solution. 11) C= µi ai | 0 ≤ µ1 , µ2 , . . , µm ∈ R 1 can be separated from C by a hyperplane.

The equivalence of (ii) and (iii) now gives Gordan’s theorem. We now proceed by using Gordan’s theorem to derive the Farkas lemma, one of the cornerstones of many approaches to optimality conditions. The proof uses the idea of the projection onto a linear subspace Y of E. Notice ﬁrst that Y becomes a Euclidean space by equipping it with the same inner product. The projection of a point x in E onto Y, written PY x, is simply the nearest point to x in Y. 1), and is characterized by the fact that x − PY x is orthogonal to Y.

D) Show the Mangasarian-Fromovitz constraint qualiﬁcation holds at ¯ by considering the direction d = −X. ¯ X ¯ must (e) Write down the Karush-Kuhn-Tucker conditions which X satisfy. (f) When {y 1 , y 2, . . , y m} is the standard basis of Rn , the optimal so¯ = I. Find the corresponding lution of the problem in part (c) is X Lagrange multiplier vector. 2), one beneﬁt of convexity in optimization: critical points of convex functions are global minimizers. In this section we extend the types of functions we consider in two important ways: (i) we do not require f to be diﬀerentiable; (ii) we allow f to take the value +∞.