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

By Borwein,Lewis

Show description

Read Online or Download Convex Analysis and Non Linear Optimization Theory and Examples PDF

Best mathematics books

Mathematik für Physiker 2: Basiswissen für das Grundstudium der Experimentalphysik

Die für Studienanfanger geschriebene „Mathematik für Physiker'' wird in Zukunftvom Springer-Verlag betreut. Erhalten bleibt dabei die Verbindung einesakademischen Lehrbuches mit einer detaillierten Studienunterstützung. DieseKombination hat bereits vielen Studienanfangern geholfen, sich die Inhalte desLehrbuches selbständig zu erarbeiten.

Extra info for Convex Analysis and Non Linear Optimization Theory and Examples

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 +∞.

Download PDF sample

Rated 4.67 of 5 – based on 23 votes

Related posts