By Prof. Dr. Josef Stoer, Dr. Christoph Witzgall (auth.)

ISBN-10: 3642462162

ISBN-13: 9783642462160

ISBN-10: 3642462189

ISBN-13: 9783642462184

Dantzig's improvement of linear programming into some of the most appropriate optimization recommendations has unfold curiosity within the algebra of linear inequalities, the geometry of polyhedra, the topology of convex units, and the research of convex capabilities. it's the aim of this quantity to supply a synopsis of those issues, and thereby the theoretical again floor for the mathematics of convex optimization to be taken care of in a sub sequent quantity. The exposition of every bankruptcy is basically autonomous, and makes an attempt to mirror a selected form of mathematical reasoning. The emphasis lies on linear and convex duality conception, as initiated by way of Gale, Kuhn and Tucker, Fenchel, and v. Neumann, since it represents the theoretical improvement whose influence on smooth optimi zation concepts has been the main mentioned. Chapters five and six are dedicated to attribute features of duality idea: conjugate capabilities or polarity at the one hand, and saddle issues at the different. The Farkas lemma on linear inequalities and its generalizations, Motzkin's description of polyhedra, Minkowski's helping aircraft theorem are crucial straight forward instruments that are contained in chapters 1, 2 and three, respectively. The therapy of extremal houses of polyhedra in addition to of common convex units relies at the a long way attaining paintings of Klee. bankruptcy 2 terminates with an outline of Gale diagrams, a lately built profitable process for exploring polyhedral structures.

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O =i X =0 holds. 10. 1 "statement I ~ statement II" means that statement II follows from statement I. 3 Stoer/Witzgall, Convexity and Optimization 24 1. 4) Transposition Theorem of Stiemke [1]. For A =1=0 the following statements are equivalent: (i) AX =0, X>O has no solution, (ii) AT U ::::; 0, AT U =1= 0 has a solution. Proof. 1) of the Kuhn-Fourier theorem, the system AX =0, X>O is solvable if and only if the implication v~O : : : } v=o holds. But this is just another way of expressing that the system AT U ::::; 0 has no solution with AT U =1= O.

16) are not reversable. The following simple program, 30 I. Inequality Systems Minimize X3 subject to Xl +x2+x3=1, X~O, provides a counterexample. The vector X =(l,O,Ol is the optimal solution, whereas U = (0) is an optimal solution of the dual program, M aximi:;e u subject to u~O, u~O, u~l. The optimal solutions X and U satisfy both X2 ~0 and u ~ 0 as equations. CHAPTER 2 Convex Polyhedra The first chapter dealt with the logical structure of linear systems. This chapter directs the attention to the geometrical properties of their solution sets.

Trivial for manifolds. Consider a halfmanifold H n M with H:= {XIA~ X ~bo} ~ M. 3), M contains points X, Y such that A~X~bo and A~Y>bo. The point V:= X +cx(Y -X), bo-A~X oc: = ----=,-------=-A~Y-A~X belongs to M and satisfies A~ V = boo Hence L:= En M - V, where E:={XIA~X=bo}, is a subspace. We define a halfline N:={X =V-f3(Y-V)If3~O} and assert HnM=N+L. Assume XEHnM. Then there exists f3 ~ 0 such that A~ (X + f3(Y - V)) = boo Thus X + f3(Y - V)EE n M, whence X =(X +f3(Y - V)- V) + (V -f3(Y - V))EL +N.

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