New PDF release: Numerical Methods of Mathematical Optimization. With ALGOL

By Hans P. Künzi, Werner Rheinboldt, H. G. Tzschach, C. A. Zehnder

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Application of Lemma 3 to the skew-symmetric matrix / 0 -A a, 0 K=[ \-al \ -ao. 0/ al now yields the existence of a vector ΖΟ^ = ( Μ ' · , Λ Α · ) ^ 0 for which the following inequalities are satisfied. 56) a%H>\ Two cases can now be distinguished for the scalar λ" introduced here. Lemma 4. Let λ" > 0 ; then optimal feasible vectors the dual programs such that floV = + α0 A^w' + > and exist for «ν Ax' x'>ao,. PROOF. 56) (cf. also Tucker [141]). 56) we can use the normalized vectors x^ and as the desired vectors x^ and w^.

This implies the second part of (b). Finally, (c) is a direct consequence of (b). Corollary. Either both the maximization and minimization problems possess optimal vectors or neither one does. In the first case the maximum and minimum are equal to each other and their common value is called the optimal value of the dual problems. PROOF. If one of the problems has an optimal vector, it follows from Lemma 5(c) that A® > 0 and from Lemma 4 that both problems have 34 Optimal 1 vectors jc^ and minimum LINEAR OPTIMIZATION and that the m a x i m u m a^x^ is e q u a l to the a^w®.

Geometrical interpretation of a convex function in the plane.

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