v2006.03.09 - Convex Optimization

Appendix CSome analytical optimal resultsinfC.1 properties of infima∅ ∆ = ∞ (1258)sup ∅ ∆ = −∞ (1259)Given f(x) : X →R defined on arbitrary set X [123,0.1.2]inf f(x) = − sup −f(x)x∈X x∈Xsupx∈Xf(x) = −infx∈X −f(x) (1260)arg inf f(x) = arg sup −f(x)x∈X x∈Xarg supx∈Xf(x) = arg infx∈X −f(x) (1261)Given g(x,y) : X × Y →R with independent variables x and y definedon mutually independent arbitrary sets X and Y [123,0.1.2]inf g(x,y) = inf inf g(x,y) = inf inf g(x,y) (1262)x∈X, y∈Y x∈X y∈Y y∈Y x∈XThe variables are independent if and only if the corresponding sets arenot interdependent. An objective function of partial infimum is notnecessarily unique.2001 Jon Dattorro. CO&EDG version 03.09.2006. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.479