The Condition Number Of Convex Set And Interrelated Properties

The condition number is a vital tool depicting error bounds. For a system of nonlinear convex inequalities, in view of the interrelation between the variableκC-1(ε) and the error bounds for s system of nonlinear inequalities, we may come to a conclusion that there must exist some connection betweenκC-1(ε) and the condition number of the solution set of a system of nonlinear inequalities. Herein we only aim at the case that the solution set is closed convex. In fact, since the introduction of oriented distance function, that is the canonical representation of convex set, we define the condition number of convex set depending upon variableκC-1(ε) derived from directional derivative of canonical representation function. In order to examine the validity of definition, we must verify the condition of the relationκC-1(ε)=(dH(C,Cε))(|ε|).In fact, we find that the relationκC-1(ε)=(dH(,C_ε))/(|ε|) isn't always right under the condition of any a closed solid convex set through reversed instance. In order to get the valid condition to make the relation true, first, herein we give some important conclusions through three lemmas:providing another expression for projection map (πc(p)) with point p belonging to the interior of C, meanwhile we deduce the one for the subdifferential focusing on the functionΔC, which is related to normal cone of C at point p above mentioned; and then we develop the effective information on the expression ofκC-1(ε); Second, through dividing the single of the set C into four situations in detail and researching the method of proof interrelated in document 1, we get much stronger condition and prove our conclusion.The fourth section in this text base on the above section that the relation set up. In the first place, we discuss the interrelation between several variables introduced newly andκC-1(ε). Next, we give an example to calculate the condc(C); Finally, we place emphasis on comparing the measurement between 2dH(C, center(C)) and diam(C) especially for a bounded convex set C; through the instances we get that both variables are not really comparable.