Miami University Convex Function and Quadratic Function Questions

Question Description

For each statement, provide an example.

(a) A convex set that is not a cone.

(b) A cone which is not convex.

(d) A strictly convex function f : IRn ? IR and a nonzero n × m matrix A so that g(x) = f(Ax) is not strictly convex.

(e) A set C and a point x ? outside this set for which PC(x ?), the projection of x ? on C (see page 46 for more information of PC(x ?)), is not unique.

(f) A function f : IRn ? IR and two convex sets C1 and C2 such that f is convex over C1 and C2 but is not convex on C1 ? C2.

(g) A function for which the steepest decent method with a fixed stepsize is not convergent.

(h) A function for which the steepest decent method with the exact line search converges for any initial point x0

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