Problem 2: Convexity in 1- and 2-Dimensions

1. Consider the objective function f(v)=(1v)2f(v)=(1v)2 for a scalar variable vv. Using first and second derivatives confirm or reject that ff is convex. If you reject convexity everywhere, please specify for which values of vv that f(v)f(v) is not convex.

2. Suppose that two functions g(v)g(v) and h(v)h(v) are both convex functions. Confirm or reject that their sum is convex by the method above.

3. Let g(v)=(av2)2g(v)=(av2)2, where aa is a fixed real number. Confirm or reject the claim that g(v)g(v) is convex for any constant aa. If you reject convexity everywhere, please specify which for values of vv that g(v)g(v) is not convex (as a function of aa).

4. For a vector v=[v1v2]Tv=[v1v2]T, let p(v)=(1v1)2+100(v2v21)2p(v)=(1v1)2+100(v2v12)2. Calculate the gradient and Hessian of pp, and confirm or reject that pp is convex everywhere in 2R2. If the function is not convex everywhere, please specify with a plot the region where convexity does not hold. What minimizing challenges arise for functions which are not convex?

Suggested Approach Compute the Hessian Hp(v)Hp(v). Function p(v)p(v) is convex around point vv if the eigenvalues of HpHp are both positive at vv. On a coarse [v1,v2][v1,v2] grid, compute the minimum eigenvalue. Look for regions where the minimum eigenvalue is 00. Produce a contour plot of the minimum eigenvalue in that region, if you notice it. Carefully label your axes.