Due in class, 12 April, 2012

Do Exercises 1 and 3 from Chapter 5 of the Convex Optimization book:

**5.1** Consider the optimization problem

minimize x

^{2}+ 1

subject to (x-2)(x-4) ≤ 0

with x ε **R**.

- Give the feasible set, the optimal value, and the optimal solution.
- Plot the objective x
^{2}+ 1 versus x. On the same plot, show the feasible set, optimal point and value, and plot the Lagrangian L(x, λ) versus x for a few positive values of &lambda. Verify the lower bound property inf_{x}L(x, λ) ≤ p^{*}for λ ≥ 0. Derive and sketch the Lagrange dual function g. - State the dual problem and verify that it is a concave maximization problem.
Find the dual optimal value and dual optimal solution λ
^{*}. Does strong duality hold? - Let p
^{*}(u) denote the optimal value of the problem

minimize x^{2}+ 1

subject to (x-2)(x-4) ≤ u

as a function of the parameter u. Plot p^{*}(u) and verify that its derivative at 0 is -λ^{*}.

**5.3** Express the dual problem of

minimize c

^{T}x

subject to f(x) ≤ 0

with c ≠ 0, in terms of the Fenchel conjugate f^{*}. Explain why
the dual problem is convex even if f isn't convex.