
# Homework 4

Please answer the following questions in complete sentences in submit the solution on Blackboard by the due date there.

## Problem 0: List your collaborators.

Please identify anyone, whether or not they are in the class, with whom you discussed your homework. This problem is worth 1 point, but on a multiplicative scale.

## Problem 1: Log-barrier terms

The basis of a class of methods known as interior point methods is that we can handle non-negativity constraints such as $\vx \ge 0$ by solving a sequence of unconstrained problems where we add the function $b(\vx; \mu) = -\mu \sum_i \log(x_i)$ to the objective. Thus, we convert into

1. Explain why this idea could work. (Hint: there's a very useful picture you should probably show here!)

2. Write a matrix expression for the gradient and Hessian of $f(\vx) + b(\vx; \mu)$ in terms of the gradient vector $g(\vx)$ and the Hessian matrix $\mH(\vx)$ of $f$.

## Problem 2: Inequality constraints

Draw a picture of the feasible region for the constraints:

## Problem 3: Necessary and sufficient conditions

Let $f(\vx) = \frac{1}{2} \vx^T \mQ \vx - \vx^T \vc$.

1. Write down the necessary conditions for the problem:

2. Write down the sufficient conditions for the same problem.

3. Consider the two-dimensional case with Determine the solution to this problem by any means you can, and justify your work.

4. Produce a Julia or hand illustration of the solution showing the function contours, and gradient. What are the active constraints at the solution? What is the value of $\lambda$ in $\mA^T \lambda = \vg$?

## Problem 4: Constraints can make a non-smooth problem smooth.

Show that

can be reformulated as a constrained optimization problem with a continuously differentiable objective function and both linear equality and inequality constraints.