Homework 5

The first six exercises are from Beck, Introduction to Nonlinear Optimization.

1. (Convexity of set operations) Exercise 6.1

2. (Convex hull) Exercise 6.4

3. (Normal cone) Exercise 6.10

4. (Convex functions) Exercise 7.1 (i–ii)

5. (Affine functions) Exercise 7.3

6. (Strict convexity) Exercise 7.7

7. (Entropy) Show that $f(x) = -\sum_{i=1}^n x_i\log x_i$ is concave over the probability simplex $\bar\Delta_n=\{ x\in \mathbb R^n_+\mid \sum_{i=1}^n x_i = 1 \}$.

8. (Log-Sum-Exp) Show that $f(x) = \log\left(\sum_{i=1}^m e^{a_i^T x}\right)$ is convex in $\mathbb R^n$, where $a_1,\ldots,a_m\in\mathbb R^n$.