Worksheet 7-3: Convex Functions

Worksheet 7-3: Convex Functions#

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Worksheet 7-3: Q1#

Consider \(\max{\{\mathbf{x}^{\top} Q \mathbf{x}\mid \|\mathbf{x}\| \leq 1}\}\) where \(\mathbf{x} \in \mathbb{R}^n\) and \(Q \succeq 0\).

(a) Does a maximizer exist? Justify by checking all the conditions of the relevant theorem.

(b) Let \(\mathbf{x} \in \mathbb{R}^2\) (that is, assume \(n=2\) above) and answer the following questions.

(i) Using \(L_1\)-norm for \(\mathbf{x}\): Sketch the feasible region and point out where a maximizer, if it exists, can be found.

(ii) Using \(L_2\)-norm for \(\mathbf{x}\): Sketch the feasible region and point out where a maximizer, if it exists, can be found.

Worksheet 7-3: Q2#

Consider the function

\[\begin{split} \begin{array}{rccc} f: & [0,\infty) & \rightarrow & \mathbb{R} \cup \{\infty\} \\ & x & \mapsto & \begin{cases} \infty & \text{if } x = 0 \\ 1/x & \text{if } x > 0 \end{cases} \end{array} \end{split}\]

  • What is the effective domain of \(f\), \(\text{dom}(f)\)?

  • Sketch the epigraph of \(f\).

  • Is \(f\) convex? Justify your answer.

Worksheet 7-3: Q3#

Epigraph problem

For each of the functions shown in the figure, answer the following:

(i) Sketch the sublevel sets at level \(\alpha\).

(ii) Sketch the epigraph.

(iii) Based only on your sketches, is the function convex? Justify your answer.

(iv) Based only on your sketches, is the function quasi-convex?

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