Worksheet 6-1: Convex Sets

Download: CMSE382-WS6_1.pdf

Warning

This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.

---

Worksheet 6-1: Q1

Let \(\mathbf{x}_1 = (2,3)^\top\) and \(\mathbf{x}_2 = (1,4)^\top\).

1. Write the equation of the line passing through \(\mathbf{x}_1\) and \(\mathbf{x}_2\) as a function of \(\lambda\):

   \[\mathbf{x}(\lambda) = (1-\lambda)\mathbf{x}_1 + \lambda \mathbf{x}_2\]

2. For each value of \(\lambda\) below, compute the point \(\mathbf{x}(\lambda)\) and state whether it lies on the segment between \(\mathbf{x}_1\) and \(\mathbf{x}_2\), or outside it:

   \(\lambda\)	\(\mathbf{x}(\lambda)\)	On segment?
   \(\lambda = 0\)
   \(\lambda = 1/2\)
   \(\lambda = 1\)
   \(\lambda = 2\)

---

Worksheet 6-1: Q2

For each of the following sets, determine whether it is convex or not convex.

(a) A finite set of points \(S = \{p_1, p_2, \ldots, p_k\} \subset \mathbb{R}^n\) (with \(k \ge 2\)).

(b) The nonnegative orthant \(\mathbb{R}^2_+ = \{(x,y) \mid x \ge 0, y \ge 0\}\) (shown below).

\[\includegraphics[width=0.25\textwidth]{../../../figures/Rplus.png}\]

(c) The set \(\{(x,y) \mid xy = 0\}\) (axes, or “L-shape”):

(d) The set shown below (defined by a nonlinear inequality):

(e) The \(\ell_1\), \(\ell_2\), and \(\ell_\infty\) unit balls in \(\mathbb{R}^2\):

---

Worksheet 6-1: Q3

(a) Draw an example of a union of two convex sets that is not convex.

(b) Draw an example of a union of two convex sets that is convex.

---

Worksheet 6-1: Q4

Let \(S = \{(1,1), (2,2), (3,1)\}\).

1. Sketch the convex hull \(\text{conv}(S)\).

2. For each of the following points, find \(\lambda_1, \lambda_2, \lambda_3 \ge 0\) with \(\lambda_1+\lambda_2+\lambda_3=1\) such that \(\mathbf{x} = \lambda_1(1,1)+\lambda_2(2,2)+\lambda_3(3,1)\):

   Point \(\mathbf{x}\)	\(\lambda_1\)	\(\lambda_2\)	\(\lambda_3\)
   \((2,2)\)
   \((2,1.5)\)
   \((1.5, 1.5)\)
   \((2, 1)\)

3. For each set of coefficients below, identify the corresponding point:

   \(\lambda_1\)	\(\lambda_2\)	\(\lambda_3\)	Point \(\mathbf{x}\)
   \(1/3\)	\(1/3\)	\(1/3\)
   \(1/2\)	\(0\)	\(1/2\)
   \(0\)	\(1/4\)	\(3/4\)

---

Worksheet 6-1: Q5

Let \(S = \{(1,1), (1,2), (2,1), (2,2)\}\).

1. Sketch \(S\) and its convex hull \(\text{conv}(S)\).

2. What is the maximum number of points from \(S\) needed to represent any point in \(\text{conv}(S)\) as a convex combination? (Hint: Carathéodory’s theorem.)

3. Write the point \(\mathbf{x} = (1.5, 1.4)\) as a convex combination of points in \(S\).