Worksheet 7-1: Convex Functions

Worksheet 7-1: Convex Functions#

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Atom functions (assume convex unless you are proving directly from the definition):


\(f(t) = mt + b\), \(m,b \in \mathbb{R}\)	\(f(t) = t^2\)	\(f(t) = e^t\)
	\(f(t) = t^4\)	\(f(t) = -\ln(t)\), \(t > 0\)

Worksheet 7-1: Q1#

Justify each step (i)–(vi) in the following proof that \(f(t) = t^2\) is convex. Pick any \(\lambda \in [0,1]\) and \(t_1, t_2 \in \mathbb{R}\). Then:

\[\begin{split} \begin{aligned} f(\lambda t_1+(1-\lambda)t_2) &= (\lambda t_1+(1-\lambda)t_2)^2 \qquad\qquad\textit{(i)}\\ &= \lambda^2 t_1^2 + 2\lambda(1-\lambda)t_1 t_2 + (1-\lambda)^2 t_2^2 \qquad\qquad\textit{(ii)}\\ &= \lambda t_1^2 - \lambda(1-\lambda)t_1^2 + 2\lambda(1-\lambda)t_1 t_2 + (1-\lambda)t_2^2 - \lambda(1-\lambda)t_2^2 \qquad\qquad\textit{(iii)}\\ &= \lambda t_1^2+(1-\lambda)t_2^2-\lambda(1-\lambda)(t_1-t_2)^2 \qquad\qquad\textit{(iv)}\\ &\le \lambda t_1^2+(1-\lambda)t_2^2 \qquad\qquad\textit{(v)}\\ &= \lambda f(t_1)+(1-\lambda)f(t_2). \qquad\qquad\textit{(vi)} \end{aligned} \end{split}\]

Worksheet 7-1: Q2#

Show that each of the following functions is convex. Use the theorems from lecture to justify your answer.

(a) \(f(x,y) = x^2 + 3y^2\)

(b) \(h(x,y) = \exp(x + y)\)

(c) \(f(x,y) = x^2 + 2xy + 3y^2 + 2x - 3y\)

(d) \(f(x,y,z) = \exp(x - y + z) + \exp(2y) + x\)

(e) \(f(x,y) = -\ln(xy)\), for \(x > 0\), \(y > 0\)

Worksheet 7-1: Q3#

Are the following functions convex? Justify your answer.

(a) \(f(x,y) = xy\)

(b) \(\|\mathbf{x}\|\) for any norm on \(\mathbb{R}^n\). Use the definition of convex function.

(c) \(h(\mathbf{x}) = \exp(\|\mathbf{x}\|)\)

(d) \(\|\mathbf{x}\|^2\) for any norm on \(\mathbb{R}^n\)