---
title: Ex. 4.7
linktitle: Ex 4.7
type: book
date: "2021-01-02T00:00:00+01:00"
# Prev/next pager order (if `docs_section_pager` enabled in `params.toml`)
weight: 1
toc: false
---
???+ question "Ex. 4.7"
    Consider the criterion
	
    \begin{equation}
		D^\ast(\beta,\beta_0) = -\sum_{i=1}^Ny_i(x_i^T\beta + \beta_0),\non
	\end{equation}
	
    a generalization of (4.41) in the textbook where we sum over all the observations. Consider minimizing $D^\ast$ subject to $\|\beta\|=1$. Describe this criterion in words. Does it solve the optimal separating hyperplane problem?
??? done "Soln. 4.7"
    When $\|\beta\| = 1$, $\beta^Tx_i + \beta_0$ is the signed distance of $x_i$ to the hyperplane $\beta^Tx + \beta_0 = 0$. This does not solve the optimal separating hyperplane problem. Optimal separating hyperplane is actually solving a max-min problem such that each point satisfies the distance requirement, however minimizing $D^\ast$ does not have such pointwise constraint.