I'm currently taking a convex optimization class and we're using the textbook by Boyd & Vandenberghe. I was solving an exercise problem when I came across a question (for anybody curious the problem is Exercise 9.1). I have two questions in total regarding the same problem. One about a detail within the problem and one regarding the solving itself. Here's the problem:

Consider the problem of minimizing a quadratic function:

$$\text{minimize}\quad f(x) = \frac{1}{2}x^T P x + q^T x + r$$

where $P \in S^n$ (i.e. an $n \times n$ symmetric matrix) but we do not assume $P$ is positive semidefinite.

Show that if $P$ is not positive semidefinite (and therefore objective function $f$ is not convex) then the problem is unbounded below.

My approach

By definition if a matrix is not positive semidefinite then that means there exists some $z$ such that

$$z^TPz \lt 0$$

Following this logic we could say that $x = \alpha z$ with $x, z \in \Bbb{R}^n$ and $\alpha \in \Bbb{R}$. The objective function changes to

$$f(x) = \frac{1}{2}\alpha^2 (z^TPz) + q^T(\alpha z) + r$$

This is where I get stuck. According to the solution manual, it states that if $\alpha$ becomes large then $f \rightarrow -\infty$ and we can therefore conclude that the function is unbounded below.

My two questions are as follows:

  1. Why does the matrix not being positive semidefinite imply the function is not convex?

  2. How was the conclusion drawn that if $\alpha$ becomes large then the function diverges to $-\infty$?

Thank you.


To answer your first question,

The definition of a convex function (in the context of twice continuously differentiable functions) is that the Hessian is positive semidefinite, and the Hessian of $f$ in this case is $P$.

As for your second question,

Since you've now fixed $z$, the expression $\frac{1}{2}z^TPz\alpha^2 + q^Tz\alpha+r$ is just a quadratic equation with variable $\alpha$, and if the second order term of a quadratic equation is negative then we know its parabola faces downwards.


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