# Minimize a quadratic form over a subspace

I'm learning optimization methods. For the minimization problem:

\begin{align} \min_{x}\,\,x^TAx+b^Tx,&&\mbox{s.t.}\,\,& Cx=d,\\& \end{align}

I was asked to minimize it over some given n-dimensional subspace L subject to the constraint.

I'm not familiar with the subspace concept here. What does it mean to "optimize over the subspace"? Could someone explain it in a more plain language? Thanks.

Updates: If L is a span of vectors $l_1,l_2,l_3...$, does it mean x must be a linear combination of those vectors?

• For your question @James, it is worth noting that minimizing over a subspace is the same as a minimizing with an equality constraint (as you have written). So, by being asked to "optimize over the subspace" you are essentially adding another equality constraint. YOu might want to see my related question for the optimality conditions of optimizing over a linearly constrained problem: math.stackexchange.com/questions/2307096/… – jaja Apr 8 '18 at 7:16

## 1 Answer

Presumably they mean that $x$ needs to be in $L,$ that is you need to find $\min_{x|x \in L}$

• In terms of your update: yes, that's how I would understand it. – Nir Dec 12 '16 at 17:41