Minimizing quadratic form with quadratic and linear constraints

Asked Apr 11 '20 at 19:09

Active Apr 12 '20 at 06:09

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I am trying to solve the following optimization problem

$$\begin{array}{ll} \text{minimize} & \mathbf{x}^T \mathbf{A} \mathbf{x}\\ \text{subject to} & \left(\mathbf{x}-m\mathbf{1}\right)^T \left(\mathbf{x}-m\mathbf{1}\right) \le a\\ & \mathbf{1}^T \mathbf{x} = b\end{array}$$

where

$\mathbf{A} \in \mathbb{R}^{n \times n}$ is a symmetric, positive definite matrix.
$\mathbf{1}$ denotes the vector of all ones.
$a$, $b$ and $m$ are scalar positive values.

I have tried to apply KKT conditions but I am not able to find a closed-form solution. Is there any way to simplify this problem?

edited Apr 12 '20 at 06:09

Rodrigo de Azevedo

23,223

asked Apr 11 '20 at 19:09

Leonardo Massai

1,943

1

Using the Schur complement, you can rewrite the quadratic constraints as LMIs. Then, you are left with a (convex) semidefinite program. – Rodrigo de Azevedo Apr 12 '20 at 06:10
Could you show me how? – Leonardo Massai Apr 12 '20 at 19:20
Take a look at this. However, if you use something like CVX, you do not need to worry about such things. Assuming a numerical solution is fine. – Rodrigo de Azevedo Apr 12 '20 at 19:24
I have already solved the problem with CVX. I was actually hoping to get some analytical results somehow but maybe it is not possible. – Leonardo Massai Apr 12 '20 at 19:47

Minimizing quadratic form with quadratic and linear constraints

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