Semidefinite relaxations of quadratic constraint


I am trying to solve an optimization problem with 174 quadratic constraints of the form
x_{1,2}(k)^TQx_{1,2}(k) +c^Tx_3(k) \leq 0, where Q is symmetric positive definite and the subscripts 1,2 are meant to indicate that at each time step k only variables 1 and 2 are included in this constraint. Currently I enforce these constraints by looping over all time steps.

I would like to try a semidefinite relaxation of these constraints which is defined as: \text{Tr}\ QX(k) + c^Tx_3(k) \leq 0, where the condition X(k) = x_{1,2}(k)x_{1,2}(k)^T is relaxed to an inequality and written as \begin{pmatrix}1& x_{1,2}(k)^T \\ x_{1,2}(k)& X(k) \end{pmatrix} \succeq 0 using the Schur complement.

My question is: is it possible to declare a variable X(k) for every timestep k within a loop so that the matrix inequality stated above can be enforced within a loop?

Sure, you can create a 3-D array variable:

variable X(N,N,M) symmetric

Each 2-D slice X(:,:,k) will be symmetric.


Does this hold for symmetric semidefinite matrices? So each 2-D slice X(:,:,k) will be symmetric semidefinite? Thanks in advance!

Yes it does. It is sufficient to declare semidefinite, since that implies symmetric.

1 Like

Thank you very much Mark! Very Helpful :slight_smile: