criticalIdeal -- ideal of critical equations of a semidefinite program

Synopsis

Usage:

(I,X,y,Z) = criticalIdeal(P)

(I,X,y,Z) = criticalIdeal(P,rk)
Inputs:
- P, an instance of the type SDP, a semidefinite programming problem
- rk, an integer, the rank of the primal matrix (optional)
Outputs:
- I, an ideal, of critical equations
- X, an $n\times n$ matrix, primal variable
- y, an $m\times 1$ matrix, dual variable
- Z, an $n\times n$ matrix, dual variable

This method computes the ideal of critical equations of an SDP. This ideal can be used to solve the SDP symbolically.

i1 : A = (-matrix{{0,1,0},{1,0,0},{0,0,1}}, matrix{{0,0,1},{0,0,0},{1,0,0}}, -matrix{{0,0,0},{0,0,1},{0,1,0}});

i2 : (C, b) = (matrix{{1/1,0,3},{0,5,0},{3,0,9}}, matrix{{-1},{-1},{-1}});

i3 : P = sdp(C,A,b);

i4 : (I,X,y,Z) = criticalIdeal P;

i5 : degree I

o5 = 6

We can restrict the rank of the primal matrix X.

i6 : rk = 1;

i7 : (J,X,y,Z) = criticalIdeal(P, rk);

i8 : degree J

o8 = 4