kronecker product in optimization

[Jack1]

I have the following problem: 

min_x  x^T C (x \Kron x)

where C is an (N x N^2) matrix. Clearly the above program is non-convex.

However, I was wondering if I could leverage the work around kronecker products from here? 

Thus, I can replace (x \Kron x) by K(x, X) where X = xx^T and have included the semidefinite restriction X >= xx^T in the overall optimization. 

Hence, the above program, 

min_x x^T C (x \Kron x ) 

becomes 

min_x x^T C K(x, X) s.t. X >= xx^T 

and would like to follow something like the above to tractably address this.  Can I do something like the above substitutions and/or leverage the RLT approach from here in someway? I see that there might be terms like xx^T  (which should be manageable), but, the term xX might be a hassle. Thank you. 

[Jack1]

such that each N x N slice of C is PSD .... can interpret it as a 3D tensor too if stacking each (N x N) slice behind one another. thank you. 

[Johan Löfberg]

Define "can"?. Both models can be thrown into a global solver such as bmibnb, or a moment relaxation in solvemoment

[Jack1]

Thank you - I was trying to look at a cvx re-formulation for this to arrive at an optimal soln. 

Is there a way for me to quantify the optimality gap from the output of the global solver? 

[Johan Löfberg]

P = optimize(Constraints,Objective,sdpsettings('savesolveroutput',1,'solver,'bmibnb'));

P.solveroutput

[Jack1]

Thank you - one quick I had re: obtaining an analytical solution for the below 

min_{a, b} abs(b - a)

s.t. a*P - b*N = T, a > 0, b > 0

where P, N, T are positive numbers. Do I need another constraint in (a, b) to obtain an analytical soln. for this?

[Jack1]

I arrive at 

a - b = \pm T

and am wondering if I need to pick an a and solve for the b subsequently? 

[Johan Löfberg]

the solution depends on N and P

you get a =(T+b*N)/P and then the minimizer in b is given by b=T/(P-N) if P>N. if P=N any feasible solution is a minimizer, and to fix case N>P you do some similiar thing I guess