Least Squares With Multiple Quadratic Constraints

[WCMU101]

Problem:

A = n by m matrix
x = m by 1 vector
y = n by 1 vector

C = c by m matrix
E = e by m matrix

Alpha, gamma and theta are constants.

norm(Ax-y) = min

subject to:

norm(Cx) = alpha
norm(Ex) = gamma
transpose(Cx)*Ex = (alpha^2)*(gamma^2)*cos(theta)

I read a paper on how to do this with 1 quadratic constraint and I got that working, but I can't figure out how to do it with 3. Any advice? (Numerical methods are fine).

Nick.

 

[AlephZero]

You can do this using Lagrange multipliers. See any optimisation textbook, or Google.

This will add 3 more equations and 3 more variables to the "standard" least squares equations. Note, the extended system of equations will be symmetric, but NOT positive definite, so be careful what library routine you use to solve them!