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I am looking for a way to optimize F(x,y,z) = f(x) + g(x,y,z) + h(x,y,z) with respect to three vectors x,y,z to where f, g, h are given by

  1. f(x) = ||d – Px||^2_2
  2. g(x,y,z) = a*||z – B(y)x ||^2_2
  3. h(x,y,z) = b*||Ax + B(y)u +z||^2_2

Knowns for this problem: vector d, square matrix (diagonal but rank-deficient) P, square and invertible matrix A, and balancing scalars a and b

Question How can I solve this optimization problem through alternating approach?

As far as I know, ADMM cannot be used because x,y,z are not separable. I am wondering if I can compute gradient w.r.t x, y, z separately by fixing the others and update alternately.

FYI B(y) is a matrix which depends on y.

Thank you.

GoodGame
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Apr 18 '23 at 22:03

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