# Minimizing the ratio of two specific non negative quadratic convex functions

$$F$$ is $$m\times m$$ diagonal, with real non negative elements

$$D$$ is $$n \times m$$ complex

$$P$$ is $$n \times 1$$ complex

$$A$$ is $$m \times 1$$ complex.

Minimize $$\Gamma(A)$$, with respect to $$A$$.

$$\Gamma(A) = \frac{m^2(DA-P)^H (DA-P) + (FA)^H(FA)}{A^HA}$$

It is known that both numerator and denominator of $$\Gamma(A)$$ are convex and non negative. Also both the terms of the numerator are individually convex and non negative.

Question: An numerical optimization algorithm to find global minimum.

Apart from a reagular solution, I am also interested in a gradient descent based method if possible as the matrices are large. Also $$m >> n$$.

PS : This question is a specific version of this question.

EDIT : more known information

No constraints on problem but

1. $$\sum P = 0$$, I mean sum of elements of matrix $$P$$ is zero.
2. Diagonal elements of $$F$$ are not all zeros.
3. Also $$P^HP \ne 0$$.
4. Rows of $$D$$ are orthogonal to each other. Also they are linearly independent.
• What is your question? – nicoguaro Feb 24 '19 at 3:47
• @nicoguaro An numerical optimization algorithm to find global minimum. – Rajesh D Feb 24 '19 at 4:30
• it can also be a reference request, if a numerical scheme already exists. – Rajesh D Feb 24 '19 at 4:55
• If your objective function is not convex you don't have guaranteed to attain the global minimum. – nicoguaro Feb 24 '19 at 13:50
• @RajeshDachiraju: This problem has a nice solution if you don't have to divide by $A^HA$. You might want to reflect on whether that's a necessary thing to do. – Richard Feb 25 '19 at 1:07