I have this problem \begin{align} \min_{\alpha,\beta,X}~&<\alpha \cdot X+\beta \cdot Y,D>-c \cdot (<\alpha \cdot X+\beta \cdot Y,H>)^{1/2}\\ &X,\alpha,\beta>=0\ \end{align} Where X,Y,D,H are semi-definite matrices. How can I solve this problem efficiently? If we assume that we know the variable X, i.e. the above problem is
\begin{align} \min_{\alpha,\beta}~&<\alpha \cdot X+\beta \cdot Y,D>-c \cdot (<\alpha \cdot X+\beta \cdot Y,H>)^{1/2}\\ &\alpha,\beta>=0\ \end{align} How to solve it efficiently? Is it possible to find a close form for the answer?
Your problem boils down to a form $$ \min_{\alpha, \beta} \;\alpha \cdot c_{1} + \beta \cdot c_{2} - c \cdot \sqrt{(\alpha \cdot c_{3} + \beta \cdot c_{4})}, $$ where $c_{1}, c_{2}, c_{3}, c_{4} \ge 0$ (by the positive semidefiniteness of the involved matrices).
The square root is a concave function over its domain (here its argument is guaranteed to be nonnegative). Hence, the entire objective is convex in $[\alpha, \beta]^{T}$. It can be solved efficiently.