0
votes
1answer
68 views

Rewriting a quadratic Matrix equation as a quadratic vector equation

Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum \begin{align} ...
1
vote
0answers
21 views

Can we express a SPD matrix $S$ in terms of $S^{2}$ in a different manner to solve a convex problem?

I have to find the Symmetric Positive Definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ which has been proven to be convex in the ...
3
votes
2answers
285 views

A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align} ...
1
vote
0answers
73 views

Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
0
votes
2answers
40 views

Convexity of a function and constraint

Consider the quadratic function $f(x_1,x_2,x_3,x_4)=x_1+2x_2+4x_4+x_1^2+5x_2^2+3x_3^2x_4^2-4x_1x_2-2x_2x_3+2x_3x_4$. Is f a convex function? Consider a constraint defined using the above function f: ...