1
vote
2answers
34 views

Convex set, quadratic form

I'm trying to answer a question concerning convex sets "Does the following constraint system define a convex set? $x^T Qx ≤ 1$ $a^T x = 0$ Here, Q is a symmetric and positive definite matrix and a ...
3
votes
2answers
312 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
86 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
41 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: ...
3
votes
1answer
429 views

Why does positive semi-definiteness in this inequality imply a convex set?

I was reading a proof that rewrote an inequality in the form: $$b^Tx +x^T A x \le \alpha$$ for $b,x \in \mathbb{R}^n$ and $\alpha \in \mathbb{R}$, and with $A$ positive semidefinite. It then ...
0
votes
1answer
142 views

General quadratic form of two variables

I was referring to this lecture http://www.stanford.edu/class/ee364a/videos/video04.html. and he gave an example of a generalized quadratic equation ...