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Consider the linear programming problem \begin{align} \max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1 \end{align} where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a positive …

Let $x_1,\dots,x_n$ be a set of given vectors in $\mathbb{R}_{+}^d$. Let $c_1,\dots,c_n$ be given positive constants. I am interested in finding the vectors $w_1,\dots,w_n$ in $\mathbb{R}_{+}^d$ that …

I had asked this question in math.se without any success Let $A$ be the symmetric $n\times n$ adjacency matrix for a graph where $A_{ij}$ is the positive edge value between node $i$ and $j$ (thus full …

Let $\mathbf{x}_1, \dots, \mathbf{x}_n \in \mathbb{R}^d$ be $n$ given vectors. Define the function $$ \mathcal{K}(\mathbf{x},\mathbf{y}) := \alpha\exp\left(-\frac{\|\mathbf{x}-\mathbf{y}\|^2}{2\sigma^ …

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as \begin{align} \mathcal{S}=\{\left(\mathbf{u}^H\ma …

I am interested in solving the following equality constrained quadratic (?) problem. \begin{align} \min_{u^{H}u=1}~(u^{H}A_1u) \\\ s.t.~ u^{H}A_2u=0 \end{align} $A_1$ and $A_2$ are $N\times N$ her …

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios …