# MatrixProblems

Matrix Problems Group

 $\mathbf{A} \mathbf{x} = \mathbf{b}$ $\mathbf{f^T x \to min,\ max}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A}_\mathbf{eq} \mathbf{x} = \mathbf{b}_\mathbf{eq}$ $\mathbf{f^T x \to min,\ max}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A}_\mathbf{eq} \mathbf{x} = \mathbf{b}_\mathbf{eq}$ $\forall \mathbf{i} \in \mathbf{intVars}: \mathbf{x_i} \in \mathbf{Z}$ $\forall \mathbf{j} \in \mathbf{boolVars}: \mathbf{x_j} \in \{0,1\}$ $\mathbf{\frac{1}{2} x^T Hx + f^T x \rightarrow min}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A_{eq} x = b_{eq}}$ $\frac{1}{2} \mathbf{\| C x - d \|^2} + \frac{1}{2} \mathbf{\mu \| x - \widehat{x} \|^2 \rightarrow min}$ subjected to $\mathbf{lb} \le \mathbf{x} \le \mathbf{ub}$ $\mathbf{A x} \le \mathbf{b}$ $\mathbf{A}_\mathbf{eq} \mathbf{x} = \mathbf{b}_\mathbf{eq}$ $\mathbf{\frac{1}{2} x^T Hx + f^T x \rightarrow min}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A_{eq} x = b_{eq}}$ $\mathbf{\forall i = 0...I: \frac{1}{2}x^T Q_i x + p_i ^T x + s_i \le 0 }$ $\mathbf{f^T x \to min}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A} \mathbf{x} \le \mathbf{b}$ $\mathbf{A}_\mathbf{eq} \mathbf{x} = \mathbf{b}_\mathbf{eq}$ $\mathbf{\forall i = 0,\dots,I: \lVert C_i x + d_i \rVert_2 \leq q_i^T x + s_i}$ $x,\ f \in \mathbb{R}^n$ $C_i \in \mathbb{R}^{{m_i}\times n}, \ d_i \in \mathbb{R}^{m_i}$ $q_i \in \mathbb{R}^n, \ s_i \in \mathbb{R}$ $A_{eq} \in \mathbb{R}^{p_{eq}\times n}, \ b_{eq} \in \mathbb{R}^{p_{eq}}$ $\mathbf{f^T x \to min}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A}_\mathbf{eq} \mathbf{x} = \mathbf{b}_\mathbf{eq}$ $\mathbf{\forall i = 0,...,I: \sum_{j=0}^{n-1} S^{ij} x_j \le d^i}$ (matrix componentwise inequalities) $\mathbf{x \in R^n;\ S^{ij}, d^i \in R^{m_i \times m_i}}$ $\mathbf{i = 0,...,I;\ j = 0,...,n-1}$ $\mathbf{S^{ij}}$are positive semidefinite matrices $\mathbf{\frac{1}{2} x^T Hx + f^T x \rightarrow min}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A_{eq} x = b_{eq}}$ $\mathbf{\forall i \in intVars: x_i \in N}$ $\mathbf{\frac{1}{2} x^T Hx + f^T x \rightarrow min}$ subjected to $\mathbf{lb \le x \le ub}$ $\mathbf{A x \le b}$ $\mathbf{A_{eq} x = b_{eq}}$ $\mathbf{\forall i = 0...I: \frac{1}{2}x^T Q_i x + p_i ^T x + s_i \le 0 }$ $\mathbf{\forall j \in intVars: x_j \in N}$ (aka LLADP - Linear Least Absolute Deviation Problem) $\mathbf{\| C x - d \|_1} + \mathbf{\mu \| x - \widehat{x} \|_1 \rightarrow min}$ subjected to $\mathbf{lb} \le \mathbf{x} \le \mathbf{ub}$ $\mathbf{find \ w, z: w = M z + q}$ subjected to $\mathbf{M \in R^{n \times n}, q \in R^n}$ $\mathbf{w \in R^n, w \ge 0 }$ $\mathbf{z \in R^n,z \ge 0}$ $\mathbf{w^T z = 0}$ $\mathbf{\| C x - d \|_{\infty} (= max |C x - d|) \rightarrow min}$ subjected to $\mathbf{lb} \le \mathbf{x} \le \mathbf{ub}$ $\mathbf{A x} \le \mathbf{b}$ $\mathbf{A}_\mathbf{eq} \mathbf{x} = \mathbf{b}_\mathbf{eq}$ search for $\mathbf{ \lambda \in C, x \in C^n}$: $\mathbf{A x = \lambda x}$ (A has to be square matrix)
##### Latest OOSuite 0.38

from 2012-03-15

2012-06-15

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