MatrixProblems

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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)
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