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This is an overview of most common numerical optimization problems. Some definitions are accompanied with usage example from OpenOpt framework, and, maybe, with some more info; some others have no OpenOpt-connected solvers yet. Also, you can view it a little bit classified:

MatrixProblems NonLinearProblems NetworkProblems OptimalControlProblems(empty yet)


Matrix Problems Group


\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\}
\frac{1}{2}\mathbf{x}^T\mathbf{Hx} + \mathbf{f}^T \mathbf{x \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}
\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{f^T x \to min}
subjected to
\mathbf{lb \le x \le ub}
\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

(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{\| 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}


NonLinear Problems Group


\mathbf {f(x) \to min,\ max}
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{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{\forall j=0,...,J: h_j(x) = 0}
\mathbf{ \{ {f, c_i, h_j :R^n \to R \} \subset C^1}}
(smooth differentiable functions)
\mathbf{x \in R^n}


\mathbf {f(x) \to min,\ max}
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{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{\forall j=0,...,J: h_j(x) = 0}
\mathbf{ \{ {f, c_i, h_j :R^n \to R \} \subset C^0}}
(continuous functions,
sometimes with some numerical noise)
\mathbf{x \in R^n}
\mathbf {f(x) \to min,\ max\ \ } (global)
subjected to
\mathbf{lb} \le \mathbf{x} \le \mathbf{ub}
\mathbf{A x} \le \mathbf{b}
\mathbf{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{ f, c_i :R^n \to R }
\mathbf{x \in R^n}
Solve set of non-linear equations
\mathbf{F(x)=0}
subjected to
\mathbf{lb} \le \mathbf{x} \le \mathbf{ub}
\mathbf{A x} \le \mathbf{b}
\mathbf{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{F: R^n \to R^n}
\mathbf{x \in R^n}
\mathbf {\sum_{k=0}^{K} f_k(x)^2 \to 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{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{\forall j=0,...,J: h_j(x) = 0}
\mathbf{ \{ {f_k, c_i, h_j :R^n \to R \} \subset C^1}}
(smooth differentiable functions)
\mathbf{x \in R^n}
\mathbf {\sum_{k=0}^{K} \|f(x, X_k)-Y_k\| ^2 \to 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{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{\forall j=0,...,J: h_j(x) = 0}
\mathbf{f, c_i, h_j :R^n \to R}
\mathbf{x \in R^n,\ X_k \in R^m, Y_k \in R^s}
\mathbf{max_{k=0,...,K}\{f_k(x)\} \to 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{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{\forall j=0,...,J: h_j(x) = 0}
\mathbf{ \{ {f, c_i, h_j :R^n \to R \} \subset C^1}}
(smooth differentiable functions)
\mathbf{x \in R^n}
\mathbf {f(x) \to min,\ max}
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{\forall i=0,...,I: c_i(x) \le 0}
\mathbf{\forall j=0,...,J: h_j(x) = 0}
\mathbf{\forall k \in \{k_1,k_2,...k_m\}: x_k \in S_k}
\mathbf{S_k\ is\ a\ set\ of\ values\ from\ R}
\mathbf{ \{ {f, c_i, h_j :R^n \to R \} \subset C^1}}
(smooth differentiable functions)
\mathbf{x \in R^n}
Retrieved from "http://openopt.org/Problems"
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