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NonLinearProblems

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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{\forall j=0,...,J: h_j(x) = 0}
\mathbf{ f, c_i, h_j :R^n \to R }
\mathbf{x \in R^n}
Solve system 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}

\begin{align} \min_{x} &\left[\mu_1(x), \mu_2(x),\dots, \mu_n(x) \right]^T & \\ \text{subjected to} & \\ g(x) & \le 0 \\ h(x) & = 0 \\ x_l \le & x \le x_u \\ x \in & R^n \end{align}

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