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MINLP

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Mixed-Integer Non-Linear Problems (MINLP)

$\mathbf {f(x) \to min,\ max}$

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)

$\mathbf{x \in R^n}$

OpenOpt MINLP example
FuncDesigner MINLP examples:
- simple (with automatic differentiation)
- example with specifiable accuracy (via interval analysis by interalg)

MINLP solvers connected to OpenOpt:

Solver	License	Algorithm	Authors	Info
(since v. 0.38) interalg	BSD	an interval algorithm	Dmitrey	searches global extremum with specifiable accuracy fTol: abs(f - f) < fTol , where f is theoretical optimal value. Can handle nonsmooth, multiextremum and even some discontinuous funcs like ceil, floor.
branb	BSD	branch-and-bound	Ingar Solberg, Institutt for teknisk kybernetikk, Norges Tekniske Hrgskole, Norway	This is translation of MATLABs fminconset routine to Python (by Dmitrey, with some minor changes). It recuires parameter "nlpSolver" like "ipopt', "scipy_lbfgsb", "scipy_tnc" etc for solving NLP subproblems.

Note!

MINLP is NP-Hard class of problems.
Please pay attention: objective function and non-linear constraints should be defined in whole R^nVars, i.e. for discrete variables as well as for continuous.
For branb It is assumed that objective function and constraints are at least unimodal (convex is preferred).