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gusmaogabriels authored Jul 1, 2017
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Expand Up @@ -323,7 +323,10 @@ Successive α-domain subsectioning following the golden-ratio.
### Non-linearly Constrained Optimization (`.constrained`)
#### fmincon (`.fminnlcon`)
[*fminnlcon*](#fminnlcon-fminnlcon) makes use of [*fminunc*](#fminunc-fminunc) routines within linearly and/or non-linearly constrained optimization domains. In essence, it all boils down to modifing the objective function to include the constraints weighted by a increasing or decreasing factor so that, when convergence is achieved, the constraint parcel of the objective function tends to zero. The increase in the *c* parameter, which weights the constraints, is given by a *beta* factor.
Since such approach relies on succesive unconstrained optimization and the solution, depending on the algorithm, must always lie within or outside the feasible set, the increase/decrease (*beta* factor) in the constraints weight should be done in a way that on one hand it is not too litle, so that the iterative process would take forever, and, on the other hand, it cannot be too large for then a succeeding solution might hop to the other side of the feasibility frontier, when constraint weighting function become numerically inconsistent.

Since such approach relies on succesive unconstrained optimization and the solution, depending on the algorithm, must always lie within or outside the feasible set, the increase/decrease (*beta* factor) in the constraints weight should be done in a way that on one hand it is not too litle, so that the iterative process would take forever, and, on the other hand, it cannot be too large for then a succeeding solution might hop to the other side of the feasibility frontier, when constraint weighting function become numerically inconsistent.

The **unconstrained optimization method that is meployed within inner optimization steps is the one defined in** [`.nonlinear.params`](#parameters-params). The converge phase will depend on the suitability of the unconstrained optimization method both to the plain objective function, f(*x*), and the its combination with the weighted constraint parcel, *P*(*x*) or *B*(*x*).

- ##### Penalty (`method='penalty'`)
The penalty algorithm starts of from an initially infeasible point with a function of the following shape:
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