A modified Nelder-Mead barrier method for constrained optimization

Figure 1.  The current polytope $ x_0 $, $ x_1 $, and $ x_2 $ is shown in solid lines. The modified Nelder Mead sub-algorithm first tries to replace the worst polytope point $ x_2 $ with one of $ x_{ic} $, $ x_{oc} $, $ x_r $ or $ x_e $. These four points lie on the dashed line through $ x_2 $ and the centroid $ \bar{x} $ of the remaining polytope points. If this fails to give sufficient descent, the fictitious polytope $ y_1 $, $ y_2 $, $ H $ is used, where the fictitious polytope's worst point $ H $ is reflected through the centroid $ \bar{y} $ of that polytope's remaining points, giving a fictitous new low point $ x_0 $. The expand step is then tried from this fictitious polytope, giving the pseudo-expand point $ x_{pe} $. The points $ y_1,\ldots,y_n $ ($ y_1 $ and $ y_2 $ here) and $ x_{pe} $ form a frame around $ x_0 $

Figure 2.  Data profiles for PENMECO and V-ORTHOMADS on 100 randomly generated cones test functions from [24], in 12, 18, and 24 dimensions with 11 constraints. Each curve gives the number of problems solved to an accuracy $ E_f < 10^{-4} $ within $ N $ simplex gradients, which is $ N(n+1) $ evaluations of $ f $