Appendix

This chapter has details on some of SACOBRA_Py’s data structures.

With

cobra = CobraInitializer(...)

we generate the main data structure cobra of SACOBRA_Py (see CobraInitializer). Some of its data frame and dictionary elements are described below.

cobra.sac_opts

SACoptions cobra.sac_opts holds the collection of all parameters (options) for SACOBRA_Py.

cobra.sac_res

cobra.sac_res is a dictionary which holds SACOBRA_Py’s results from the initialization phase and from the optimization phase.

cobraInit.CobraInitializer.get_sac_res(self)

Return dictionary sac_res with the following elements, accessible with e.g. sac_res['fn']:

  • fn: function returning (1+nConstraints)-dim vector: [objective to minimize, constraints], optionally rescaled

  • originalfn: the same, but before rescaling

  • lower: lower-bound vector with dimension \(d\), optionally rescaled

  • originalL: the same, but before rescaling

  • upper: upper-bound vector with dimension \(d\), optionally rescaled

  • originalU: the same, but before rescaling

  • x0: initial start point, either given with the problem formulation or a random point, optionally resacled

  • originalx0: the same, but before rescaling

  • xStart: start point for each sequential optimization: initially x0, but in later iterations replaced by xbest or random start

  • dimension: input space dimension \(d\)

  • nConstraints: number of constraints

  • is_equ: boolean vector of size nConstraints: True if n’th constraint is equality constraint

  • A: matrix (p2.num, dimension) holding the initial design + all infill points

  • Fres: vector of size p2.num: objective value for each point in A

  • Gres: matrix (p2.num, nConstraints): constraint values for each point in A

  • predC: matrix (p2.num, nConstraints): predicted constraint values for each point in A. The prediction of the current infill point is made with the current constraint surrogates which are formed from the p2.num - 1 previous points.

  • muVec: vector of size p2.num: \(\mu\) in the i’th iteration

  • numViol, trueNumViol, maxViol, trueMaxViol: same as columns nViolations, trueNViol, maxViolation, trueMaxViol of cobra.df

  • fbestArray: vector of size p2.num: best feasible objective value in the i’th iteration

  • …

p2.num counts the number of real function evaluations, see Phase2Vars p2.

Returns:

SACOBRA results

Return type:

dict

cobra.df

cobra.df is a data frame which holds diagnostic information about the optimization process. df is only for diagnostics, its elements are not used by the optimization process in any form.

cobraPhaseII.CobraPhaseII.get_df(self) → DataFrame

Return data frame cobra.df with the following elements, accessible with e.g. cobra.df['iter']. Data frame cobra.df contains feval rows, one for each true function evaluation.

The contents of a specific row of cobra.df holds the results of a specific iteration (current iteration):

  • iter: the iteration number

  • y: the objective function value at the current infill point

  • predY: the fitness surrogate value at the current infill point

  • predSolu: the fitness surrogate value at the true solution (if provided, else none)

  • feasible: is the current infill point feasible on the true objective?

  • feasPred: is the current infill point predicted to be feasible by the surrogate models?

  • nViolations: the number of violations in the constraint surrogates’ prediction at the infill point

  • trueNViol: the number of violations in the true constraints at the infill point

  • maxViolation: the maximum violation of the constraint surrogates’ prediction at the infill point

  • trueMaxViol: the maximum violation of the true constraints at the infill point

  • feMax: the number of iterates that the sequential optimizer took when producing this infill point

  • fBest: the all-time best feasible objective value. As long as no feasible point is found, the fitness of the one with the least maximum violation

  • dist: distance of the true solution to infill point, in rescaled space. Minimum distance for multiple solu’s, None if no solu is provided

  • distOrig: the same, but in original space

  • RS: True if it is an iteration with a random start

  • XI: DRC for this iteration

  • optimConv: convergence of sequential optimization

  • optimTime: time of this sequential optimization

  • optimizer: optimization algorithm

  • seed: `sac_opts.cobraSeed

  • …

Returns:

SACOBRA diagnostic information

Return type:

pd.DataFrame

cobra.df2

cobra.df2 is a data frame which holds diagnostic information about the optimization process. df2 is only for diagnostics, its elements are not used by the optimization process in any form.

cobraPhaseII.CobraPhaseII.get_df2(self) → DataFrame

Return data frame cobra.df2 with the following elements, accessible with e.g. cobra.df2['iter']. Data frame cobra.df2 contains feval - initDesPoints rows, one for each true function evaluation in phase II.

The contents of a specific row of cobra.df2 holds the results of a specific iteration, the current iteration:

  • iter: the iteration number

  • predY: the fitness surrogate value at the current infill point

  • predVal: surrogate fitness + penalty at xNew

  • predSolu: the fitness surrogate value at the true solution (if provided, else none)

  • predSoluPenal: surrogate fitness + penalty at the true solu (only diagnostics)

  • sigmaD: …

  • penaF: …

  • XI: DRC for this iteration

  • rho: smoothing factor \(\rho\) (interpolating / approximating RBFs)

  • fBest: the all-time best feasible objective value. As long as no feasible point is found, the fitness of the one with the least maximum violation

  • EPS: the current safety margin EPS in constraint surrogates

  • muVec: artificial equality constraint enlargement \(\mu\)

  • PLOG: whether \(plog\) transformation is used in the current iteration (see p-Effect)

  • pshift: shift used in \(plog\) transformation (see p-Effect)

  • pEffect: the p-Effect value

  • state: name of iteration state [‘initialized’ | ‘optimized’ | ‘refined’]

Only if the COP contains equality constraints and if equality handling is active (EQUoptions EQU.active==True), then the following row elements are not None, instead they contain these attributes evaluated for the current infill point:

  • nv_cB: number of (artificial) constraint violations (> conTol) before refine on surrogates

  • nv_cA: number of (artificial) constraint violations (> conTol) after refine on surrogates

  • nv_tB: number of (artificial) constraint violations (> conTol) before refine on true constraints

  • nv_tA: number of (artificial) constraint violations (> conTol) after refine on true constraints

Returns:

SACOBRA diagnostic information

Return type:

pd.DataFrame