Common input for all CGO solvers

Input parameters

Prob structure

The following fields are used in the problem description structure Prob
Field	Description
Name	Name of the problem. Used for security when doing warm starts.
FUNCS.f	The routine to compute the function, given as a string, e.g. CGOF.
FUNCS.c	The routine to compute the nonlinear constraint, e.g. CGOC.
x_L	Lower bounds for each element in x. Must be finite.
x_U	Upper bounds for each element in x. Must be finite.
b_L	Lower bounds for the linear constraints.
b_U	Upper bounds for the linear constraints.
A	Linear constraint matrix.
c_L	Lower bounds for the nonlinear constraints.
c_U	Upper bounds for the nonlinear constraints.
WarmStart	If true, >0, the solver reads the output from the last run from the mat-file cgoSave.mat, and continues from the last run. If Prob.CGO.WarmStartInfo has been defined through a call to WarmDefGLOBAL, this field is used instead of the cgoSave.mat file. In the last case, exactly the same points are used. In the first case, a new test is made which points to use.
MaxCPU	Maximal CPU Time (in seconds) to be used.
user	User field used to send information to low-level functions.
PriLevOpt	Print Level. value output 0 Silent 1 Summary 2 Printing each iteration 3 Info about local / global solution 4 Progress in x.
PriLevSub	Print Level in subproblem solvers, see help in snSolve and gnSolve.
f_Low	Lower bound on the optimal function value. If defined, used to restrict the target values into interval ['f_Low,min(surface)].
optParam	Structure with optimization parameters. See #optParam structure.

optParam structure

fields used in Prob.optParam:
Field	Description
MaxFunc	Maximal number of costly function evaluations, default 300 for rbfSolve and arbfMIP, and default 200 for ego. MaxFunc must be <= 5000. If WarmStart = 1 and MaxFunc <= nFunc (Number of f(x) used) then set MaxFunc := MaxFunc + nFunc.
IterPrint	Print one information line each iteration, and the new x tried. Default IterPrint = 1. fMinI means the best f(x) is infeasible. fMinF means the best f(x) is feasible (also integer feasible).
fGoal	Goal for function value, not used if inf or empty.
eps_f	Relative accuracy for function value, fTol == eps_f. Stop if |f - f Goal| <= |fGoal| * fTol, if fGoal ≠ 0. Stop if |f - fGoal| <= fTol, if fGoal = 0. See the output field maxTri.
bTol	Linear constraint tolerance.
cTol	Nonlinear constraint tolerance.
MaxIter	Maximal number of iterations used in the local optimization on the re- sponse surface in each step. Default 1000, except for pure IP problems, then max(GO.MaxFunc, MaxIter);.
CGO	Structure (Prob.CGO) with parameters concerning global optimization options.
Percent	Type of strategy to get the initial sampled values: Percent Experimental Design ExD Corner strategies 900 All Corners 1 997 xL + xU + adjacent corners 2 998 xU + adjacent corners 3 999 xL + adjacent corners 4 Deterministic Strategies|| 0 User given initial points 5 94 DIRECT solver glbFast 6 95 DIRECT solver glcFast 6 96 DIRECT solver glbSolve 6 97 DIRECT solver glcSolve 6 98 DIRECT solver glbDirect 6 99 DIRECT solver glcDirect 6 Latin Based Sampling|| 1 Maximin LHD 1-norm 7 2 Maximin LHD 2-norm 8 3 Maximin LHD Inf-norm 9 4 Minimal Audze-Eglais 10 5 Minimax LHD (only 2 dim) 11 6 Latin Hypercube 12 7 Orthogonal Samling 13 Random Strategies (pp in %) 1pp Circle surrounding 14 2pp Ellipsoid surrounding 15 3pp Rectangle surrounding 16 Negative values of Percent result in constrained versions of the experimental design methods 7-16. It means that all points sampled are feasible with respect to all given constraints. For ExD 5,6-12,14-16 user defined points are used.
nSample	Number of sample points to be used in initial experimental design. nSample is used differently dependent on the value of Percent: (n)Sample: ExD < 0 = 0 > 0 [] 1 2d 6 |n| iterations 7-11 d+1 d+1 max(d + 1,n) (d + 1)(d + 2) / 2 12 LATIN(k) 13 |n| 14-16 d + 1 where LATIN = [21 21 33 41 51 65 65] and k = |nSample|. Otherwise nSample as input does not matter. Description of the experimental designs: ExD 1, All Corners. Initial points is the corner points of the box given by Prob.x_L and Prob.x_U. Generates 2d points, which results in too many points when the dimension is high. ExD 2, Lower and Upper Corner point + adjacent points. Initial points are 2 * d + 2 corners: the lower left corner xL and its d adjacent corners xL + (xU(i) - xL (i)) * ei, i = 1, ..., d and the upper right corner xU and its d adjacent corners xU - (xU (i) - xL (i)) * ei, i = 1, ..., d ExD 3. Initial points are the upper right corner xU and its d adjacent corners xU - (xU (i) - xL (i)) * ei , i = 1, ..., d ExD 4. Initial points are the lower left corner xL and its d adjacent corners xL + (xU (i) - xL (i)) * ei , i = 1, ..., d ExD 5. User given initial points, given as a matrix in CGO.X. Each column is one sampled point. If d >= length(Prob.x L), then size(X,1) = d, size(X,2) = d + 1. CGO.F should be defined as empty, or contain a vector of corresponding f (x) values. Any CGO.F value set as NaN will be computed by solver routine. ExD 6. Use determinstic global optimization methods to find the initial design. Current methods available (all DIRECT methods), dependent on the value of Percent: 99 = glcDirect, 98 = glbDirect, 97 = glcSolve, 96 = glbSolve, 95 = glcFast, 94 = glbFast. ExD 7-11. Optimal Latin Hypercube Designs (LHD) with respect to different norms. The following norms and designs are available, dependent on the value of Percent: 1 = Maximin 1-Norm, 2 = Maximin 2-Norm, 3 = Maximin Inf-Norm, 4 = Audze-Eglais Norm, 5 = Minimax 2-Norm. All designs taken from: http://www.spacefillingdesigns.nl/ Constrained versions will try bigger and bigger designs up to M = max(10 * d, nTrial) different designs, stopping when it has found nSample feasible points. ExD 12. Latin hypercube space-filling design. For nSample < 0, k = |nSample| should in principle be the problem dimension. The number of points sampled is: k : 2 3 4 5 6 > 6 Points : 21 33 41 51 65 65 The call made is: X = daceInit(abs(nSample),Prob.x_L,Prob.x_U); Set nSample = [] to get (d+1)*(d+2)/2 sampled points: d : 1 2 3 4 5 6 7 8 9 10 Points : 3 6 10 15 21 28 36 45 55 66 This is a more efficient number of points to use. If CGO.X is nonempty, these points are verified as in ExD 5, and treated as already sampled points. Then nSample additional points are sampled, restricted to be close to the given points. Constrained version of Latin hypercube only keep points that fulfill the linear and nonlinear constraints. The algorithm will try up to M = max(10 * d, nTrial) points, stopping when it has found nSample feasible points (d + 1 points if nSample < 0). ExD 13. Orthogonal Sampling, LH with subspace density demands. ExD 14-16. Random strategies, the |Percent| value gives the percentage size of an ellipsoid, circle or rectangle around the so far sampled points that new points are not allowed in. Range 1%-50%. Recommended values 10% - 20%. If CGO.X is nonempty, these points are verified as in ExD 5, and treated as already sampled points. Then nSample additional points are sampled, restricted to be close to the given points.
X,F,CX	The fields X,F,CX are used to define user given points. ExD = 5 (Percent = 0) needs this information. If ExD == 6-12,14-16 these points are included into the design.
X	A matrix of initial x values. One column for every x value. If ExD == 5, size(X,2) >= dim(x)+1 needed.
F	A vector of initial f (x) values. If any element is set to NaN it will be computed.
CX	Optionally a matrix of nonlinear constraint c(x) values. If nonempty, then size(CX,2) == size(X,2). If any element is set as NaN, the vector c(x) = CX(:,i) will be recomputed.
RandState	If >= 0, rand('state', RandState) is set to initialize the pseudo-random generator. If < 0, rand('state', 100 * clock) is set to give a new set of random values each run. If isnan(RandState), the random state is not initialized. RandState will influence if a stochastic initial experimental design is applied, see input Percent and nSample. RandState will also influence if using the multiMin solver, but the random state seed is not reset in multiMin. The state of the random generator is saved in the warm start output rngState, and the random generator is reinitialized with this state if warm start is used. Default RandState = 0.
AddMP	If = 1, add the midpoint as extra point in the corner strategies. Default 1 for any corner strategy, i.e. Percent is 900, 997, 998 or 999.
nTrial	For experimental design CLH, the method generates M = max(10 * d, nTrial) trial points, and evaluate them until nSample feasible points are found. In the random designs, nTrial is the maximum number of trial points randomly generated for each new point to sample.
CLHMethod	Different search strategies for finding feasible LH points. First of all, the least infeasible point is added. Then the linear feasible points are considered. If more points are needed still, the nonlinear infeasible points are added. 1 - Take the sampled infeasible points in order. 2 - Take a random sample of the infeasible points. 3 - Use points with lowest constraint error (cErr).
SCALE	0 - Original search space (default if any integer values). 1 - Transform search space to unit cube (default if no integers).
REPLACE	0 - No replacement, default for constrained problems. 1 - Large function values are replaced by the median. > 1 - Large values Z are replaced by new values. The replacement is defined as Z := FMAX + log10(Z - FMAX + 1), where FMAX = 10REPLACE , if min(F ) < 0 and FMAX = 10(ceil(log10(min(F )))+REPLACE), if min(F ) = 0. A new replacement is computed in every iteration, because min(F ) may change. Default REPLACE = 5, if no linear or nonlinear constraints.
LOCAL	0 - No local searches after global search. If RBF surface is inaccurate, might be an advantage. 1 - Local search from best points after global search. If equal best function values, up to 20 local searches are done.
SMOOTH	1 - The problem is smooth enough for local search using numerical gradient estimation methods (default). 0 - The problem is nonsmooth or noisy, and local search methods using numer- ical gradient estimation are likely to produce garbage search directions.
globalSolver	Global optimization solver used for subproblem optimization. Default glcCluster (SMOOTH=1) or glcDirect (SMOOTH=0). If the global- Solver is glcCluster, the fields Prob.GO.maxFunc1, Prob.GO.maxFunc2, Prob.GO.maxFunc3, Prob.GO.localSolver, Prob.GO.DIRECT and other fields set in Prob.GO are used. See the help for these parameters in glcCluster.
localSolver	Local optimization solver used for subproblem optimization. If not defined, the TOMLAB default constrained NLP solver is used. - Special RBF algorithm parameters in Prob.CGO -
rbfType	Type of radial basis function: 1 - thin plate spline; 2 - Cubic Spline (default); 3 - Multiquadric; 4 - Inverse multiquadric; 5 - Gaussian; 6 - Linear.
idea	Type of search strategy on the response surface. idea = 1 - cycle of N+1 points in target value fnStar. if fStarRule =3, then N=1 default, otherwise N=4 default. By default idea =1, fStarRule =1, i.e. N =4. To change N, see below. idea = 2 - cycle of 4 points (N+1, N=3 always) in alpha. alpha is a bound on an algorithmic constraint that implicitly sets a target value fStar.
N	Cycle length in idea 1 (default N=1 for fStarRule 3, otherwise default N=4) or idea 2 (always N=3).
infStep	If =1, add search step with target value -8 first in cycle. Default 0. Always =1 for the case idea =1, fStarRule =3.
fStarRule	Global-Local search strategy in idea 1, where N is the cycle length. Define minsn as the global minimum on the RBF surface. The following strategies for setting the target value fStar is defined: 1: fStar = minsn - ((N - (n - nInit))/N )2 * Δn (Default), 2: fStar = minsn - (N - (n - nInit))/N * Δn . Strategy 1 and 2 depends on Δ n estimate (see DeltaRule). If infStep =1, add $-\infty$-step first in cycle. 3: fStar = $-\infty$-step, minsn-k *0.1*|minsn|k = N, ..., 0. These strategies had the following names in Gutmanns thesis: III, II, I.
DeltaRule	1 = Skip large f(x) when computing f(x) interval ?. 0 = Use all points. Default 1.
AddSurfMin	Add up to AddSurfMin interior local minima on RBF surface as search points, based on estimated Lipschitz constants. AddSurfMin=0 implies no additional minimum added (Default). This option is only possible if globalSolver = multiMin. Test for additional minimum is done in the local step (modN == N) If these additional local minima are used, in the printout modN = -2, -3, -4, ... are the iteration steps with these search points.
TargetMin	Which minimum, if several minima found, to select in the target value problem: =0 Use global minimum. =1 Use best interior local minima, if none use global minimum. =2 Use best interior local minima, if none use RBF interior minimum. =3 Use best minimum with lowest number of coefficients on bounds. Default is TargetMin = 3.
eps_sn	Relative tolerance used to test if the minimum of the RBF surface, minsn , is sufficiently lower than the best point (fM in ) found (default is 10-7 ).
MaxCycle	Max number of cycles without progress before stopping, default 10.
GO	Structure Prob.GO (Default values are set for all fields). The following fields are used:
MaxFunc	Maximal number of function evaluations in each global search.
MaxIter	Maximal number of iterations in each global search.
DIRECT	DIRECT solver used in glcCluster, either glcSolve or glcDirect(default).
maxFunc1	glcCluster parameter, maximum number of function evaluations in the first call. Only used if globalSolver is glcCluster, see help globalSolver.
maxFunc2	glcCluster parameter, maximum number of function evaluations in the second call. Only used if globalSolver is glcCluster, see help globalSolver.
maxFunc3	glcCluster parameter, maximum sum of function evaluations in repeated first calls to DIRECT routine when trying to get feasible. Only used if globalSolver is glcCluster, see help globalSolver.
localSolver	The local solver used by glcCluster. If not defined, then Prob.CGO.localSolver is used
MIP	Structure in Prob, Prob.MIP. Defines integer optimization parameters. Fields used:
IntVars	If empty, all variables are assumed non-integer. If islogical(IntVars) (=all elements are 0/1), then 1 = integer variable, 0 = continuous variable. If any element > 1, IntVars is the indices for integer variables.
varargin	Other parameters directly sent to low level routines.