CGO arbfMIP
This page is part of the CGO Manual. See CGO Manual. |
Purpose
Solve general constrained mixed-integer global black-box optimization problems with costly objective functions.
The optimization problem is of the following form
where ; ;
the linear constraints are defined by , ;
and the nonlinear constraints are defined by .
The variables are restricted to be integers,
where is an index subset of possibly empty.
It is assumed that the function is continuous with respect to all
variables, even if there is a demand that some variables only take integer values.
Otherwise it would not make sense to do the surrogate modeling of used by all CGO solvers.
f (x) is assumed to be a costly function while c(x) is assumed to be cheaply computed. Any costly constraints can be treated by adding penalty terms to the objective function in the following way:
where weighting parameters wj have been added. The user then returns p(x) instead of f (x) to the CGO solver.
Calling Syntax
Result = arbfMIP(Prob,varargin) Result = tomRun('arbfMIP', Prob);
Description of Inputs
Problem structure
The following fields are used in the problem description structure Prob:
Input | Description |
---|---|
Name | See Common input for all CGO solvers |
FUNCS.f | |
FUNCS.c | |
x_L | |
x_U | |
b_L | |
b_U | |
A | |
c_L | |
c_U | |
WarmStart | |
MaxCPU | |
user | |
PriLevOpt | |
f_Low | |
optParam | |
CGO | See the table below but also this table for input common to all CGO solvers |
GO | See common input for all CGO solvers |
MIP | See common input for all CGO solvers |
varargin | Additional parameters to arbfmip are sent to the costly f(x) |
- Special ARBF algorithm parameters in Prob.CGO - | |||||||||||||||
rbfType | Selects type of radial basis function
| ||||||||||||||
infStep | If =1, add search step with target value -inf first in cycle. Default 0 | ||||||||||||||
TargetMin | Which minimum of several to pick in target value problem:
Default is TargetMin = 3. | ||||||||||||||
fStarRule | Global-Local search strategy. N = cycle length. Define min_sn as the global minimum on surface.
Strategy names in Gutmanns thesis: III, II, I | ||||||||||||||
DeltaRule | 1 = Skip large f(x) when computing f(x) interval Delta. 0 = Use all points. If objType > 0, default DeltaRule = 0, otherwise default is 1. | ||||||||||||||
eps_sn | Relative tolerance used to test if the minimum of surface, min_sn, is sufficiently lower than the best point (fMin) found. Default is eps_sn = 10-7. |
Description of Outputs
Result structure
The output structure Result contains results from the optimization.
The following fields are set:
Field | Description | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
x_k | See Common output for all CGO solvers for details. | ||||||||||||||||
f_k | |||||||||||||||||
Iter | |||||||||||||||||
FuncEv | |||||||||||||||||
ExitText | |||||||||||||||||
ExitFlag | Always 0 | ||||||||||||||||
Inform | Information parameter.
| ||||||||||||||||
CGO | Subfield WarmStartInfo saves warm start information, the same information as in cgoSave.mat, see Common output for all CGO solvers#WSInfo. |
Output printing
PRINTING in MATLAB window in Iteration 0 after Experimental Design | |
---|---|
If IterPrint >= 1 or PriLev > 1 | |
Row 1 | |
Iter | Number of iterations |
n | Number of trial x, n-Iter is number of points in initial design |
nFunc | Number of costly f(x) computed, nFunc <= n, n-nFunc = rejected points |
--->> | Time stamp (date and exact time of this printout) |
Cycle | Cycle steps global to local. infStep is marked -1, 0 to N-1 are global steps. Last step N in cycle is surface minimum |
R | If the letter R is printed, the current step is a RESCUE step, i.e. the new point is already sampled in a previous step, instead the surface minimum is used as a rescue |
fnStar | Target value fn_star (if set) |
fGoal | Goal value (if set) |
fMin | Best f(x) found so far. E.g. at 27/It 12 means n=27, Iter=12 fMinI means the best f(x) is infeasible fMinF means the best f(x) is feasible (also integer feasible) |
Row 2 | |
max(F) | Maximum of all f(x) in the initial set of points X |
med(X) | Median of all f(x) in the initial set of points X |
rng(F) | maxF-fMin, the range of f(x) values in the initial set X |
pDist | The size of the simply bounded region, ||x_U-x_L||2 |
LipU | Maximum Lipschitz constant for initial set X |
LipUFt | Maximum Lipschitz constant for initial set X, using transform F |
objType | Function transformation used during run, one of 0 to 8. |
Row 3 | |
xMin | Best point in initial set X |
Row 4 | |
xOptS | User-given global optimum Prob.x_opt (if defined) |
If PriLev > 2 and global optimum xOptS known and given in Prob.x_opt | |
Row 5 | |
SumXO | Sum of distances from global optimum xOptS to all sampled points X in experimental design |
MeanXO | Mean of distances from global optimum xOptS to all sampled points X in experimental design |
doO | Distance from global optimum xOptS to closest point of sampled points X in experimental design |
If PriLev > 3 | |
dXO | Minimal distance from global optimum xOptS to closest point of all sampled points X in experimental design |
snOptf | Surface value at xOptS, sn_f(xOptS) |
snOptg | Surface gradient at xOptS, sn_g(xOptS) |
snOptE | Sum of negative eigenvalues of surface Hessian at xOptS, sum((eig(sn_H(xOptS)) < 0)) |
PRINTING in MATLAB window in Iteration 1,2, ... | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
if IterPrint >= 1 or PriLev > 1 | |||||||||||||
Row 1 | |||||||||||||
Iter | Number of iterations | ||||||||||||
n | Number of trial x, n-Iter is number of points in initial design | ||||||||||||
nFunc | Number of costly f(x) computed, nFunc <= n, n-nFunc = rejected pnts | ||||||||||||
--->> | Time stamp (date and exact time of this printout) | ||||||||||||
fGoal | Goal value (if set) | ||||||||||||
fMin | Best f(x) found so far. E.g. at 27/It 12 means n=27, Iter=12 fMinI means the best f(x) is infeasible fMinF means the best f(x) is feasible (also integer feasible) IT implies reduction in last step, It no reduction last step | ||||||||||||
Row 2 Header line | |||||||||||||
| |||||||||||||
Row 3 to m+2 with m new sample points xNew(1,1:m) obtained in last iteration | |||||||||||||
# | ith new point x = xNew(:,i) | ||||||||||||
f(x) | Costly f(x) value at x | ||||||||||||
Task | Which method that gave the new point
| ||||||||||||
onB | Number of coordinates on bound for new point, onB(x) | ||||||||||||
fnStar | Target value used obtaining new point | ||||||||||||
doX | Minimal distance from x to sample set X, min||x-X|| | ||||||||||||
doM | Distance from x to (xMin,fMin), best point found, min||x-xMin|| | ||||||||||||
doS | Distance from x to minimum on surface, min||x-min_sn_y|| | ||||||||||||
surfErr | Error between predicted and actual value of f(x), i.e. Costly f(x) - Surface value at x | ||||||||||||
f-Reduc | Function value reduction if fNew < fMin | ||||||||||||
doO | Distance from x to global optimum xOptS ||x-xOptS|| (if Prob.x_opt specified) | ||||||||||||
ln10(my) | Coefficient my in RBF interpolation | ||||||||||||
x: | x values for i:th point, scaled back i SCALE == 1 | ||||||||||||
Row m+3 with values for current global surface minimum (min_sn_y, min_sn) | |||||||||||||
Sn | f(x) = min_sn, onB, doX, doM, doO values for min_sn_y, and x: are values for min_sn_y transformed back to original coordinates if SCALE == 1 | ||||||||||||
NOTE: All distances measured are in SCALED space [0,1]d, if SCALE' == 1 |
Description
arbfMIP implements the Adaptive Radial Basis Function (ARBF) algorithm. The ARBF method handles linear equality and inequality constraints, and nonlinear equality and inequality constraints, as well as mixed-integer problems.
M-files Used
daceInit.m, iniSolve.m, endSolve.m, conAssign.m, glcAssign.m, snSolve.m, gnSolve.m, expDesign.m.
MEX-files Used
tomsol
See Also
rbfSolve.m and ego.m
Warnings
Observe that when cancelling with CTRL+C during a run, some memory allocated by arbfMIP will not be deallocated. To deallocate, do:
''>> ''clear cgolib