CGO rbfSolve

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

${\begin{array}{cccccc}\min \limits _{x}&f(x)&&&&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&c_{L}&\leq &c(x)&\leq &c_{U}\\&~x_{j\in \mathbb {N} \ ~~\forall j\in \mathbb {I} },\\\end{array}}$

where $f(x)\in \mathbb {R}$; $x_{L},~x,~x_{U}\in \mathbb {R} ^{d}$; the $m_{1}$ linear constraints are defined by $A\in \mathbb {R} ^{m_{1}\times d}$, $b_{L},~b_{U}\in \mathbb {R} ^{m_{1}}$; and the $m_{2}$ nonlinear constraints are defined by $c_{L},~c(x),~c_{U}\in ~\mathbb {R} ^{m_{2}}$. The variables $x_{I}$ are restricted to be integers, where $\mathbb {I}$ is an index subset of $\{1,\ldots ,d\},$ possibly empty. It is assumed that the function $f(x)$ 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 $f(x)$ 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:

$\min _{x}\quad p(x)=f(x)+\sum _{j}w_{j}\max \left(0,c^{j}(x)-c_{U}^{j},c_{L}^{j}-c^{j}(x)\right),$

where weighting parameters w_j have been added. The user then returns p(x) instead of f (x) to the CGO solver.

Calling Syntax

Result = rbfSolve(Prob,varargin) 
Result = tomRun('rbfSolve', Prob);

Description of Inputs

The following fields are used in the problem description structure Prob:

Field	Description
Name	See Common input for all CGO solvers
FUNCS.f
FUNCS.c
x_L
x_U
b_U
b_L
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 RBF algorithm parameters in Prob.CGO -
rbfType	Selects type of radial basis function Value Type 1 Thin Plate Spline 2 Cubic Spline (default) 3 Multiquadric 4 Inverse multiquadric 5 Gaussian 6 Linear.
infStep	If =1, add search step with target value -inffirst in cycle. Default 0. Always =1 for the case fStartRule == 3
fStarRule	Global-Local search strategy. N = cycle length. Define min_sn as the global minimum on surface. Value fStar target value 1 min_sn - ((N - (n - nInit))/N )2 * Deltan (Default) 2 min_sn - (N - (n - nInit))/N * Deltan. Strategy 1 and 2 depends on Deltan estimate (see DeltaRule). 3 -inf-step, min_sn-k *0.1*|min_sn| k = N,...,0. If infStep true, addition of -inf-step first in cycle.
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.
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 AddSurfMin==1). Only possible if globalSolver = multiMin or glcDirect. Test for additional minimum in local step (modN == N) modN = -2,-3,-4,... are iteration steps with these search points.
TargetMin	Which minimum of several to pick in target value problem: Value Minimum picked 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 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, except 1 = Initial interpolation failed, normally because too huge f(x).
Inform	Information parameter. Value Signification 0 Normal termination. 1 Function value f(x) is less than fGoal. 2 Error in function value f (x), |f - fGoal| <= fTol, fGoal = 0. 3 Relative Error in function value f (x) is less than fTol, i.e. |f - fGoal|/|fGoal| <= fTol. 6 All sample points same as previous point for the last 11 iterations. 7 All feasible integers tried. 9 Max CPU Time reached.
CGO	Subfield WarmStartInfo saves warm start information, the same information as in cgoSave.mat, see Common output for all CGO solvers#WSInfo.

Output printing

IterPrint == 1 or PriLev > 0
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
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
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
-- Additional information in iteration 1,2,...
fNew	f(xNew), the costly function value for point tried in current Iter
RelErr	Relative distance to known or assumed global minimum (Prob.x_opt)
-- Additional information in iteration 0, on next 3 lines
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
LipL	Minimum Lipschitz constant for initial set X
objType	Function transformation used during run, one of 0 to 8.
xMin	Best point in initial set X
xOpt	User-given global optimum Prob.x_opt (if defined)

Description

rbfSolve implements the Radial Basis Function (RBF) algorithm based on the work by Gutmann. The RBF method is enhanced to handle linear equality and inequality constraints, and nonlinear equality and inequality constraints, as well as mixed-integer problems.

A response surface based on radial basis functions is fitted to a collection of sampled points. The algorithm then balances between minimizing the fitted function and adding new points to the set.

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

ego.m

Warnings

Observe that when cancelling with CTRL+C during a run, some memory allocated by rbfSolve will not be deal- located. To deallocate, do:

>> clear cgolib