CGO ego

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:

${\begin{array}{ll}\min \limits _{x}&p(x)=f(x)+\sum \limits _{j}w_{j}\max \left(0,~c^{j}(x)-c_{U}^{j},~~c_{L}^{j}-c^{j}(x)\right),\end{array}}$

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

Usage

See CGO solver usage

Description of Inputs

Problem structure

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_L
b_U
A
c_L
c_U
WarmStart
user
MaxCPU
PriLevOpt
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 ego are sent to the costly f(x)

- Special EGO algorithm parameters in Prob.CGO -
EGOAlg	Main algorithm in the EGO solver =1 Run expected improvement steps (modN = 0,1,2,...). If no f (x) improvement, use DACE surface minimum (modN = -1) in 1 step =2 Run expected improvement steps (modN=0) until ExpI / \|yMin\| < TolExpI for 3 successive steps (modN = 1,2,3) without f (x) improvement (fRed <= 0). After 2 such steps (when modN = 2), 1 step using the DACE surface minimum (modN = -1) is tried. If then fRed > TolExpI, reset to modN = 0 steps. =3 Compute trial points from both Expexted Improvement and DACE surface minimum in every step =4 Compute DACE surface minimum in every step Default EGOAlg = 1, but if any x is integer valued, EGOAlg = 3 is default
pEst	Norm parameters, fixed or estimated, also see p0, pLow, pUpp (default pEst = 0). 0 = Fixed constant p-value for all components (default, p0=1.99). 1 = Estimate one p-value valid for all components. 2 = Estimate d\|\| \|\|_p parameters, one for each component.
p0	Fixed p-value (pEst==0, default = 1.99) or initial p-value (pEst == 1, default 1.9) or d-vector of initial p-values (pEst > 1, default 1.9*ones(d,1))
pLow	If pEst == 0, not used if pEst == 1, lower bound on p-value (default 1.98) if pEst == 2, lower bounds on p (default 1.99*ones(d,1))
pUpp	If pEst == 0, not used if pEst == 1, upper bound on p-value (default 1.99999) if pEst == 2, upper bounds on p (default 1.99999*ones(d,1))
snPLim	Avoid the time consuming global optimization in DACEFit for iterations > snPLim. Instead just do a local search from the previous DACE parameter model This part is the most time consuming in ego Default 15 if pEst = 0, Default 20 if pEst = 1, Default 25 if pEst > 1
TolExpI	Convergence tolerance for expected improvement (default 10^-7).
SAMPLEF	Sample criterion function: 0 = Expected improvment (default) 1 = Kushner's criterion (related option: KEPS) 2 = Lower confidence bounding (related option: LCBB) 3 = Generalized expected improvement (related option: GEIG) 4 = Maximum variance 5 = Watson and Barnes 2
KEPS	The ε parameter in the Kushner's criterion If set to a positive value, the epsilon is taken as KEPS. If set to a negative value, then epsilon is taken as \|KEPS\|*f_min Default: -0.01
GEIG	The exponent g in the generalized expected improvement function Default: 2
LCBB	Lower Confidence Bounding parameter b Default 2
AddSurfMin	Add up to AddSurfMin global or local minima on DACE surface as search points, based on estimated Lipschitz constants, number of components on bounds, and distance to sampled set X. AddExpIMin=0 implies no additional minimum added.
AddExpIMin	Add up to AddExpIMin global or local minima on ExpI surface as search points, based on estimated Lipschitz constants, number of components on bounds, and distance to sampled set X. AddExpIMin=0 implies no additional minimum added. Only possible if globalSolver = 'multiMin' or 'glcCluster'. Default AddExpIMin = 1

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.

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.
10	Expected improvement low for three iterations

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 0
	Initial row with values for some important parameters from Prob.CGO for the current run
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)
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)
dacePtr	Number used internally in CGOLIB (if PriLev > 2)
Row 2
max(F)	maximum of all f(x) in the initial set of points X
med(F)	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	Maximum Lipschitz constant for initial set X, using transformed F
objType	Function transformation used during run, one of 0 to 8.
DACEfk	Best DACE fitness value found
Row 3
	Information about estimation of p and θ in DACE model
Row 4
xMin	Best point in initial set X
Row 5
xOptS	User-given known global optimum Prob.x_opt
if PriLev > 2 and global optimum xOptS known and given in Prob.x_opt
Row 6
SumXO	Sum of distances from global optimum xOptS to all sampled points X in experimental design
MeanX0	Mean of distances from global optimum xOptS to all sampled points X in experimental design
doO	Distance from global optimum xOptS to x with best f(x) in Experimental Design, xMin

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

#	f(x)	Task	onB	ExpI-Sc	doX	doM	doS	surfErr	f-Reduc	doO	ExpImpr

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

Value	Method
0	Global minimum of Exponential Improvement
1	Global minimum of Kriging Variance
2	Global minimum of DACE surface
999	Rescue
-3	Additional local Exponential Improvement Minima
-7	Additional local surface minima
-8	Sample point inside the full box defined by [x_L,x_U]
-9	Sample point inside a trust region box

onB

Number of coordinates on bound for new point, onB(x)

ExpI-Sc

Expected improvement, scaled by |fMin|

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)

ExpImpr

Expected improvement

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

Additional rows if PriLev > 2

One row with DACEFit information, result of p, theta estimation dependent on pEst value

DACEfk

Optimal DACEFit objective -log₁₀(-ML), where ML = Maximum Likelihood DACE fitness objective value (Step >0)

p

Exponential parameters in DACE model

theta

Weights in DACE model

One row with DACEupd information when updating with new sample points

minDist

Minimal distance between points in sample set X, if small value ill-conditioning might occur

NOTE: All distances measured are in SCALED space [0,1]^d, if SCALE' == 1

Description

ego implements the algorithm EGO by D. R. Jones, Matthias Schonlau and William J. Welch presented in the paper "Efficient Global Optimization of Expensive Black-Box Functions".

Please note that Jones et al. has a slightly different problem formulation. The TOMLAB version of ego treats linear and nonlinear constraints separately.

ego samples points to which a response surface is fitted. The algorithm then balances between sampling new points and minimization on the surface.

ego and rbfSolve use the same format for saving warm start data. This means that it is possible to try one solver for a certain number of iterations/function evaluations and then do a warm start with the other. Example:

>> Prob	= probInit('glc_prob',1);		%   Set up problem structure
>> Result_ego = tomRun('ego',Prob);		%   Solve for a while with  ego
>> Prob.WarmStart = 1; 				%   Indicate a warm start
>> Result_rbf = tomRun('rbfSolve',Prob);	%   Warm start with rbfSolve

M-files Used

iniSolve.m, endSolve.m, conAssign.m, glcAssign.m

Warnings

Observe that when cancelling with CTRL+C during a run, some memory allocated by ego will not be deallocated.

To deallocate, do:

>> clear egolib

CGO ego

Contents

Purpose

Calling Syntax

Usage

Description of Inputs

Problem structure

Description of Outputs

Result structure

Output printing

Description

M-files Used

See Also

Warnings

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