CGO rbfSolve: Difference between revisions
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& b_{L} & \leq & Ax & \leq & b_{U} \\ | & b_{L} & \leq & Ax & \leq & b_{U} \\ | ||
& c_{L} & \leq & c(x) & \leq & c_{U} \\ | & c_{L} & \leq & c(x) & \leq & c_{U} \\ | ||
& | & ~x_{j \in \mathbb{N}\ ~~\forall j \in \mathbb{I}}, \\ | ||
\end{array} | \end{array} | ||
</math> | </math> | ||
where <math>f(x) \in \ | where <math>f(x) \in \mathbb{R}</math>; <math>x_L,~x,~x_U \in \mathbb{R}^d</math>; | ||
the <math>m_1</math> linear constraints are defined by <math>A \in | the <math>m_1</math> linear constraints are defined by <math>A \in | ||
\ | \mathbb{R}^{m_1 \times d}</math>, <math>b_L,~b_U \in \mathbb{R}^{m_1}</math>; | ||
and the <math>m_2</math> nonlinear constraints are defined by <math>c_L,~c(x),~c_U \in~ | and the <math>m_2</math> nonlinear constraints are defined by <math>c_L,~c(x),~c_U \in~ | ||
\ | \mathbb{R}^{m_2}</math>. | ||
The variables <math>x_I</math> are restricted to be integers, | The variables <math>x_I</math> are restricted to be integers, | ||
where <math>\ | where <math>\mathbb{I}</math> is an index subset of <math>\{1,\ldots,d\},</math> possibly empty. | ||
It is assumed that the function <math>f(x)</math> is continuous with respect to all | It is assumed that the function <math>f(x)</math> is continuous with respect to all | ||
variables, even if there is a demand that some variables only take integer values. | variables, even if there is a demand that some variables only take integer values. |
Revision as of 09:21, 12 December 2011
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 = rbfSolve(Prob,varargin) Result = tomRun('rbfSolve', Prob);
Description of Inputs
Problem description structure. The following fields are used:
Field | Description | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Name | Name of the problem. Used for security when doing warm starts. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
FUNCS.f | Name of function to compute the objective function. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
FUNCS.c | Name of function to compute the nonlinear constraint vector. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
x_L | Lower bounds on the variables. Must be finite. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
x_U | Upper bounds on the variables. Must be finite. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
b_U | Upper bounds for the linear constraints. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
b_L | Lower 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 | Set true (non-zero) to load data from previous run from cgoSave.mat and resume optimization from where the last run ended. If Prob.CGO.WarmStartInfo has been defined through a call to WarmDefGLOBAL, this field is used instead of the cgoSave.mat file. All CGO solvers uses the same mat-file and structure field and can read the output of one another. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
MaxCPU | Maximal CPU Time (in seconds) to be used. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
user | User field used to send information to low-level functions. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
PriLevOpt | Print Level. 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. The following fields are used: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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:
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:
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 -step first in cycle. 3: fStar = -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. |
Description of Outputs
Structure with result from optimization. The following fields are changed:
Field | Description |
---|---|
x_k | Matrix with the best points as columns. |
f_k | The best function value found so far. |
Iter | Number of iterations. |
FuncEv | Number of function evaluations. |
ExitText | Text string with information about the run. |
ExitFlag | Always 0. |
CGO | Subfield WarmStartInfo saves warm start information, the same information as in cgoSave.mat, see below. |
Inform | Information parameter.
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. 4 = No new point sampled for MaxCycle iteration steps. 5 = All sample points same as the best point for MaxCycle last iterations. 6 = All sample points same as previous point for MaxCycle last iterations. 7 = All feasible integers tried. 8 = No progress for MaxCycle * (N + 1) + 1 function evaluations (> MaxCycle cycles, input CGO.MaxCycle). 9 = Max CPU Time reached. |
cgoSave.mat | To make a warm start possible, all CGO solvers saves information in the file cgoSave.mat. The file is created independent of the solver, which enables the user to call any CGO solver using the warm start information. cgoSave.mat is a MATLAB mat-file saved to the current directory. If the parameter SAVE is 1, the CGO solver saves the mat file every iteration, which enables the user to break the run and restart using warm start from the current state. SAVE = 1 is currently always set by the CGO solvers. If the cgoSave.mat file fails to open for writing, the information is also available in the output field Result.CGO.WarmStartInfo, if the run was concluded without interruption. Through a call to WarmDefGLOBAL, the Prob structure can be setup for warm start. In this case, the CGO solver will not load the data from cgoSave.mat. The file contains the following variables: |
Name | Problem name. Checked against the Prob.Name field if doing a warmstart. |
O | Matrix with sampled points (in original space). |
X | Matrix with sampled points (in unit space if SCALE==1) |
F | Vector with function values (penalty added for costly Cc(x)) |
F_m | Vector with function values (replaced). |
F00 | Vector of pure function values, before penalties. Cc MMatrix with costly constraint values, C c(x). nInit Number of initial points. |
Fpen | Vector with function values + additional penalty if infeasible using the linear constraints and noncostly nonlinear c(x). |
fMinIdx | Index of the best point found. |
rngState | Current state of the random number generator used. |
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