MATLAB/NOMAD for global optima?

Question

basak ozturk 2020-4-17

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/518691-matlab-nomad-for-global-optima

评论： Walter Roberson 2022-9-15

Here comes my nonsmooth nonconvex MINLP, in fact this is a maximization problem. MATLAB/NOMAD from opti toolbox does not find the global optima? Do you have any idea why?

tnank you very much, basak

clc
fun=@(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+...
    ((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*... 
    (x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+... 
    (1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*... 
    (x(6)*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+... 
    (1-x(6))*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))
x0 = [1.0000 0.8629 0.5863 0 0 0]';
lb = [-1;-1;-1;0;0;0];
ub = [1;1;1;1;1;1];
xtype='CCCBBB';
opts=optiset('solver','nomad','display','iter','solverOpts',nomadset('direction_type','lt 2n'))
Opt=opti('fun',fun,'bounds',lb,ub,'xtype',xtype,'options',opts)
[x,fval,exitflag,info] = solve(Opt,x0)

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Walter Roberson 2022-9-15

编辑：Walter Roberson 2022-9-15

NOMAD appears to refer to the third party toolbox, one source of which is at https://github.com/jonathancurrie/OPTI

Note that third party toolboxes are not created by Mathworks, and volunteers here might not be familiar with them.

basak ozturk 2022-9-15

I am a mathematician who uses MATLAb because of its gui, I am not a software engineer, hence github sources becomes practically out of scope, thank you

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Answer 1

Abdolkarim Mohammadi 2020-7-20

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/518691-matlab-nomad-for-global-optima#answer_468249

Your problem is a bound-constrained problem with five decision variables. I think many solvers in the Global Optimization toolbox like GA and surrogate optimization can handle such problems efficiently.

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basak ozturk 2022-9-15

Thank you very much, I would be very happy if we answer the questions, surragate optimization is a new tool and I definitely would concentrate on it when it becomes a classic

Walter Roberson 2022-9-15

NOMAD is third party code. The volunteers generally do not know anything about it.

I see that the third-party toolbox historically had a user forum. However, that appears to be closed now, as the toolbox is no longer being developed.

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Answer 2

Walter Roberson 2022-9-15

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https://ww2.mathworks.cn/matlabcentral/answers/518691-matlab-nomad-for-global-optima#answer_1054415

在 MATLAB Online 中打开

Your function can return complex values. The ^(1/3) generates a complex result when the base expression is negative.

format long g

fun=@(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+...

((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*...

(x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+...

(1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*...

(x(6)*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+...

(1-x(6))*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))

fun = function_handle with value:

@(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*(x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+(1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*(x(6)*((140.2333+5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+(1-x(6))*((140.2333+5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))

x0 = [1.0000 0.8629 0.5863 0 0 0]';

lb = [-1;-1;-1;0;0;0];

ub = [1;1;1;1;1;1];

ga_opts = optimoptions('ga', 'HybridFcn', 'fmincon')

ga_opts =

ga options: Set properties: HybridFcn: 'fmincon' Default properties: ConstraintTolerance: 0.001 CreationFcn: [] CrossoverFcn: [] CrossoverFraction: 0.8 Display: 'final' EliteCount: '0.05*PopulationSize' FitnessLimit: -Inf FitnessScalingFcn: @fitscalingrank FunctionTolerance: 1e-06 InitialPopulationMatrix: [] InitialPopulationRange: [] InitialScoresMatrix: [] MaxGenerations: '100*numberOfVariables' MaxStallGenerations: 50 MaxStallTime: Inf MaxTime: Inf MutationFcn: [] NonlinearConstraintAlgorithm: 'auglag' OutputFcn: [] PlotFcn: [] PopulationSize: '50 when numberOfVariables <= 5, else 200' PopulationType: 'doubleVector' SelectionFcn: [] UseParallel: 0 UseVectorized: 0

[ga_x, ga_fval, ga_exitflag, ga_info] = ga(fun, length(x0), [], [], [], [], lb, ub, [], ga_opts)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

Error using optimfcnchk/checkfun
Supplied function '@(x)FitnessFcn(x,options.FitnessFcnArgs{:})' returned a complex value when evaluated;
FMINCON cannot continue.

Error in finDiffEvalAndChkErr

Error in finitedifferences

Error in computeFinDiffGradAndJac

Error in barrier

Error in fmincon (line 886)
[X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = barrier(funfcn,X,A,B,Aeq,Beq,l,u,confcn,options.HessFcn, ...

Error in callHybrid (line 61)
[x,fval,eflag,output] = fmincon(FitnessHybridFcn,x0,Aineq,bineq,Aeq,beq,lb,ub,nonlcon,HybridFcnArgs{:});

Error in galincon/callHybridFunction (line 163)
callHybrid(hfunc,FitnessHybridFcn,x,options.HybridFcnArgs,Aineq,bineq,Aeq,beq,lb,ub,ConstrHybridFcn);

Error in galincon (line 129)
[x,fval, output.hybridflag] = callHybridFunction;

Error in ga (line 416)
[x,fval,exitFlag,output,population,scores] = galincon(FitnessFcn,nvars, ...

[fmc_x, fmc_fval, fmc_exitflag, fmc_info] = fmincon(fun, x0, [], [], [], [], lb, ub, [], [])

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Walter Roberson 2022-9-15

在 MATLAB Online 中打开

You can see from the below that if you permit continuous variables, then there are positions well within the bounds (not just right at the bounds) that produce complex results from the function. In my tests, roughly 45.3 % of all random configurations in-bounds produce complex results.

format long g
lb = [-1;-1;-1;0;0;0];
ub = [1;1;1;1;1;1];
N = 100;
M = length(lb);
trials = rand(M, N) .* (ub - lb) + lb;
fun=@(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+...
    ((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*... 
    (x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+... 
    (1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*... 
    (x(6)*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+... 
    (1-x(6))*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))
fun = function_handle with value:
    @(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*(x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+(1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*(x(6)*((140.2333+5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+(1-x(6))*((140.2333+5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))
results = zeros(1, N);
for K = 1 : N
    results(K) = fun(trials(:,K));
end
mask = imag(results) ~= 0;
tp = trials(:,mask).';
x1 = tp(:,1); x2 = tp(:,2); x3 = tp(:,3); x4 = tp(:,4); x5 = tp(:,5); x6 = tp(:,6);
fval = results(mask);
fval_real = real(fval(:)); fval_imag = imag(fval(:));
T = table(fval_real, fval_imag, x1, x2, x3, x4, x5, x6);
if height(T) == 0
    fprintf('Excellent, %d trials produced no complex results!\n', N);
else
    fprintf('Opps, %d trials produced %d complex results!\n', N, height(T));
    T
end
Opps, 100 trials produced 48 complex results!
T = 48×8 table
fval_realfval_imagx1x2x3x4x5x6_____________________________________________________________________________________________________________________________________________

    0.302036267914207    0.523142161755892     0.855313778814469     0.167890364370559    0.0722028543147295     0.773636892775948    0.748790637365133     0.854477000319776
    0.692218663072773     1.19895789438944    -0.127728919215863     0.446704147486387      0.89215122469133      0.55399930615609     0.84959267633737     0.130278628196272
     1.47901888917058     2.56173586139753    -0.846266323190261     0.359603615851935    -0.837466766233311     0.816154971675825    0.997270145172786     0.118301564702515
     1.93873384508983     3.35798552204895     -0.52851829722236    -0.165609523897328    -0.247584322035819     0.981078408094133    0.935259559058378     0.988714463088133
     2.98701427061245     5.17366047963406     0.742228322148513    -0.601178031646714    -0.457535097626576     0.230150076364672    0.867498124458657     0.176543685453211
     1.01066193871383     1.75051782712842    -0.926501607159432     0.981109366147993    -0.585150676993805     0.467285660681846    0.697523023659257    0.0310193291640264
      1.9926964985532     3.45145157955874     0.721081459950179    -0.823430131174107     0.285245074613108     0.909312596962406    0.842095142509196      0.97458936025113
     2.35519833860522     4.07932318436605     0.656375255614923    -0.807051456913801    -0.755082562064693     0.262775457151212    0.946356055134857     0.550240301407202
     1.26525127466414     2.19147949205958    -0.672576092089491    -0.584268041932339     0.632762234384127     0.589726591783466    0.998031020753582    0.0409425876625146
     1.07933008780239     1.86945455021151     0.850412009173303    -0.277293521650321    -0.899023656825273     0.397590493694744    0.585689720864097     0.213107752684854
     1.96197542672752     3.39824112229369    -0.652323322641124    -0.933325765084956     0.360206217988309     0.218356813737419    0.821102016732672     0.505683211972857
     1.09188172385272      1.8911946215688    -0.299030526140808     0.735402756680416    -0.996004317623713     0.968077990873665    0.646739059040298    0.0786006489183513
     2.61989576202502     4.53779257036171    -0.292347117069892    -0.526828500143656    -0.849161461114467     0.929864435897615    0.935262613586016     0.904328205780685
     2.76324420891784     4.78607936356616    -0.687546163744532    -0.625164102281192     -0.33742676532606    0.0694593496238728    0.164381555058081     0.916729011788188
    0.538851774640784    0.933318651426492    -0.828445883690186     0.779047799800357    -0.634560984566284    0.0799828530965894    0.531022688412337     0.295810685714847
    0.370719751499232    0.642105444965979     0.778552376779659     0.605276118857012    -0.409681794006963     0.740212211899616    0.748395409921125     0.664631819919668

Walter Roberson 2022-9-15

在 MATLAB Online 中打开

If you interpret the lb = 0 ub = 1 as being the locations of binary variables, and assume the -1 to 1 locations are continuous, then half of the random configurations lead to complex results.

format long g
lb = [-1;-1;-1;0;0;0];
ub = [1;1;1;1;1;1];
N = 1000;
M = length(lb);
trials = rand(M, N) .* (ub - lb) + lb;
trials(4:6, :) = randi([0 1], 3, N);
fun=@(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+...
    ((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*... 
    (x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+... 
    (1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*... 
    (x(6)*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+... 
    (1-x(6))*((140.2333+ 5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))
fun = function_handle with value:
    @(x)((((x(4)*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-185)/(190-185))+((1-x(4))*(174.9333+23.3750*x(2)+3.6250*x(3)-19.0000*x(2)*x(3)-195)/(190-195)))*(x(5)*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-185)/(190-185))+(1-x(5))*((154.8571+8.5000*x(1)+30.6250*x(2)+7.8750*x(3)-12.8571*x(1)^2+11.2500*x(1)*x(2)-195)/(190-195)))*(x(6)*((140.2333+5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-170)/(185-170))+(1-x(6))*((140.2333+5.3437*x(1)+18.2500*x(2)+19.5938*x(3)-195)/(185-195))))^(1/3))
results = zeros(1, N);
for K = 1 : N
    results(K) = fun(trials(:,K));
end
mask = imag(results) ~= 0;
tp = trials(:,mask).';
x1 = tp(:,1); x2 = tp(:,2); x3 = tp(:,3); x4 = tp(:,4); x5 = tp(:,5); x6 = tp(:,6);
fval = results(mask);
fval_real = real(fval(:)); fval_imag = imag(fval(:));
T = table(fval_real, fval_imag, x1, x2, x3, x4, x5, x6);
if height(T) == 0
    fprintf('Excellent, %d trials produced no complex results!\n', N);
else
    fprintf('Opps, %d trials produced %d complex results!\n', N, height(T));
    T
end
Opps, 1000 trials produced 499 complex results!
T = 499×8 table
fval_realfval_imagx1x2x3x4x5x6______________________________________________________________________________________________

    0.765174135811443    1.32532047986303     -0.627620563025766     0.713613270253988    -0.177459935981493    0     0     1 
     1.26385834049921    2.18906685931431      0.988053679562041     0.448849483369191    -0.968492181331381    1     0     1 
     1.54116641560331    2.66937853474375     0.0384926002740478     0.383317513499709     0.410954890582803    0     1     0 
     2.09028656313053    3.62048252972061     -0.754458217427597    -0.570283573899732     0.539703303215139    1     1     1 
     1.24452585998435    2.15558202082624     -0.923287437838263     0.194904492171103     0.332043662367417    1     1     1 
     1.48391019468221    2.57020785105902      0.744910334393307    0.0192149467987215     0.523744810222613    0     0     1 
     1.93047141268254    3.34367456932542     -0.206862798379165    -0.395886710834123     0.802344396662161    0     0     1 
      3.3227904702965    5.75524191745922      0.192426680560663    -0.903049473500569     0.765153552797197    0     1     0 
     2.06436948987523    3.57559284205894      -0.17419580241817     0.967258097554198    -0.859584603396955    0     0     0 
     2.26499593568055    3.92308803953572      0.377452942354955    -0.832288862539742     0.883172432788291    0     0     1 
     4.04268026603771    7.00212761953339     -0.340830379823908    -0.875518756058784     0.267215839543906    0     1     0 
     1.63217213867249     2.8270050708791      0.412029568246556     0.794782749154116    -0.925573891137459    1     1     0 
     5.62736559718543    9.74688312709033       -0.6661651489624    -0.847055069507603     -0.88853980262778    0     1     0 
     1.26087517686777    2.18389986833737     -0.877452310421777       0.5367768798219    0.0327113404834443    1     1     0 
     3.41773575432019     5.9196919733273     -0.173206949030919    -0.715038942268025     0.318751010348369    1     0     0 
     3.76902585182056    6.52814427039378    -0.0954255833206981    -0.839929714651277    -0.303185036333343    0     0     1 

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MATLAB/NOMAD for global optima?

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