MATLAB/NOMAD for global optima?

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)

[Walter Roberson]

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]

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

Answer 1

[Abdolkarim Mohammadi]

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.

[basak ozturk]

İ have never hreard surragate optimization, genetic algorithm or nomad what i know, please answer accordingly

[Walter Roberson]

See surrogateopt and ga

[basak ozturk]

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]

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.

Answer 2

[Walter Roberson]

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, [], [])

[basak ozturk]

Base expression is defined as between (0,1) because of the values given to the lb and ub, it never is negative

[basak ozturk]

Also the genetic algorithm under Matlab cannot produce binary variables at the results, even if the variable types are defined as binary at the beginning

[Walter Roberson]

Your nomad configuration does not define any variables as constrained to binary.

[Walter Roberson]

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_real            fval_imag                x1                    x2                    x3                    x4                   x5                    x6        
    _________________    _________________    __________________    __________________    __________________    __________________    _________________    __________________

    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]

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_real           fval_imag                x1                     x2                    x3            x4    x5    x6
    _________________    ________________    ___________________    __________________    __________________    __    __    __

    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