A study using the MATLAB Optimization Toolbox

(last updated on 14/Dec/2005)


In this study we use the function FMINCON on the NLP test function suite. Since FMINCON is a local optimizer it is decided (as commonly done in practice) to start from random points uniformly distributed in the search space, i.e. a multi-start hill-climber.

When using FMINCON from the OPTIM toolbox in MATLAB one will usually create two M-files (one for the objective function and another for the non-linear constraints), for example consider test function G01: 

function f = myfun(x),
 [f, c, ceq] = mlbsuite(x, 9, 0, 'g01');
% if the problem is a maximization problem you must set f = -f

function [c, ceq] = mycon(x)
%  [f, c, ceq] = mlbsuite(x, 9, 0, 'g01');
% because FMINCON has a separate input for linear constraints
% and all constraints are linear for G01, just return empty matrices
c = []; ceq = [];

Now run the optimization from MATLAB like this:

lb = zeros(13,1);
ub = [ones(9,1);100*ones(3,1);1];
A = [ 2  2  0  0  0  0  0  0  0  1  1  0  0;
      2  0  2  0  0  0  0  0  0  1  0  1  0;
      0  2  2  0  0  0  0  0  0  0  1  1  0;
     -8  0  0  0  0  0  0  0  0  1  0  0  0;
      0 -8  0  0  0  0  0  0  0  0  1  0  0;
      0  0 -8  0  0  0  0  0  0  0  0  1  0;
      0  0  0 -2 -1  0  0  0  0  1  0  0  0;
      0  0  0  0  0 -2 -1  0  0  0  1  0  0;
      0  0  0  0  0  0  0 -2 -1  0  0  1  0];
b = [10 10 10 0 0 0 0 0 0]';
FcnCount = 0; % count the number of function evaluations
for i = 1:100, % or we can use FcnCount as a stop criteria
  x0 = lb + rand(13,1).*(ub-lb); % random starting point
  [X01(:,i),F01(i),exitflag,output] = fmincon('myfun01',x0,A,b,[],[],lb,ub,'mycon01');
  FcnCount = output.funcCount + FcnCount; % increment function evaluation counter
end
% now pick out the best solution found from all these hill-climbs
[dummy,i] = min(F01); x01 = X01(:,i), f01 = F01(i), FcnCount
x01 =
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    3.0000
    3.0000
    3.0000
    1.0000
f01 =
   -15
FcnCount =
        4636



MATLAB functions myfunXX and myconXX  [Download All]

The functions myfunXX and myconXX where XX is 01 to 24 can be downloaded from here. To be fair to FMINCON the linear constraints are treated separately and so the linear constraint values returned from fcnsuite are not used.

Results are currently only based on a single run of 100 random starting points. Take a look at the histograms of f(x) found, this gives some idea about the number of local minima/maxima.

  1. myfun01.m, mycon01.m: best found is f(x) = -15 (see help myfun01 and hist(F01))

    x01 =

1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
3.00000000000000
3.00000000000000
3.00000000000000
1.00000000000000


f01 =

-15.00000000000000


g01 =

0
0
0
-5.00000000000000
-5.00000000000000
-5.00000000000000
0
-0.00000000000000
0


h01 =

Empty matrix: 0-by-1

  2. myfun02.m, mycon02.m: best found is f(x) =-3.169202012269866e-006 (see help myfun02 and hist(F02))

    x02 =

4.77528004987257
4.72194998494275
4.77416704155021
1.51113496956842
7.89953528875943
4.78981998299744
4.64925653766998
7.84312684528473
7.85047077060430
4.63476662602853
1.61109265662308
1.61760952760552
7.88500283707216
7.80476655036788
4.64772035754067
7.88935885887532
1.61814553318049
1.47518779988136
4.70182684462553
7.80768202561237


f02 =

-3.169202012269866e-006


g02 =

1.0e+012 *

-4.20852079637390
-0.00000000004949


h02 =

Empty matrix: 0-by-1

  3. myfun03.m, mycon03.m: best found is f(x) = -1.00000191871532 (see help myfun03 and hist(F03))

    x03 =

0.31613375019488
0.31634882742300
0.31627145290089
0.31617610146500
0.31622869011441
0.31626214515567
0.31623366279673
0.31592348198058
0.31628839448495
0.31641201266618


f03 =

-1.00000191871532


g03 =

Empty matrix: 0-by-1


h03 =

7.027896724576266e-007

  4. myfun04.m, mycon04.m: best found is f(x) =-3.066553867178332e+004 (see help myfun04 and hist(F04))

    x04 =

78.00000000000000
33.00000000000000
29.99525602568159
45.00000000000000
36.77581290578821


f04 =

-3.066553867178332e+004


g04 =

0
-92.00000000000000
-11.15949969107314
-8.84050030892686
-5.00000000000000
0


h04 =

Empty matrix: 0-by-1

  5. myfun05.m, mycon05.m: best found is f(x) = 5.126498109001777e+003 (see help myfun05 and hist(F05))

    x05 =

1.0e+003 *

0.67994710914725
1.02606522055003
0.00011887508847
-0.00039623415883


f05 =

5.126498109001777e+003


g05 =

-0.03489075270334
-1.06510924729666


h05 =

1.0e-007 *

0.57592728808231
0.10398480299045
0.57657189245219

  6. myfun06.m, mycon06.m: best found is f(x) =-6.961813875581064e+003 (see help myfun06 and hist(F06))

    x06 =

14.09499999999999
0.84296078921546


f06 =

-6.961813875580155e+003


g06 =

1.0e-014 *

-0.35527136788005
0


h06 =

Empty matrix: 0-by-1

  7. myfun07.m, mycon07.m: best found is f(x) = 24.30620897255022 (see help myfun07 and hist(F07))

    no feasible solution is found by fmincon.
    x07 =

2.17197976836280
2.36371956277467
8.77392356889764
5.09594446746571
0.99064016307554
1.43052328860937
1.32161453645052
9.82870296911411
8.27999973971528
8.37579983861169


f07 =

24.30620897255022


g07 =

0
-0.00000000000000
0.00000000000000
0.00000061481619
0.00000010423835
0.00000036940871
-6.14868224183549
-50.02342254783467


h07 =

Empty matrix: 0-by-1

  8. myfun08.m, mycon08.m: best found is f(x) = -0.09582504135112 (see help myfun08 and hist(F08))

    x08 =

1.22797326711932
4.24536843253200


f08 =

-0.09582504135112


g08 =

-1.73745008777231
-0.16776759943611


h08 =

Empty matrix: 0-by-1

  9. myfun09.m, mycon09.m: best found is f(x) = 680.6300564022225 (see help myfun09 and hist(F09))

    no feasible solution is found by fmincon.

    x09 =

2.33043648894835
1.95141224114835
-0.47762925103314
4.36564222851942
-0.62447622288225
1.03815066596664
1.59416653756602


f09 =

6.806300565486768e+002


g09 =

1.0e+002 *

0.00000000785690
-2.52561292388090
-1.44878742488331
0.00000000595225


h09 =

Empty matrix: 0-by-1

  10. myfun10.m, mycon10.m: best found is f(x) = 7.049248020518209e+003 (see help myfun10 and hist(F10))

    x10 =

1.0e+003 *

0.57930684132680
1.35997021338417
5.10997096580724
0.18201771268567
0.29560116136771
0.21798228731433
0.28641655131796
0.39560116136771


f10 =

7.049248020518209e+003


g10 =

1.0e-006 *

0
0
0
0.41359453462064
0.73344563134015
-0.00034924596548


h10 =

Empty matrix: 0-by-1

  11. myfun11.m, mycon11.m: best found is f(x) = 0.75 (see help myfun11 and hist(F11))

    x11 =

-0.70710677538567
0.49999999179631


f11 =

0.75000000000000


g11 =

Empty matrix: 0-by-1


h11 =

1.110223024625157e-016

  12. myfun12.m, mycon12.m: best found is f(x) = -1 (see help myfun12 and hist(F12))

    x12 =

5.00000002316183
4.99999996666389
4.99999996045174


f12 =

-1


g12 =

-0.06250000000000


h12 =

Empty matrix: 0-by-1

  13. myfun13.m, mycon13.m: best found is f(x) = 0.05394981807739 (see help myfun13 and hist(F13))

    x13 =

-1.71717389291460
1.59574482401267
1.82718961052188
-0.76365962601746
-0.76361983995613


f13 =

0.05394981807739


g13 =

Empty matrix: 0-by-1


h13 =

1.0e-006 *

0.87905356238593
0.15647793638607
0.15854859913844

  14. myfun14.m: best found is f(x) = -47.76109085069346 (see help myfun14 and hist(F14))

    x14 =

0.04066308411210
0.14774055796946
0.78313627518151
0.00141533299404
0.48524720540836
0.00069540575037
0.02739485043886
0.01795578743362
0.03731262155521
0.09688784383558


f14 =

-47.76109085069346


g14 =

Empty matrix: 0-by-1


h14 =

1.0e-015 *

0
-0.11102230246252
0

  15. myfun15.m, mycon15.m: best found is f(x) = 961.7151721300522 (see help myfun15 and hist(F15))

    x15 =

-0.80939768623134
4.86721180667644
-0.80939768623134


f15 =

9.545946345891907e+002


g15 =

Empty matrix: 0-by-1


h15 =

1.0e-011 *

0.37907454952801
-0.00071054273576

    For some reason FMINCON is returning values for x that are out of bounds! The other legal solution returned is:

    x15 =
   3.51212177435569
   0.21698790713563
   3.55217072932224
f15 =
    9.617151721300522e+002
g15 =
   Empty matrix: 0-by-1
h15 =
  1.0e-012 *
   0.18118839761883
                  0

  16. myfun16.m, mycon16.m: best found is f(x) = -1.90399603217138 (see help myfun16 and hist(F16))

    x16 =

1.0e+002 *

7.04910558969095
0.68600000000000
1.02900000000000
2.82361659323949
0.33077959262027


f16 =

-1.90399603217138


g16 =

1.0e+006 *

-0.00008175012763
0
-0.00000000000000
-0.00000768919863
0
-0.00019213000000
-0.00000029094609
-0.00103587075391
-0.00000009062813
-0.00002366437187
-0.00000171893448
-0.00044963806552
-0.00033761950332
-0.00023938549668
-0.00013155949304
-0.00013339550696
-0.00000100751814
-0.00000442648186
-0.00000000097718
-0.00000007502282
-0.00004666296334
-0.00011871303666
-0.00004675555437
-0.00031665644563
-0.00018754910577
-0.00032966489423
-0.00000013216858
-0.00051824283142
-0.00185929540594
-0.00031558059406
-0.00586812280354
-0.01201451519646
-0.00000018269180
-0.00000014030820
-0.06891567000000
-0.00000000000000
-9.27624613265450
-0.06714886734550


h16 =

Empty matrix: 0-by-1

  17. myfun17.m, mycon17.m: best found is f(x) = 8.927597735318928e+003 (see help myfun17 and hist(F17))

    x17 =

1.0e+002 *

1.07811879409624
1.96318667305285
3.73830721425211
4.20000000000000
0.21307156039082
0.00153292005057


f17 =

8.927597735318928e+003


g17 =

Empty matrix: 0-by-1


h17 =

1.0e-006 *

0.19419022123657
-0.16030219285312
0.18757076603038
0.20139873413427

  18. myfun18.m, mycon18.m: best found is f(x) =-0.86602540378444 (see help myfun18 and hist(F18))

    x18 =

-0.58034855586849
0.17726425432159
0.41508926784198
-0.90978068770579
-0.58034855350470
-0.81436819464233
0.41508926520124
0.08185175836876
0.99163244727939


f18 =

-0.86602540378444


g18 =

0.00000000000000
-0.01666508950269
0.00000000000000
-0.00000000000000
-0.01666508616183
0.00000000000000
0.00000000000000
-0.01666509189220
0.00000000000000
-0.45440941872619
-0.41161598650955
-0.57549245638692
-0.29053294594622


h18 =

Empty matrix: 0-by-1

  19. myfun19.m, mycon19.m: best found is f(x) =  32.65559295024639 (see help myfun19 and hist(F19))

    x19 =

0
0
3.94599019469641
0
3.28317748680954
10.00000000000000
0
0
-0.00000000000000
0
0.37076487209734
0.27845598338938
0.52383846630380
0.38862011053770
0.29815677523079


f19 =

32.65559295024639


g19 =

1.0e-013 *

-0.19539925233403
-0.03552713678801
-0.28421709430404
-0.19095836023553
-0.01776356839400


h19 =

Empty matrix: 0-by-1

  20. myfun20.m, mycon20.m: best found is f(x) = 0.18844594779460 (see help myfun20 and hist(F20))

    NO FEASIBLE SOLUTION FOUND, best known is also slightly INFEASIBLE
    x20 (exceed bounds)=

-0.00000701588170
-0.00002747800776
-0.00001904673189
0.00004198825941
-0.00021374845827
0.00002228257815
-0.00001750338799
0.00008693344289
0.00029391785656
-0.00543221144799
0.00072615016482
0.00096441951758
0.00763807334431
0.00766611084426
0.00766068598340
-0.00543221144799
0.15933789607244
-0.00543221144799
0.00765613623374
-0.00543221144799
-0.00543221144799
0.84621567830600
-0.00543221144799
-0.00543221144799


f20 =

0.18844594779460


g20 =

0.00693732496600
0.00587587141270
0.00545831375108
0.00587587141980
-0.00334079875319
-0.00395253353187


h20 =

1.0e-006 *

0.01891638330177
-0.01859445981391
-0.00820340556075
-0.00116443867005
0.35316809716968
-0.00093047227171
-0.00397489029885
-0.00160875350194
-0.00071936149789
-0.35187571811868
0.00021434688746
0.01077328313105
-0.00000000011102
0

  21. myfun21.m, mycon21.m: best found is f(x) = 1.937957105345119e+002 (see help myfun21 and hist(F21))

    x21 =

1.0e+002 *

1.93795710534512
0.00000001000000
0.17328488765172
1.00002034656743
0.06684609184344
0.05991469633737
0.06214599959762


f21 =

1.937957105345119e+002


g21 =

-5.684341886080802e-014


h21 =

1.0e-010 *

0.03183231456205
0.12732925824821
0.00000888178420
0.00001776356839
-0.00000888178420

  22. myfun22.m, mycon22.m: best found is f(x) =2.363703133145658e+002 (see help myfun22 and hist(F22))

    Seems to be difficult to find a feasible solution:

    x22 =

1.0e+007 *

0.00002363703133
0.00001354324412
0.00002004283126
0.00064625524595
0.30000000000000
0.40000000000000
3.30000000000000
0.00001300000000
0.00001700000000
0.00003000000000
0.00004000000000
0.00003300000000
0.00001845938814
0.00002494657534
0.00001276585382
0.00002700000000
0.00001600000000
0.00000052983174
0.00000051357984
0.00000055984220
0.00000054380793
0.00000050751738


f22 =

2.363703133145658e+002


g22 =

0


h22 =

1.0e-008 *

0
0
0
0
0
0.74505805969238
0
0.04656612873077
-0.37252902984619
0
-0.00000284217094
0
0
0
0
0
0
0
0

  23. myfun23.m, mycon23.m: best found is f(x) = -4.000000000000005e+002 (see help myfun23 and hist(F23))

    x23 =

1.0e+002 *

0.00000000000000
1.00000000000000
0.00000000000000
1.00000000000000
0.00000000000000
0.00000000000000
1.00000000000000
2.00000000000000
0.00010000000000


f23 =

-4.000000000000005e+002


g23 =

1.0e-026 *

-0.52657027486897
0


h23 =

1.0e-015 *

0
0.22204460492503
0.00000000043381
0

  24. myfun24.m, mycon24.m: best found is f(x) = -5.50801327159531(see help myfun24 and hist(F24))

    x24 =

2.32952019747761
3.17849307411770


f24 =

-5.50801327159531


g24 =

1.0e-013 *

-0.07993605777301
-0.63948846218409


h24 =

Empty matrix: 0-by-1