MATLAB Examples

runBeamGV

Script to Run Gary Vanderplaats cantilever Beam with SQP and SLP. N beam segments and 2N design variables, taken from Vanderplaats (1984) Example 5-1, pp. 147-150.

Contents

Initialize variables

clear; clc
options.Display = 'iter';
options.MaxIter = 40;
options.TolX   = 0.5;
options.TolFun = 0.5;
options.TolCon = 1e-4;
options.MoveLimit = 0.2;
for N = [5 200]; % number of beam segments
   x0 = [5*ones(1,N), 40*ones(1,N)];
   xlb = [ ones(1,N),  5*ones(1,N)];
   options.TypicalX = x0;
   disp(' ')
   disp(['N=',num2str(N),' beam segments'])
 
N=5 beam segments
 
N=200 beam segments

SQP with BFGS

   disp('SQP with BFGS'), tic
   [dv2,out2,lam2] = sqp(@fbeamGV,x0,options,xlb,[],@gbeamGV); toc
   disp(out2)
SQP with BFGS
 
                                       Termination Criteria
                                     0.0001           0.5       0.5
                                -----------------------------------
f-CNT         FUNC      STEP NAC     max{g}  j        KTO    max(S)
    1        1e+05         0  20      0.562  6   1.35e+05        35
    3        93901       0.1  21      0.514  6   8.27e+04      19.7
    5        90002       0.1  21      0.468  6   7.17e+04      14.6
    6        53951         1  21      0.585  4    1.9e+04      5.73
    7        62838         1  21      0.119  4   4.82e+03      5.74
    8        64748         1  21     0.0434  6   1.86e+03       6.6
   10        64685     0.356  21     0.0359  6   2.16e+03      5.05
   12        64582     0.489  21     0.0278  6   1.64e+03     0.777
   13        65394         1  20   0.000862  6       51.1     0.214
   14        65418         1  21   4.93e-05  6       2.51    0.0757
   15        65419         1  21   7.37e-06  6      0.457  0.000298
Optimization Terminated Successfully from sqp
 
Elapsed time is 0.262413 seconds.
          fval: 6.5419e+04
     funcCount: 15
     gradCount: 11
    iterations: 10
       options: [1x18 double]

SQP with BFGS
 
                                       Termination Criteria
                                     0.0001           0.5       0.5
                                -----------------------------------
f-CNT         FUNC      STEP NAC     max{g}  j        KTO    max(S)
    1        1e+05         0 759      0.562 201   1.08e+05      25.2
    3        72442     0.477 784      0.526 201   4.27e+04      15.6
    4        55333         1 799       2.31 176      1e+04      14.5
    5        55503         1 798      0.769 176   1.31e+04      25.1
    7        55925     0.121 798      0.369 201    1.4e+04      22.1
    9        56337       0.1 798      0.336 201   1.33e+04      19.9
   10        62122         1 801       8.24 177   1.69e+04      9.97
   12        59702     0.255 801       4.87 177   1.26e+04      10.8
   14        57161     0.423 801       2.14 178   7.99e+03      25.6
   17        56699     0.159 801       1.42 177   1.23e+04      12.7
   18        62210         1 801       3.62 178   2.79e+03      17.1
   19        62204         1 801       6.21 177   1.31e+04      9.85
   22        61906    0.0328 801       4.52 177   3.39e+03      8.56
   23        62826         1 801       1.86 177   3.54e+03      7.92
   24        62512         1 801      0.619 177   3.08e+03      3.35
   25        63708         1 801      0.195 200        233      3.57
   26        63758         1 801     0.0099 177       92.4      2.13
   27        63797         1 801    0.00186 200       5.12     0.438
   28        63798         1 801   2.62e-05 201       9.57     0.803
   29        63789         1 801   5.91e-05 201       17.5       1.6
   30        63774         1 801   0.000147 201       18.3      1.78
   31        63759         1 801   0.000179 201       16.3      1.57
   32        63748         1 801   0.000496 178       35.7      4.34
   33        63709         1 801   0.000454 201       54.5      4.36
   34        63679         1 801   0.000286 201       13.9      1.11
   35        63675         1 801   7.94e-05 201       14.5      1.38
   36        63664         1 801   0.000178 201       10.7       0.3
   37        63669         1 801   7.63e-06 201       0.34     0.143
Optimization Terminated Successfully from sqp
 
Elapsed time is 8.187800 seconds.
          fval: 6.3669e+04
     funcCount: 37
     gradCount: 28
    iterations: 27
       options: [1x18 double]

SQP with Exact Hessian

   disp('SQP with exact Hessians')
   options.HessFun=@HbeamGV; tic
   [dv3,out3,lam3] = sqp(@fbeamGV,x0,options,xlb,[],@gbeamGV); toc
   disp(out3)
   options=rmfield(options,'HessFun');
SQP with exact Hessians
 
                                       Termination Criteria
                                     0.0001           0.5       0.5
                                -----------------------------------
f-CNT         FUNC      STEP NAC     max{g}  j        KTO    max(S)
    1        1e+05         0   6      0.562  6    9.1e+04      16.3
    2        70046         1  21       2.23  5   3.61e+04      9.63
    3        61442         1   9      0.744  5   1.32e+04      5.24
    4        63790         1   8      0.176  5    3.4e+03       1.7
    5        65245         1   8     0.0162  5        350     0.199
    6        65418         1   8   0.000171  5       3.68   0.00204
    7        65420         1   8   1.95e-08  5    0.00039  2.28e-07
Optimization Terminated Successfully from sqp
 
Elapsed time is 0.326080 seconds.
          fval: 6.5420e+04
     funcCount: 7
     gradCount: 7
    iterations: 6
       options: [1x18 double]

SQP with exact Hessians
 
                                       Termination Criteria
                                     0.0001           0.5       0.5
                                -----------------------------------
f-CNT         FUNC      STEP NAC     max{g}  j        KTO    max(S)
    1        1e+05         0 211      0.562 201   2.13e+05        35
    2        68325         1 801       26.9 188   1.71e+05      7.87
    3        70421         1 801       11.4 188   9.63e+04      11.1
    4        60010         1 801       4.56 188   4.23e+04      9.99
    5        61919         1 801        1.7 188   1.33e+04      3.08
    6        63907         1 801      0.523 188   1.88e+03      4.44
    7        64244         1 801       0.22 197        306      3.74
    8        64022         1 801      0.516 194        565      5.27
   10        63987       0.1 801      0.461 194        535      5.35
   12        63922     0.192 801      0.362 194        472       5.3
   14        63812     0.375 801      0.197 194        342      5.03
   15        63605         1 801      0.181 188        138      2.22
   16        63653         1 801   0.000544 201       32.1     0.795
   17        63668         1 801   3.19e-05 201       2.02    0.0712
   18        63669         1 801   1.57e-07 201    0.00998  0.000496
Optimization Terminated Successfully from sqp
 
Elapsed time is 12.562141 seconds.
          fval: 6.3669e+04
     funcCount: 18
     gradCount: 15
    iterations: 14
       options: [1x18 double]

SLP Trust

with and without active set strategy
   if N>50
      options.TrustRegion='TRAMfilter';
      tic
      [~,fa,stat,outa] = slp_trust(@fbeamGVa,x0,options,xlb,[],@gbeamGVa), toc
   else
      options.TrustRegion='filter'; tic
      [~,f1,sta1,out1] = slp_trust(@fbeamGV,x0,options,xlb,[],@gbeamGV)
   end
 
         Sequential Linear Programming Iteration History
Iteration      Objective MaxConstraint    Index   Step-size   Merit      MoveLimit  TrustRatio
        0          1e+05        0.5625      6           0   1.396e+05
        1          86438        0.3037      6           8   1.068e+05        0.2      0.4601  g *
        2          73122        0.2401      5           8   1.195e+05        0.2       0.483  g *
        3          64957        0.0784      6           8   8.156e+04        0.2      0.4687  g *
        4          64871       0.04005      6       5.804   7.061e+04        0.2      0.3879  f *
        5          64121       0.06975      2       6.896   7.061e+04        0.2     -0.4509  g +
        6          64646       0.03038      6           4   6.603e+04        0.1      0.4772  g +
        7          65009       0.02236      6           4   6.688e+04        0.1      0.2641  g +
        8          64817       0.02686      6           4   6.688e+04        0.1     -0.2015  g +
        9          65210      0.007001      6           2   6.575e+04       0.05      0.7393  g +
       10          65392       0.00504      6           2   6.603e+04       0.05        0.28  g +
       11          65255      0.006365      6           2   6.603e+04       0.05     -0.2628  g +
       12          65381      0.001588      6           1   6.558e+04      0.025      0.7505  g +
       13          65391      0.001616      6           1   6.558e+04      0.025    -0.01749  g - Rejected
       14          65408     0.0004104      6         0.5   6.546e+04     0.0125      0.7416  g +
       15          65416     0.0004046      6         0.5   6.547e+04     0.0125     0.01425  g +
       16          65418     0.0001006      6        0.25   6.543e+04    0.00625      0.7512  g +
       17          65417     0.0001009      6        0.25   6.543e+04    0.00625   -0.002696  g +
       18          65419      2.53e-05      6       0.125   6.542e+04   0.003125      0.7493  g +
       19          65419     2.562e-05      6       0.125   6.542e+04   0.003125    -0.01246  g - Rejected
       20          65420     6.391e-06      6      0.0625   6.542e+04   0.001563      0.7474  g +
       21          65420     6.411e-06      6      0.0625   6.542e+04   0.001563   -0.003155  g +
       22          65420     1.601e-06      6     0.03125   6.542e+04  0.0007813      0.7503  g + Bound
              ----------  ------------         ----------
    Criteria         0.5        0.0001                0.5
SLP converged. Final objective function value = 65419.6141
               Lagrangian gradient   2-norm = 0.75712
               Lagrangian gradient inf-norm = 0.5419
Trust Region Strategy uses filter
* Dominates prior points
+ Nondominated
- Dominated by prior point(s)
f/g/m Objective/Constraint/Merit governs Trust Ratio

f1 =

   6.5420e+04


sta1 =

     1


out1 = 

             f: [1x23 double]
             g: [11x23 double]
    iterations: 22
       message: 1

 
         Sequential Linear Programming Iteration History
Iteration      Objective MaxConstraint    Index   Step-size   Merit      MoveLimit  TrustRatio
        0          1e+05        0.5625    201           0   1.749e+05
        1          85009         0.363    201           8   1.406e+05        0.2      0.3546  g *
        2          71739         0.342    157           8   2.023e+05        0.2      0.2449  g *
        3          64553        0.6741    185           8   4.656e+05        0.2     -0.2295  g + _g
        4          68620        0.2154    195       3.253   1.168e+05    0.08133      0.6985  g +
        5          67308         0.233    198       3.253   1.844e+05    0.08133      0.6032  g +
        6          65393        0.3759    200       3.253    5.09e+05    0.08133    -0.05922  g + _g
        7          65186       0.01285    189       1.536   6.666e+04    0.03839      0.7233  g +
        8          64519       0.01201    200       1.536   7.818e+04    0.03839     0.01833  g * _g
        9          64292      0.003121    189      0.7822   6.476e+04    0.01955      0.6988  g *
       10          64070      0.002132    201      0.7822   6.417e+04    0.01955     -0.0263  g * _g
       11          64026     0.0005307    201      0.3811   6.405e+04   0.009527      0.7511  g *
       12          63956      0.000536    201      0.3811   6.405e+04   0.009527   -0.009858  g + _g
       13          63935     0.0001318    201      0.1887   6.396e+04   0.004717      0.7541  g *
       14          63907     0.0001317    201      0.1887   6.393e+04   0.004717    0.001118  g * _g
       15          63897      3.35e-05    201     0.09445    6.39e+04   0.002361      0.7456  g *
       16          63885     3.321e-05    201     0.09445   6.389e+04   0.002361    0.008494  g * _g
       17          63880     8.449e-06    201     0.04763   6.388e+04   0.001191      0.7456  g *
       18          63874     8.451e-06    201     0.04763   6.388e+04   0.001191  -0.0002235  g + _g
       19          63872     2.121e-06    201     0.02381   6.387e+04  0.0005952       0.749  g *
       20          63869     2.113e-06    201     0.02381   6.387e+04  0.0005952    0.004079  g * _g
       21          63867     5.328e-07    201     0.01195   6.387e+04  0.0002988      0.7478  g *
       22          63866     5.351e-07    201     0.01195   6.387e+04  0.0002988   -0.004285  g + _g
       23          63865      1.32e-07    201    0.005951   6.387e+04  0.0001488      0.7533  g *
       24          63865     1.323e-07    201    0.005951   6.386e+04  0.0001488   -0.002401  g + _g
       25          63864     3.286e-08    201    0.002968   6.386e+04  7.421e-05      0.7517  g * Bound
              ----------  ------------         ----------
    Criteria         0.5        0.0001                0.5
SLP slowed.    Final objective function value = 63864.251
               Lagrangian gradient   2-norm = 183.9117
               Lagrangian gradient inf-norm = 64.267
Trust Region Strategy uses filter
* Dominates prior points
+ Nondominated
- Dominated by prior point(s)
! Trust Radius set by Merit function minimization
_ Trust Radius set by target Trust Ratio
f/g/m Objective/Constraint/Merit governs Trust Ratio

fa =

   6.3864e+04


stat =

     1


outa = 

             f: [1x26 double]
             g: [401x26 double]
    iterations: 25
       message: 1

Elapsed time is 0.929543 seconds.

fmincon with analytic gradients

   A = [diag(-20.*ones(N,1)),diag(ones(N,1))]; % Linear Constraints
   b_U     = zeros(N,1);                       % Bound on linear constraints
   disp(' ')
   disp('fmincon fails with looser tolerances used for SQP and SLP_Trust')
   Options=optimoptions(@fmincon,'Display','iter','GradObj','on','GradConstr','on',...
                                 'TolX',0.5,'TolFun',0.5,'TolCon',1e-4);
   [~,~,flag,out]=fmincon(@GVbeam_obj,x0,A,b_U,[],[],xlb,[],...
                          @GVbeam_nlcon,Options)
   Options=optimoptions(Options,'Algorithm','sqp');
   [~,~,flag,out]=fmincon(@GVbeam_obj,x0,A,b_U,[],[],xlb,[],...
                          @GVbeam_nlcon,Options)
 
fmincon fails with looser tolerances used for SQP and SLP_Trust
                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       1    1.000000e+05    5.625e-01    3.357e+03
    1       2    6.242015e+04    3.265e+01    8.209e+04    3.728e+01
    2       3    5.691764e+04    2.984e+01    5.167e+04    8.345e+00

Converged to an infeasible point.

fmincon stopped because the size of the current step is less than
the selected value of the step size tolerance but constraints are not
satisfied to within the selected value of the constraint tolerance.




flag =

    -2


out = 

         iterations: 2
          funcCount: 3
    constrviolation: 29.8449
           stepsize: 8.3448
          algorithm: 'interior-point'
      firstorderopt: 5.1674e+04
       cgiterations: 0
            message: 'Converged to an infeasible point....'

                                                          Norm of First-order
 Iter F-count            f(x) Feasibility  Steplength        step  optimality
    0       1    1.000000e+05   5.625e-01                           4.000e+03
    1       2    4.589283e+04   8.471e+01   1.000e+00   4.113e+01   3.567e+05
    2       4    5.445246e+04   4.157e+01   7.000e-01   1.236e+00   4.196e+03

Converged to an infeasible point.

fmincon stopped because the size of the current step is less than
the selected value of the step size tolerance but constraints are not
satisfied to within the selected value of the constraint tolerance.




flag =

    -2


out = 

         iterations: 2
          funcCount: 4
          algorithm: 'sqp'
            message: 'Converged to an infeasible point....'
    constrviolation: 41.5701
           stepsize: 1.2357
       lssteplength: 0.7000
      firstorderopt: 4.1964e+03

 
fmincon fails with looser tolerances used for SQP and SLP_Trust
                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       1    1.000000e+05    5.625e-01    8.522e+01
    1       2    7.813599e+04    6.381e-01    3.458e+01    2.709e+01
    2       3    5.554585e+04    6.280e-01    6.971e+01    1.019e+02
    3       4    5.544879e+04    6.078e-01    4.666e+01    4.257e+00

Converged to an infeasible point.

fmincon stopped because the size of the current step is less than
the selected value of the step size tolerance but constraints are not
satisfied to within the selected value of the constraint tolerance.




flag =

    -2


out = 

         iterations: 3
          funcCount: 4
    constrviolation: 0.6078
           stepsize: 4.2567
          algorithm: 'interior-point'
      firstorderopt: 46.6635
       cgiterations: 0
            message: 'Converged to an infeasible point....'

                                                          Norm of First-order
 Iter F-count            f(x) Feasibility  Steplength        step  optimality
    0       1    1.000000e+05   5.625e-01                           1.000e+02
    1       2    4.468310e+04   2.473e+00   1.000e+00   1.894e+02   4.777e+02
    2       3    5.092954e+04   7.679e-01   1.000e+00   6.480e+01   7.992e+01
    3       4    5.318400e+04   6.726e-01   1.000e+00   3.162e+01   7.324e+01
    4       6    5.858385e+04   1.085e+00   7.000e-01   5.379e+01   1.044e+02
    5       8    5.858385e+04   1.085e+00   7.000e-01   3.811e+01   1.044e+02

Converged to an infeasible point.

fmincon stopped because the size of the current step is less than
the selected value of the step size tolerance but constraints are not
satisfied to within the selected value of the constraint tolerance.




flag =

    -2


out = 

         iterations: 5
          funcCount: 8
          algorithm: 'sqp'
            message: 'Converged to an infeasible point....'
    constrviolation: 1.0852
           stepsize: 38.1065
       lssteplength: 0.7000
      firstorderopt: 104.3944

fmincon, tight tolerances

    disp('fmincon with tighter default tolerances')
    Options=optimoptions(@fmincon,'Display','iter','GradObj','on','GradConstr','on');
    tic
    [~,~,flag,out]=fmincon(@GVbeam_obj,x0,A,b_U,[],[],xlb,[],...
       @GVbeam_nlcon,Options)%#ok<*ASGLU,*NOPTS>
    toc
%     disp('fmincon treating all constraints as nonlinear')
%     tic
%     [~,~,flag,out]=fmincon(@GVbeam_obj,x0,[],[],[],[],xlb,[],...
%        @GVbeam_con,Options);
%     toc
    Options=optimoptions(Options,'Algorithm','sqp'); tic
    [~,~,flag,out]=fmincon(@GVbeam_obj,x0,A,b_U,[],[],xlb,[],...
                           @GVbeam_nlcon,Options), toc
fmincon with tighter default tolerances
                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       1    1.000000e+05    5.625e-01    3.357e+03
    1       2    6.242015e+04    3.265e+01    8.209e+04    3.728e+01
    2       3    5.691764e+04    2.984e+01    5.167e+04    8.345e+00
    3       4    5.678414e+04    2.581e+01    4.067e+04    2.202e+00
    4       5    5.675424e+04    2.571e+01    4.050e+04    1.142e-01
    5       6    5.538877e+04    2.258e+01    3.292e+04    4.519e+00
    6       7    5.535629e+04    1.378e+01    1.459e+04    7.071e+00
    7       8    5.548575e+04    1.329e+01    1.385e+04    1.666e-01
    8       9    5.810871e+04    6.668e+00    5.282e+03    3.328e+00
    9      10    6.009383e+04    3.442e+00    2.548e+03    2.029e+00
   10      11    6.274813e+04    1.366e+00    2.640e+03    2.699e+00
   11      12    6.278678e+04    1.347e+00    2.640e+03    3.932e-02
   12      13    6.278701e+04    1.346e+00    2.640e+03    3.650e-04
   13      14    6.552079e+04    7.564e-01    3.319e+03    3.009e+00
   14      15    6.999767e+04    1.829e-01    3.120e+03    5.824e+00
   15      16    6.898496e+04    1.565e-01    4.392e+03    8.816e+00
   16      17    7.030941e+04    4.594e-02    3.285e+03    7.045e+00
   17      18    7.153474e+04    3.986e-03    1.827e+03    2.222e+00
   18      26    7.160226e+04    7.074e-05    5.177e+02    2.454e+00
   19      27    7.122449e+04    6.415e-03    6.574e+02    2.916e+00
   20      28    7.070723e+04    4.826e-03    6.109e+02    1.312e+00
   21      29    6.516536e+04    4.019e-01    2.420e+03    6.930e+00
   22      30    6.669049e+04    8.065e-02    4.170e+02    2.730e+00
   23      31    6.668830e+04    8.034e-02    3.990e+02    3.393e-02
   24      32    6.477656e+04    1.890e-01    1.675e+03    5.467e+00
   25      33    6.477471e+04    1.868e-01    8.412e+02    6.234e-02
   26      34    6.479281e+04    1.763e-01    7.851e+02    1.822e-01
   27      35    6.518328e+04    1.625e-02    1.221e+02    3.028e+00
   28      36    6.541804e+04    1.693e-04    3.189e+01    2.626e-01
   29      37    6.541968e+04    1.386e-07    8.001e-01    1.829e-02
   30      38    6.541966e+04    2.216e-09    1.108e-01    2.076e-03

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
   31      39    6.541966e+04    0.000e+00    3.023e-03    4.837e-05

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.




flag =

     1


out = 

         iterations: 31
          funcCount: 39
    constrviolation: 0
           stepsize: 4.8367e-05
          algorithm: 'interior-point'
      firstorderopt: 0.0030
       cgiterations: 6
            message: 'Local minimum found that satisfies the constraints....'

Elapsed time is 0.090671 seconds.
                                                          Norm of First-order
 Iter F-count            f(x) Feasibility  Steplength        step  optimality
    0       1    1.000000e+05   5.625e-01                           4.000e+03
    1       2    4.589283e+04   8.471e+01   1.000e+00   4.113e+01   3.567e+05
    2       3    5.445246e+04   4.157e+01   1.000e+00   8.644e+00   4.196e+03
    3       4    5.648206e+04   2.680e+01   1.000e+00   4.740e+00   2.186e+03
    4       7    5.697127e+04   1.755e+01   4.900e-01   5.220e-01   1.823e+03
    5       8    5.795648e+04   7.159e+00   1.000e+00   3.107e+00   3.200e+02
    6       9    5.917717e+04   2.818e+00   1.000e+00   4.456e+00   1.932e+02
    7      10    6.001910e+04   1.133e+00   1.000e+00   7.191e+00   6.012e+02
    8      11    6.386001e+04   3.272e-01   1.000e+00   7.470e+00   1.319e+03
    9      13    6.421652e+04   1.291e-01   7.000e-01   6.001e+00   1.170e+03
   10      16    6.459999e+04   6.838e-02   4.900e-01   4.034e+00   1.018e+03
   11      17    6.478693e+04   3.235e-02   1.000e+00   6.313e+00   6.911e+02
   12      18    6.510309e+04   1.048e-02   1.000e+00   3.566e+00   1.714e+02
   13      19    6.540801e+04   4.186e-04   1.000e+00   7.129e-01   9.455e+01
   14      20    6.541801e+04   5.636e-05   1.000e+00   2.651e-01   3.078e+01
   15      21    6.541958e+04   2.693e-06   1.000e+00   5.783e-02   2.438e-01
   16      22    6.541966e+04   9.953e-10   1.000e+00   1.113e-03   6.804e-02
   17      23    6.541966e+04   3.913e-12   1.000e+00   6.979e-05   5.419e-04

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.




flag =

     1


out = 

         iterations: 17
          funcCount: 23
          algorithm: 'sqp'
            message: 'Local minimum found that satisfies the constraints....'
    constrviolation: 3.9126e-12
           stepsize: 6.9790e-05
       lssteplength: 1
      firstorderopt: 5.4193e-04

Elapsed time is 0.041683 seconds.
fmincon with tighter default tolerances
                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       1    1.000000e+05    5.625e-01    8.522e+01
    1       2    7.813599e+04    6.381e-01    3.458e+01    2.709e+01
    2       3    5.554585e+04    6.280e-01    6.971e+01    1.019e+02
    3       4    5.544879e+04    6.078e-01    4.666e+01    4.257e+00
    4       5    5.566049e+04    4.279e-01    3.304e+01    3.908e+01
    5       6    5.734346e+04    2.724e-01    1.990e+01    2.991e+01
    6       7    5.980387e+04    1.480e-01    2.147e+01    2.161e+01
    7       8    6.258730e+04    3.752e-02    1.909e+01    2.209e+01
    8       9    6.326099e+04    1.557e-02    1.559e+01    1.346e+01
    9      11    6.338186e+04    1.028e-02    1.591e+01    4.051e+00
   10      13    6.346022e+04    7.328e-03    7.327e+00    1.772e+00
   11      14    6.355833e+04    4.920e-02    3.274e+00    1.793e+00
   12      15    6.367092e+04    7.490e-05    3.407e+00    1.972e+00
   13      16    6.367322e+04    4.999e-07    3.349e-01    2.478e-01
   14      17    6.366933e+04    1.855e-07    3.284e-01    1.020e-01
   15      18    6.366921e+04    3.811e-08    2.101e-01    6.229e-02
   16      19    6.366921e+04    4.636e-09    6.315e-02    2.163e-02
   17      20    6.366921e+04    0.000e+00    5.297e-03    3.711e-03
   18      21    6.366920e+04    0.000e+00    1.882e-03    8.901e-04
   19      22    6.366920e+04    0.000e+00    5.871e-04    2.165e-04
   20      23    6.366920e+04    0.000e+00    2.135e-04    9.138e-05
   21      24    6.366920e+04    0.000e+00    3.475e-05    1.832e-05

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.




flag =

     1


out = 

         iterations: 21
          funcCount: 24
    constrviolation: 0
           stepsize: 1.8319e-05
          algorithm: 'interior-point'
      firstorderopt: 3.4753e-05
       cgiterations: 0
            message: 'Local minimum found that satisfies the constraints....'

Elapsed time is 1.066116 seconds.
                                                          Norm of First-order
 Iter F-count            f(x) Feasibility  Steplength        step  optimality
    0       1    1.000000e+05   5.625e-01                           1.000e+02
    1       2    4.468310e+04   2.473e+00   1.000e+00   1.894e+02   4.777e+02
    2       3    5.092954e+04   7.679e-01   1.000e+00   6.480e+01   7.992e+01
    3       4    5.318366e+04   6.726e-01   1.000e+00   3.162e+01   7.324e+01
    4       5    5.390000e+04   3.971e-01   1.000e+00   1.304e+01   1.958e+01
    5       7    5.940931e+04   1.556e-01   7.000e-01   3.623e+01   5.942e+01
    6      11    6.058786e+04   1.067e-01   3.430e-01   1.321e+01   2.010e+01
    7      12    6.342722e+04   2.871e-02   1.000e+00   1.973e+01   2.888e+01
    8      13    6.364517e+04   8.270e-04   1.000e+00   6.326e+00   1.475e+01
    9      14    6.366731e+04   6.522e-05   1.000e+00   1.856e+00   2.696e+00
   10      15    6.366886e+04   1.477e-05   1.000e+00   9.178e-01   1.865e+00
   11      16    6.366908e+04   3.658e-06   1.000e+00   4.353e-01   5.654e-01
   12      17    6.366920e+04   7.352e-08   1.000e+00   6.439e-02   1.911e-01
   13      18    6.366920e+04   4.965e-09   1.000e+00   1.744e-02   3.118e-02
   14      19    6.366920e+04   2.367e-10   1.000e+00   4.400e-03   1.104e-02
   15      20    6.366920e+04   2.217e-11   1.000e+00   1.458e-03   4.797e-03
   16      21    6.366920e+04   2.233e-12   1.000e+00   4.669e-04   9.730e-04
   17      22    6.366920e+04   2.127e-13   1.000e+00   1.241e-04   3.226e-04
   18      23    6.366920e+04   2.127e-13   7.000e-01   1.618e-05   1.159e-05

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.




flag =

     1


out = 

         iterations: 18
          funcCount: 23
          algorithm: 'sqp'
            message: 'Local minimum found that satisfies the constraints....'
    constrviolation: 2.1272e-13
           stepsize: 1.6184e-05
       lssteplength: 0.7000
      firstorderopt: 1.1591e-05

Elapsed time is 56.301147 seconds.
end