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.

## 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. ```

``` 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 ```
``` 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 ```