# 使用 Parallel Computing Toolbox 最小化高成本优化问题

### 高成本优化问题

```function f = expensive_objfun(x) %EXPENSIVE_OBJFUN An expensive objective function used in optimparfor example. % Copyright 2007-2013 The MathWorks, Inc. % Simulate an expensive function by pausing pause(0.1) % Evaluate objective function f = exp(x(1)) * (4*x(3)^2 + 2*x(4)^2 + 4*x(1)*x(2) + 2*x(2) + 1); ```
```function [c,ceq] = expensive_confun(x) %EXPENSIVE_CONFUN An expensive constraint function used in optimparfor example. % Copyright 2007-2013 The MathWorks, Inc. % Simulate an expensive function by pausing pause(0.1); % Evaluate constraints c = [1.5 + x(1)*x(2)*x(3) - x(1) - x(2) - x(4); -x(1)*x(2) + x(4) - 10]; % No nonlinear equality constraints: ceq = []; ```

### 使用 `fmincon` 进行最小化

```startPoint = [-1 1 1 -1]; options = optimoptions('fmincon','Display','iter','Algorithm','interior-point'); startTime = tic; xsol = fmincon(@expensive_objfun,startPoint,[],[],[],[],[],[],@expensive_confun,options); time_fmincon_sequential = toc(startTime); fprintf('Serial FMINCON optimization takes %g seconds.\n',time_fmincon_sequential); ```
``` First-order Norm of Iter F-count f(x) Feasibility optimality step 0 5 1.839397e+00 1.500e+00 3.211e+00 1 11 -9.760099e-01 3.708e+00 7.902e-01 2.362e+00 2 16 -1.480976e+00 0.000e+00 8.344e-01 1.069e+00 3 21 -2.601599e+00 0.000e+00 8.390e-01 1.218e+00 4 29 -2.823630e+00 0.000e+00 2.598e+00 1.118e+00 5 34 -3.905338e+00 0.000e+00 1.210e+00 7.302e-01 6 39 -6.212992e+00 3.934e-01 7.372e-01 2.405e+00 7 44 -5.948761e+00 0.000e+00 1.784e+00 1.905e+00 8 49 -6.940062e+00 1.233e-02 7.668e-01 7.553e-01 9 54 -6.973887e+00 0.000e+00 2.549e-01 3.920e-01 10 59 -7.142993e+00 0.000e+00 1.903e-01 4.735e-01 11 64 -7.155325e+00 0.000e+00 1.365e-01 2.626e-01 12 69 -7.179122e+00 0.000e+00 6.336e-02 9.115e-02 13 74 -7.180116e+00 0.000e+00 1.069e-03 4.670e-02 14 79 -7.180409e+00 0.000e+00 7.797e-04 2.815e-03 15 84 -7.180410e+00 0.000e+00 6.368e-06 3.120e-04 Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. Serial FMINCON optimization takes 18.2876 seconds. ```

### 使用遗传算法进行最小化

```rng default % for reproducibility try gaAvailable = false; nvar = 4; gaoptions = optimoptions('ga','MaxGenerations',15,'Display','iter'); startTime = tic; gasol = ga(@expensive_objfun,nvar,[],[],[],[],[],[],[],gaoptions); time_ga_sequential = toc(startTime); fprintf('Serial GA optimization takes %g seconds.\n',time_ga_sequential); gaAvailable = true; catch ME warning(message('optim:optimdemos:optimparfor:gaNotFound')); end ```
```Single objective optimization: 4 Variable(s) Options: CreationFcn: @gacreationuniform CrossoverFcn: @crossoverscattered SelectionFcn: @selectionstochunif MutationFcn: @mutationgaussian Best Mean Stall Generation Func-count f(x) f(x) Generations 1 100 -5.546e+05 1.483e+15 0 2 147 -5.581e+17 -1.116e+16 0 3 194 -7.556e+17 6.679e+22 0 4 241 -7.556e+17 -7.195e+16 1 5 288 -9.381e+27 -1.876e+26 0 6 335 -9.673e+27 -7.497e+26 0 7 382 -4.511e+36 -9.403e+34 0 8 429 -5.111e+36 -3.011e+35 0 9 476 -7.671e+36 9.346e+37 0 10 523 -1.52e+43 -3.113e+41 0 11 570 -2.273e+45 -4.67e+43 0 12 617 -2.589e+47 -6.281e+45 0 13 664 -2.589e+47 -1.015e+46 1 14 711 -8.149e+47 -5.855e+46 0 15 758 -9.503e+47 -1.29e+47 0 Optimization terminated: maximum number of generations exceeded. Serial GA optimization takes 81.5878 seconds. ```

### 设置 Parallel Computing Toolbox

```if max(size(gcp)) == 0 % parallel pool needed parpool % create the parallel pool end ```

### 使用并行 `fmincon` 进行最小化

```options = optimoptions(options,'UseParallel',true); startTime = tic; xsol = fmincon(@expensive_objfun,startPoint,[],[],[],[],[],[],@expensive_confun,options); time_fmincon_parallel = toc(startTime); fprintf('Parallel FMINCON optimization takes %g seconds.\n',time_fmincon_parallel); ```
``` First-order Norm of Iter F-count f(x) Feasibility optimality step 0 5 1.839397e+00 1.500e+00 3.211e+00 1 11 -9.760099e-01 3.708e+00 7.902e-01 2.362e+00 2 16 -1.480976e+00 0.000e+00 8.344e-01 1.069e+00 3 21 -2.601599e+00 0.000e+00 8.390e-01 1.218e+00 4 29 -2.823630e+00 0.000e+00 2.598e+00 1.118e+00 5 34 -3.905338e+00 0.000e+00 1.210e+00 7.302e-01 6 39 -6.212992e+00 3.934e-01 7.372e-01 2.405e+00 7 44 -5.948761e+00 0.000e+00 1.784e+00 1.905e+00 8 49 -6.940062e+00 1.233e-02 7.668e-01 7.553e-01 9 54 -6.973887e+00 0.000e+00 2.549e-01 3.920e-01 10 59 -7.142993e+00 0.000e+00 1.903e-01 4.735e-01 11 64 -7.155325e+00 0.000e+00 1.365e-01 2.626e-01 12 69 -7.179122e+00 0.000e+00 6.336e-02 9.115e-02 13 74 -7.180116e+00 0.000e+00 1.069e-03 4.670e-02 14 79 -7.180409e+00 0.000e+00 7.797e-04 2.815e-03 15 84 -7.180410e+00 0.000e+00 6.368e-06 3.120e-04 Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. Parallel FMINCON optimization takes 8.79291 seconds. ```

### 使用并行遗传算法进行最小化

```rng default % to get the same evaluations as the previous run if gaAvailable gaoptions = optimoptions(gaoptions,'UseParallel',true); startTime = tic; gasol = ga(@expensive_objfun,nvar,[],[],[],[],[],[],[],gaoptions); time_ga_parallel = toc(startTime); fprintf('Parallel GA optimization takes %g seconds.\n',time_ga_parallel); end ```
```Single objective optimization: 4 Variable(s) Options: CreationFcn: @gacreationuniform CrossoverFcn: @crossoverscattered SelectionFcn: @selectionstochunif MutationFcn: @mutationgaussian Best Mean Stall Generation Func-count f(x) f(x) Generations 1 100 -5.546e+05 1.483e+15 0 2 147 -5.581e+17 -1.116e+16 0 3 194 -7.556e+17 6.679e+22 0 4 241 -7.556e+17 -7.195e+16 1 5 288 -9.381e+27 -1.876e+26 0 6 335 -9.673e+27 -7.497e+26 0 7 382 -4.511e+36 -9.403e+34 0 8 429 -5.111e+36 -3.011e+35 0 9 476 -7.671e+36 9.346e+37 0 10 523 -1.52e+43 -3.113e+41 0 11 570 -2.273e+45 -4.67e+43 0 12 617 -2.589e+47 -6.281e+45 0 13 664 -2.589e+47 -1.015e+46 1 14 711 -8.149e+47 -5.855e+46 0 15 758 -9.503e+47 -1.29e+47 0 Optimization terminated: maximum number of generations exceeded. Parallel GA optimization takes 15.2253 seconds. ```

### 比较串行和并行时间

```X = [time_fmincon_sequential time_fmincon_parallel]; Y = [time_ga_sequential time_ga_parallel]; t = [0 1]; plot(t,X,'r--',t,Y,'k-') ylabel('Time in seconds') legend('fmincon','ga') ax = gca; ax.XTick = [0 1]; ax.XTickLabel = {'Serial' 'Parallel'}; axis([0 1 0 ceil(max([X Y]))]) title('Serial Vs. Parallel Times') ```