Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox

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This example shows how to speed up the minimization of an expensive optimization problem using functions in Optimization Toolbox™ and Global Optimization Toolbox. In the first part of the example we solve the optimization problem by evaluating functions in a serial fashion, and in the second part of the example we solve the same problem using the parallel for loop (parfor) feature by evaluating functions in parallel. We compare the time taken by the optimization function in both cases.

Expensive Optimization Problem

For the purpose of this example, we solve a problem in four variables, where the objective and constraint functions are made artificially expensive by pausing.

functionf = expensive_objfun(x)%EXPENSIVE_OBJFUN An expensive objective function used in optimparfor example.% Copyright 2007-2013 The MathWorks, Inc.% Simulate an expensive function by pausingpause(0.1)% Evaluate objective functionf = 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 pausingpause(0.1);% Evaluate constraintsc = [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 = [];

Minimizing Using`fmincon`

我们一个re interested in measuring the time taken byfminconin serial so that we can compare it to the parallel time.

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);

一阶范数的Iter F-count f (x)的可行性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 -3.905339e+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 -5.948762e+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 7.799e-04 2.815e-03 15 84 -7.180410e+00 0.000e+00 6.189e-06 3.122e-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 17.0722 seconds.

Minimizing Using Genetic Algorithm

Sincegausually takes many more function evaluations thanfmincon, we remove the expensive constraint from this problem and perform unconstrained optimization instead. Pass empty matrices[]for constraints. In addition, we limit the maximum number of generations to 15 forgaso thatga可以在合理的时间内终止。我们一个re interested in measuring the time taken bygaso that we can compare it to the parallelgaevaluation. Note that runninggarequires Global Optimization Toolbox.

rngdefault% for reproducibilitytrygaAvailable = 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;catchME warning(message('optimdemos:optimparfor:gaNotFound'));end

Best Mean Stall Generation Func-count f(x) f(x) Generations 1 100 -5.546e+05 1.483e+15 0 2 150 -5.581e+17 -1.116e+16 0 3 200 -7.556e+17 6.679e+22 0 4 250 -7.556e+17 -7.195e+16 1 5 300 -9.381e+27 -1.876e+26 0 6 350 -9.673e+27 -7.497e+26 0 7 400 -4.511e+36 -9.403e+34 0 8 450 -5.111e+36 -3.011e+35 0 9 500 -7.671e+36 9.346e+37 0 10 550 -1.52e+43 -3.113e+41 0 11 600 -2.273e+45 -4.67e+43 0 12 650 -2.589e+47 -6.281e+45 0 13 700 -2.589e+47 -1.015e+46 1 14 750 -8.149e+47 -5.855e+46 0 15 800 -9.503e+47 -1.29e+47 0 Optimization terminated: maximum number of generations exceeded. Serial GA optimization takes 80.2351 seconds.

Setting Parallel Computing Toolbox

The finite differencing used by the functions in Optimization Toolbox to approximate derivatives is done in parallel using theparforfeature if Parallel Computing Toolbox™ is available and there is a parallel pool of workers. Similarly,ga,gamultiobj, andpatternsearchsolvers in Global Optimization Toolbox evaluate functions in parallel. To use theparforfeature, we use theparpoolfunction to set up the parallel environment. The computer on which this example is published has four cores, soparpoolstarts four MATLAB® workers. If there is already a parallel pool when you run this example, we use that pool; see the documentation forparpoolfor more information.

ifmax(size(gcp)) == 0% parallel pool neededparpool% create the parallel poolend

Minimizing Using Parallel`fmincon`

To minimize our expensive optimization problem using the parallelfmincon函数,我们需要明确表明objective and constraint functions can be evaluated in parallel and that we wantfminconto use its parallel functionality wherever possible. Currently, finite differencing can be done in parallel. We are interested in measuring the time taken byfminconso that we can compare it to the serialfminconrun.

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);

一阶范数的Iter F-count f (x)的可行性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 -3.905339e+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 -5.948762e+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 7.799e-04 2.815e-03 15 84 -7.180410e+00 0.000e+00 6.189e-06 3.122e-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 8.11945 seconds.

Minimizing Using Parallel Genetic Algorithm

To minimize our expensive optimization problem using thega函数,我们需要明确表明objective function can be evaluated in parallel and that we wantgato use its parallel functionality wherever possible. To use the parallelgawe also require that the 'Vectorized' option be set to the default (i.e., 'off'). We are again interested in measuring the time taken bygaso that we can compare it to the serialgarun. Though this run may be different from the serial one becausegauses a random number generator, the number of expensive function evaluations is the same in both runs. Note that runninggarequires Global Optimization Toolbox.

rngdefault% to get the same evaluations as the previous runifgaAvailable 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

Best Mean Stall Generation Func-count f(x) f(x) Generations 1 100 -5.546e+05 1.483e+15 0 2 150 -5.581e+17 -1.116e+16 0 3 200 -7.556e+17 6.679e+22 0 4 250 -7.556e+17 -7.195e+16 1 5 300 -9.381e+27 -1.876e+26 0 6 350 -9.673e+27 -7.497e+26 0 7 400 -4.511e+36 -9.403e+34 0 8 450 -5.111e+36 -3.011e+35 0 9 500 -7.671e+36 9.346e+37 0 10 550 -1.52e+43 -3.113e+41 0 11 600 -2.273e+45 -4.67e+43 0 12 650 -2.589e+47 -6.281e+45 0 13 700 -2.589e+47 -1.015e+46 1 14 750 -8.149e+47 -5.855e+46 0 15 800 -9.503e+47 -1.29e+47 0 Optimization terminated: maximum number of generations exceeded. Parallel GA optimization takes 15.6984 seconds.

Compare Serial and Parallel Time

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')

Utilizing parallel function evaluation viaparforimproved the efficiency of bothfminconandga. The improvement is typically better for expensive objective and constraint functions.