Store 
Optimization Toolbox 4.3
Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox™
This is a demonstration of how to speedup the minimization of an expensive optimization problem using functions in Optimization Toolbox™ and Genetic Algorithm and Direct Search Toolbox™. In the first part of the demo we will solve the optimization problem by evaluating functions in a serial fashion and in the second part of the demo we will solve the same problem using the parallel for loop (PARFOR) feature by evaluating functions in parallel. We will compare the time taken by the optimization function in both cases.
Contents
Expensive Optimization Problem
For the purpose of this demo, we solve a problem in four variables, where the objective and constraint functions are made artificially expensive by having them compute the eigenvalues of a large matrix.
type expensive_objfun.m type expensive_confun.m
function f = expensive_objfun(x)
%EXPENSIVE_OBJFUN An expensive objective function used in optimparfor demo.
% Copyright 2007 The MathWorks, Inc.
% $Revision: 1.1.4.1 $ $Date: 2007/11/29 15:21:36 $
% Simulate an expensive function by performing an expensive computation
eig(magic(300));
% 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 demo.
% Copyright 2007 The MathWorks, Inc.
% $Revision: 1.1.4.1 $ $Date: 2007/11/29 15:21:35 $
% Simulate an expensive function by performing an expensive computation
eig(magic(450));
% 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 = [];
We can measure the approximate time taken by the objective function and the constraint function to evaluate a point. This can give us an estimate of total time taken to minimize the problem.
startTime = tic; for i = 1:10 expensive_objfun(rand(1,4)); end stopTime = toc(startTime); averageTime = stopTime/10; fprintf('Objective function evaluation: %g seconds.\n', averageTime); startTime = tic; for i = 1:10 expensive_confun(rand(1,4)); end stopTime = toc(startTime); averageTime = stopTime/10; fprintf('Constraint function evaluation: %g seconds.\n',averageTime);
Objective function evaluation: 0.188224 seconds. Constraint function evaluation: 0.221115 seconds.
Minimizing Using FMINCON
We are interested in measuring the time taken by FMINCON so that we can compare it to the parallel FMINCON evaluation.
startPoint = [1 -2 0 5]; options = optimset('Display','iter','Algorithm','active-set'); startTime = tic; fmincon(@expensive_objfun,startPoint,[],[],[],[],[],[],@expensive_confun,opt ions); time_fmincon_sequential = toc(startTime); fprintf('Serial FMINCON optimization takes %g seconds.\n',time_fmincon_seque ntial);
Max Line search Directional First-orde
r
Iter F-count f(x) constraint steplength derivative optimalit
y Procedure
0 5 106.013 -2.5
1 16 40.1132 -1.131 0.0156 -83.7 24.
6
2 23 23.3775 -2.136 0.25 -12.3 20.
5
3 31 14.3993 -1.307 0.125 -9.59 25.
8
4 36 6.2782 -0.9803 1 -6.95 5.8
7
5 41 0.412294 -0.4266 1 -2.56 3.
7
6 46 -1.71467 -0.06477 1 -1.92 2.6
2
7 52 -3.08134 -2.682 0.5 -1.83 9.3
4
8 57 -5.97906 0.121 1 -2.82 5.7
5
9 62 -6.95348 0.01616 1 -1.77 0.63
8
10 67 -7.05833 -0.02524 1 -0.375 0.85
2
11 72 -7.08266 -0.0003593 1 -0.267 0.42
6
12 77 -7.0948 6.967e-05 1 -0.235 0.50
9
13 82 -7.12753 -0.002446 1 -0.296 0.34
4
14 87 -7.15265 -0.009521 1 -0.161 0.080
9
15 92 -7.16982 -0.01013 1 -0.107 0.29
3
16 97 -7.17988 -0.0005696 1 -0.107 0.15
4
17 102 -7.18041 -2.953e-06 1 -0.0535 0.0098
1
18 107 -7.18041 -3.2e-08 1 -0.00416 0.00011
2 Hessian modified
Local minimum possible. Constraints satisfied.
fmincon stopped because the predicted change in the objective function
is less than the default value of the function tolerance and constraints
were satisfied to within the default value of the constraint tolerance.
Active inequalities (to within options.TolCon = 1e-06):
lower upper ineqlin ineqnonlin
2
Serial FMINCON optimization takes 41.0629 seconds.
Minimizing Using Genetic Algorithm
Since GA usually takes more function evaluations than FMINCON we will remove the expensive constraint from this problem and perform unconstrained optimization instead; we pass empty ([]) for constraints. In addition, we limit the maximum generations to 15 for GA so that GA can terminate in a reasonable amount of time. We are interested in measuring the time taken by GA so that we can compare it to the parallel GA evaluation.
nvar = 4; gaoptions = gaoptimset('Generations',15,'Display','iter'); startTime = tic; ga(@expensive_objfun,nvar,[],[],[],[],[],[],[],gaoptions); time_ga_sequential = toc(startTime); fprintf('Serial GA optimization takes %g seconds.\n',time_ga_sequential);
Best Mean Stall
Generation f-count f(x) f(x) Generations
1 40 1.469 23.18 0
2 60 1.236 20.34 0
3 80 0.655 7.149 0
4 100 -0.9582 5.889 0
5 120 -1.876 4.473 0
6 140 -1.876 2.757 1
7 160 -1.958 0.2561 0
8 180 -2.387 2.363 0
9 200 -5.238 3.562 0
10 220 -23.18 2.033 0
11 240 -44.76 -1.239 0
12 260 -44.76 -12.72 1
13 280 -44.76 -23.8 2
14 300 -64.49 -34.12 0
15 320 -64.49 -42.11 1
Optimization terminated: maximum number of generations exceeded.
Serial GA optimization takes 41.4153 seconds.
Setting Parallel Computing Toolbox
The finite differencing used by the functions in Optimization Toolbox to approximate derivatives is done in parallel using the PARFOR feature if Parallel Computing Toolbox is available and MATLABPOOL is running. Similarly, GA, GAMULTIOBJ, and PATTERNSEARCH solvers in Genetic Algorithm and Direct Search Toolbox evaluate functions in parallel. To use the PARFOR feature, we can use the MATLABPOOL function to setup the parallel environment. MATLABPOOL will start four MATLAB® workers on the local machine by default. The computer on which this demo is published has two cores so we will start only two workers. Also, note that if MATLABPOOL is already open we will get an error trying to open again; see documentation for MATLABPOOL for more information.
matlabpool open 2
Starting matlabpool using the 'local' configuration ... connected to 2 labs.
Minimizing Using Parallel FMINCON
To minimize our expensive optimization problem using the parallel FMINCON function, we need to explicitly indicate that our objective and constraint functions can be evaluated in parallel and that we want FMINCON to use its parallel functionality wherever possible. Currently, finite differencing can be done in parallel. We are interested in measuring the time taken by FMINCON so that we can compare it to the serial FMINCON run.
options = optimset(options,'UseParallel','always'); startTime = tic; fmincon(@expensive_objfun,startPoint,[],[],[],[],[],[],@expensive_confun,opt ions); time_fmincon_parallel = toc(startTime); fprintf('Parallel FMINCON optimization takes %g seconds.\n',time_fmincon_par allel);
Max Line search Directional First-orde
r
Iter F-count f(x) constraint steplength derivative optimalit
y Procedure
0 5 106.013 -2.5
1 16 40.1132 -1.131 0.0156 -83.7 24.
6
2 23 23.3775 -2.136 0.25 -12.3 20.
5
3 31 14.3993 -1.307 0.125 -9.59 25.
8
4 36 6.2782 -0.9803 1 -6.95 5.8
7
5 41 0.412294 -0.4266 1 -2.56 3.
7
6 46 -1.71467 -0.06477 1 -1.92 2.6
2
7 52 -3.08134 -2.682 0.5 -1.83 9.3
4
8 57 -5.97906 0.121 1 -2.82 5.7
5
9 62 -6.95348 0.01616 1 -1.77 0.63
8
10 67 -7.05833 -0.02524 1 -0.375 0.85
2
11 72 -7.08266 -0.0003593 1 -0.267 0.42
6
12 77 -7.0948 6.967e-05 1 -0.235 0.50
9
13 82 -7.12753 -0.002446 1 -0.296 0.34
4
14 87 -7.15265 -0.009521 1 -0.161 0.080
9
15 92 -7.16982 -0.01013 1 -0.107 0.29
3
16 97 -7.17988 -0.0005696 1 -0.107 0.15
4
17 102 -7.18041 -2.953e-06 1 -0.0535 0.0098
1
18 107 -7.18041 -3.2e-08 1 -0.00416 0.00011
2 Hessian modified
Local minimum possible. Constraints satisfied.
fmincon stopped because the predicted change in the objective function
is less than the default value of the function tolerance and constraints
were satisfied to within the default value of the constraint tolerance.
Active inequalities (to within options.TolCon = 1e-06):
lower upper ineqlin ineqnonlin
2
Parallel FMINCON optimization takes 26.5757 seconds.
Minimizing Using Parallel Genetic Algorithm
To minimize our expensive optimization problem using the GA function, we need to explicitly indicate that our objective function can be evaluated in parallel and that we want GA to use its parallel functionality wherever possible. To use the parallel GA we also require that the 'Vectorized' option be set to the default (i.e., 'off'). We are again interested in measuring the time taken by GA so that we can compare it to the serial GA run. Though this run may be different from the serial one because GA uses a random number generator, the number of expensive function evaluations is the same in both runs.
gaoptions = gaoptimset(gaoptions,'UseParallel','always'); startTime = tic; ga(@expensive_objfun,nvar,[],[],[],[],[],[],[],gaoptions); time_ga_parallel = toc(startTime); fprintf('Parallel GA optimization takes %g seconds.\n',time_ga_parallel);
Best Mean Stall
Generation f-count f(x) f(x) Generations
1 40 2.084 59.95 0
2 60 1.636 21.14 0
3 80 -0.1087 17.08 0
4 100 -0.4921 15.78 0
5 120 -0.4921 19.7 1
6 140 -192.4 -7.161 0
7 160 -416.4 -66.16 0
8 180 -416.4 -98.58 1
9 200 -416.4 -157 2
10 220 -680.2 -287.2 0
11 240 -1250 -406 0
12 260 -1250 -586.4 1
13 280 -1250 -725.5 2
14 300 -1298 -923.3 0
15 320 -1298 -1099 1
Optimization terminated: maximum number of generations exceeded.
Parallel GA optimization takes 22.2307 seconds.
Utilizing parallel function evaluation via PARFOR improved the efficiency of both FMINCON and GA. The improvement is typically better for expensive objective and constraint functions. Also, using more than two workers and/or using dedicated clusters with MATLABPOOL gives better performance. At last we close the parallel environment by calling MATLABPOOL.
matlabpool close
Sending a stop signal to all the labs ... stopped.
