多维函数的替代优化
此示例展示了三个推荐求解器在最小化问题上的行为。目标函数是 multirosenbrock
函数:
type multirosenbrock
function F = multirosenbrock(x) % This function is a multidimensional generalization of Rosenbrock's % function. It operates in a vectorized manner, assuming that x is a matrix % whose rows are the individuals. % Copyright 2014 by The MathWorks, Inc. N = size(x,2); % assumes x is a row vector or 2-D matrix if mod(N,2) % if N is odd error('Input rows must have an even number of elements') end odds = 1:2:N-1; evens = 2:2:N; F = zeros(size(x)); F(:,odds) = 1-x(:,odds); F(:,evens) = 10*(x(:,evens)-x(:,odds).^2); F = sum(F.^2,2);
multirosenbrock
函数在点 [1,1,...,1]
处有一个局部最小值 0。看看针对一般非线性问题的三个最佳求解器在 20 维中对此函数的运行情况,而最大函数数量只有 200 个,这很有挑战性。
设置问题。
N = 20; % any even number mf = 200; % max fun evals fun = @multirosenbrock; lb = -3*ones(1,N); ub = -lb; rng default x0 = -3*rand(1,N);
设置 surrogateopt
的选项以仅使用 200 个函数计算,然后运行求解器。
options = optimoptions('surrogateopt','MaxFunctionEvaluations',mf); [xm,fvalm,~,~,pop] = surrogateopt(fun,lb,ub,options);
surrogateopt stopped because it exceeded the function evaluation limit set by 'options.MaxFunctionEvaluations'.
为 patternsearch
设置类似的选项,包括一个绘图函数来监控优化。
psopts = optimoptions('patternsearch','PlotFcn','psplotbestf','MaxFunctionEvaluations',mf); [psol,pfval] = patternsearch(fun,x0,[],[],[],[],lb,ub,[],psopts);
patternsearch stopped because it exceeded options.MaxFunctionEvaluations.
为 fmincon
设置类似的选项。
opts = optimoptions('fmincon','PlotFcn','optimplotfval','MaxFunctionEvaluations',mf); [fmsol,fmfval,eflag,fmoutput] = fmincon(fun,x0,[],[],[],[],lb,ub,[],opts);
Solver stopped prematurely. fmincon stopped because it exceeded the function evaluation limit, options.MaxFunctionEvaluations = 2.000000e+02.
对于这个极其有限的函数计算次数,surrogateopt
解最接近 0
的真实最小值。
table(fvalm,pfval,fmfval,'VariableNames',{'surrogateopt','patternsearch','fmincon'})
ans=1×3 table
surrogateopt patternsearch fmincon
____________ _____________ _______
8.8498 774.8 493.7
再进行 200 次函数计算表明其他求解器可以快速接近真实解,而 surrogateopt
并没有显著改善。从先前的解重新启动求解器,这会为每次优化添加 200 个函数计算。
options = optimoptions(options,'InitialPoints',pop);
[xm,fvalm,~,~,pop] = surrogateopt(fun,lb,ub,options);
surrogateopt stopped because it exceeded the function evaluation limit set by 'options.MaxFunctionEvaluations'.
[psol,pfval] = patternsearch(fun,psol,[],[],[],[],lb,ub,[],psopts);
patternsearch stopped because it exceeded options.MaxFunctionEvaluations.
[fmsol,fmfval,eflag,fmoutput] = fmincon(fun,fmsol,[],[],[],[],lb,ub,[],opts);
Solver stopped prematurely. fmincon stopped because it exceeded the function evaluation limit, options.MaxFunctionEvaluations = 2.000000e+02.
table(fvalm,pfval,fmfval,'VariableNames',{'surrogateopt','patternsearch','fmincon'})
ans=1×3 table
surrogateopt patternsearch fmincon
____________ _____________ _______
8.3754 326.73 8.5989