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 multirosenbrockfunction 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 300 个，这很有挑战性。
设置问题。
N = 20; % Any even number mf = 300; % 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 = 3.000000e+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
____________ _____________ _______
8.682 539.57 99.869
再进行 300 次函数计算后可以发现，其他求解器在向真实解靠拢方面取得了良好进展，而 surrogateopt 则没有显著改善。从先前的解重新启动求解器，这会为每次优化添加 300 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 = 3.000000e+02.
table(fvalm,pfval,fmfval,VariableNames=["surrogateopt","patternsearch","fmincon"])
ans=1×3 table
surrogateopt patternsearch fmincon
____________ _____________ _______
7.8981 215.99 9.4401
