使用基于问题的方法比较全局求解器

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此示例说明如何使用几个求解器最小化拉斯特里金函数。每个求解器都有自己的特征。这些特征会得到不同的解和运行时间。比较求解器和解中总结的结果可以帮助您为自己的问题选择合适的求解器。

拉斯特里金函数有许多局部最小值，全局最小值位于 (0,0)：

ras = @(x, y) 20 + x.^2 + y.^2 - 10*(cos(2*pi*x) + cos(2*pi*y));

绘制在每个方向上缩放 10 倍的函数。

rf3 = @(x, y) ras(x/10, y/10);
fsurf(rf3,[-30 30],"ShowContours","on")
title("rastriginsfcn([x/10,y/10])")
xlabel("x")
ylabel("y")

Figure contains an axes object. The axes object with title rastriginsfcn([x/10,y/10]), xlabel x, ylabel y contains an object of type functionsurface.

通常您不知道目标函数的全局最小值的位置。为了展示求解器如何寻找全局解，此示例将围绕远离全局最小值的点 [20,30] 启动所有求解器。

fminunc 求解器

要使用默认 Optimization Toolbox™ 求解器 fminunc 求解优化问题，请输入：

x = optimvar("x");
y = optimvar("y");
prob = optimproblem("Objective",rf3(x,y));
x0.x = 20;
x0.y = 30;
[solf,fvalf,eflagf,outputf] = solve(prob,x0)
Solving problem using fminunc.

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

<stopping criteria details>

solf = struct with fields:
    x: 19.8991
    y: 29.8486


fvalf = 
12.9344

eflagf = 
    OptimalSolution


outputf = struct with fields:
             iterations: 3
              funcCount: 5
               stepsize: 1.7773e-06
           lssteplength: 1
          firstorderopt: 2.0461e-09
              algorithm: 'quasi-newton'
                message: 'Local minimum found.↵↵Optimization completed because the size of the gradient is less than↵the value of the optimality tolerance.↵↵<stopping criteria details>↵↵Optimization completed: The first-order optimality measure, 1.278784e-09, is less ↵than options.OptimalityTolerance = 1.000000e-06.'
    objectivederivative: "reverse-AD"
                 solver: 'fminunc'

fminunc 经过很少的函数计算（只有五次，如 outputf 结构体中所示）就完成了问题求解，并且在起点附近达到局部最小值。此退出标志表示该解是局部最小值。

patternsearch 求解器

要使用 Global Optimization Toolbox 求解器 patternsearch 求解优化问题，请输入：

x0.x = 20;
x0.y = 30;
[solp,fvalp,eflagp,outputp] = solve(prob,x0,"Solver","patternsearch")
Solving problem using patternsearch.
patternsearch stopped because the mesh size was less than options.MeshTolerance.

solp = struct with fields:
    x: 19.8991
    y: -9.9496


fvalp = 
4.9748

eflagp = 
    SolverConvergedSuccessfully


outputp = struct with fields:
         function: @(x)fun(x,extraParams)
      problemtype: 'unconstrained'
       pollmethod: 'gpspositivebasis2n'
    maxconstraint: []
     searchmethod: []
       iterations: 48
        funccount: 174
         meshsize: 9.5367e-07
         rngstate: [1×1 struct]
          message: 'patternsearch stopped because the mesh size was less than options.MeshTolerance.'
           solver: 'patternsearch'

与 fminunc 一样，patternsearch 找到局部最优，如退出标志 exitflagp 所示。该解优于 fminunc 解，因为其目标函数值更低。不过，patternsearch 需要更多的函数计算，如输出结构体中所示。

ga 求解器

要使用 Global Optimization Toolbox 求解器 ga 求解优化问题，请输入：

rng default % For reproducibility
x0.x = 10*randn(20) + 20;
x0.y = 10*randn(20) + 30; % Random start population near [20,30];
[solg,fvalg,eflagg,outputg] = solve(prob,"Solver","ga")
Solving problem using ga.
ga stopped because it exceeded options.MaxGenerations.

solg = struct with fields:
    x: 0.0064
    y: 7.7057e-04


fvalg = 
8.1608e-05

eflagg = 
    SolverLimitExceeded


outputg = struct with fields:
      problemtype: 'unconstrained'
         rngstate: [1×1 struct]
      generations: 200
        funccount: 9453
          message: 'ga stopped because it exceeded options.MaxGenerations.'
    maxconstraint: []
       hybridflag: []
           solver: 'ga'

ga 比上述求解器执行更多的函数计算，并得到接近全局最小值的解。该求解器是随机求解器，可以得到次优解。

particleswarm 求解器

要使用 Global Optimization Toolbox 求解器 particleswarm 求解优化问题，请输入：

rng default % For reproducibility
[solpso,fvalpso,eflagpso,outputpso] = solve(prob,"Solver","particleswarm")
Solving problem using particleswarm.
Optimization ended: relative change in the objective value 
over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

solpso = struct with fields:
    x: 7.1467e-07
    y: 1.4113e-06


fvalpso = 
4.9631e-12

eflagpso = 
    SolverConvergedSuccessfully


outputpso = struct with fields:
      rngstate: [1×1 struct]
    iterations: 120
     funccount: 2420
       message: 'Optimization ended: relative change in the objective value ↵over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.'
    hybridflag: []
        solver: 'particleswarm'

与 ga 相比，该求解器执行的函数计算次数更少，并得到更准确的解。同样，该求解器是随机求解器，可能无法得到全局解。

simulannealbnd 求解器

要使用 Global Optimization Toolbox 求解器 simulannealbnd 求解优化问题，请输入：

rng default % For reproducibility
x0.x = 20;
x0.y = 30;
[solsim,fvalsim,eflagsim,outputsim] = solve(prob,x0,"Solver","simulannealbnd")
Solving problem using simulannealbnd.
simulannealbnd stopped because the change in best function value is less than options.FunctionTolerance.

solsim = struct with fields:
    x: 0.0025
    y: 0.0018


fvalsim = 
1.8386e-05

eflagsim = 
    SolverConvergedSuccessfully


outputsim = struct with fields:
     iterations: 1967
      funccount: 1986
        message: 'simulannealbnd stopped because the change in best function value is less than options.FunctionTolerance.'
       rngstate: [1×1 struct]
    problemtype: 'unconstrained'
    temperature: [2×1 double]
      totaltime: 0.4597
         solver: 'simulannealbnd'

求解器执行与 particleswarm 相同的函数计算次数，并得到一个良解。此求解器也是随机求解器。

surrogateopt 求解器

surrogateopt 不需要起点，但需要有限边界。在每个分量中将边界设置为 -70 到 130。要获得与其他求解器相同的输出，请禁用默认绘图函数。

rng default % For reproducibility
x = optimvar("x","LowerBound",-70,"UpperBound",130);
y = optimvar("y","LowerBound",-70,"UpperBound",130);
prob = optimproblem("Objective",rf3(x,y));
options = optimoptions("surrogateopt","PlotFcn",[]);
[solsur,fvalsur,eflagsur,outputsur] = solve(prob,...
    "Solver","surrogateopt",...
    "Options",options)
Solving problem using surrogateopt.
surrogateopt stopped because it exceeded the function evaluation limit set by 
'options.MaxFunctionEvaluations'.

solsur = struct with fields:
    x: 9.9494
    y: -9.9502


fvalsur = 
1.9899

eflagsur = 
    SolverLimitExceeded


outputsur = struct with fields:
        elapsedtime: 2.0956
          funccount: 200
    constrviolation: 0
               ineq: [1×1 struct]
           rngstate: [1×1 struct]
            message: 'surrogateopt stopped because it exceeded the function evaluation limit set by ↵'options.MaxFunctionEvaluations'.'
             solver: 'surrogateopt'

该求解器执行相对较少的函数计算即可得到接近全局最优的解。然而，每次函数计算都比其他求解器花费更多时间。

比较求解器和解

如果一个解的目标函数值小于另一个解的目标函数值，则此解优于另一个解。下表对结果进行了总结。

sols = [solf.x solf.y;
    solp.x solp.y;
    solg.x solg.y;
    solpso.x solpso.y;
    solsim.x solsim.y;
    solsur.x solsur.y];
fvals = [fvalf;
    fvalp;
    fvalg;
    fvalpso;
    fvalsim;
    fvalsur];
fevals = [outputf.funcCount;
    outputp.funccount;
    outputg.funccount;
    outputpso.funccount;
    outputsim.funccount;
    outputsur.funccount];
stats = table(sols,fvals,fevals);
stats.Properties.RowNames = ["fminunc" "patternsearch" "ga" "particleswarm" "simulannealbnd" "surrogateopt"];
stats.Properties.VariableNames = ["Solution" "Objective" "# Fevals"];
disp(stats)
                              Solution            Objective     # Fevals
                      ________________________    __________    ________

    fminunc               19.899        29.849        12.934         5  
    patternsearch         19.899       -9.9496        4.9748       174  
    ga                 0.0063672    0.00077057    8.1608e-05      9453  
    particleswarm     7.1467e-07    1.4113e-06    4.9631e-12      2420  
    simulannealbnd      0.002453     0.0018028    1.8386e-05      1986  
    surrogateopt          9.9494       -9.9502        1.9899       200

以下这些结果非常典型：

  • fminunc 很快得到其起始盆内的局部解，但根本不探索该盆以外的区域。由于目标函数具有解析导数，因此 fminunc 使用自动微分，并且只需执行很少次数的函数计算就得到精确的局部最小值。

  • patternsearch 比 fminunc 执行更多的函数计算次数并且搜索几个盆，因此得到的解优于 fminunc。

  • ga 比 patternsearch 执行更多的函数计算次数。它会偶然找到一个更好的解。在本例中，ga 找到接近全局最优的点。ga 是随机求解器，其结果随着每次运行而变化。ga 需要额外的步骤才能使初始种群接近 [20,30]。

  • particleswarm 执行的函数计算次数比 ga 少，但比 patternsearch 多。在本例中，particleswarm 找到一个其目标函数值比 patternsearch 或 ga 低的点。由于 particleswarm 是随机求解器，其结果随着每次运行而变化。particleswarm 需要额外的步骤来使初始种群接近 [20,30]。

  • simulannealbnd 执行的函数计算次数与 particleswarm 大致相同。在本例中，simulannealbnd 找到一个良解，但不如 particleswarm 找到的解。该求解器是随机求解器，可以得到次优解。

  • surrogateopt 在达到函数计算限制时停止，对于二变量问题，默认为 200 次。surrogateopt 需要有限边界。surrogateopt 尝试求全局解，在本例中已成功。surrogateopt 中执行的每次函数计算都比大多数其他求解器花费更长的时间，因为 surrogateopt 在其算法中执行许多辅助计算。

另请参阅

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