paretosearch
在帕累托集中查找点
语法
说明
示例
在二维变量的双目标函数的帕累托前沿上找到点。
fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; rng default % For reproducibility x = paretosearch(fun,2);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
将解绘制为散点图。
plot(x(:,1),x(:,2),'m*') xlabel('x(1)') ylabel('x(2)')
理论上，这个问题的解是从 [-2,-1] 到 [1,2] 的一条直线。paretosearch 返回靠近这条线的等距点。
为受线性约束 x(1) + x(2) <= 1 约束的二维双目标问题创建帕累托前沿。
fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; A = [1,1]; b = 1; rng default % For reproducibility x = paretosearch(fun,2,A,b);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
将解绘制为散点图。
plot(x(:,1),x(:,2),'m*') xlabel('x(1)') ylabel('x(2)')
理论上，这个问题的解是从 [-2,-1] 到 [0,1] 的一条直线。paretosearch 返回靠近这条线的等距点。
为二维双目标问题创建帕累托前沿，受限于边界 x(1) >= 0 和 x(2) <= 1。
fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; lb = [0,-Inf]; % x(1) >= 0 ub = [Inf,1]; % x(2) <= 1 rng default % For reproducibility x = paretosearch(fun,2,[],[],[],[],lb,ub);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
将解绘制为散点图。
plot(x(:,1),x(:,2),'m*') xlabel('x(1)') ylabel('x(2)')
所有解都在约束边界 x(1) = 0 或 x(2) = 1 上。
为二维双目标问题创建帕累托前沿，受边界 -1.1 <= x(i) <= 1.1 和非线性约束 norm(x)^2 <= 1.2 约束。非线性约束函数出现在此示例的末尾，如果您将此示例作为实时脚本运行，则该函数有效。要以其他方式运行此示例，请将非线性约束函数作为文件包含在您的 MATLAB® 路径中。
为了更好地看到非线性约束的效果，请设置选项以使用较大的帕累托集大小。
rng default % For reproducibility fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; lb = [-1.1,-1.1]; ub = [1.1,1.1]; options = optimoptions('paretosearch','ParetoSetSize',200); x = paretosearch(fun,2,[],[],[],[],lb,ub,@circlecons,options);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
将解绘制为散点图。包括圆形约束边界的图。
figure plot(x(:,1),x(:,2),'k*') xlabel('x(1)') ylabel('x(2)') hold on rectangle('Position',[-1.2 -1.2 2.4 2.4],'Curvature',1,'EdgeColor','r') xlim([-1.2,0.5]) ylim([-0.5,1.2]) axis square hold off
具有正 x(1) 值或负 x(2) 值的解点接近非线性约束边界。
function [c,ceq] = circlecons(x) ceq = []; c = norm(x)^2 - 1.2; end
为了监控 paretosearch 的进度，请指定 'psplotparetof' 绘图函数。
fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; options = optimoptions('paretosearch','PlotFcn','psplotparetof'); lb = [-4,-4]; ub = -lb; x = paretosearch(fun,2,[],[],[],[],lb,ub,[],options);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
该解看起来像一个半径为 18 的四分之一圆弧，可以证明其为解析解。
通过使用 paretosearch 和 x 输出调用 fval，在函数空间和参数空间中获得帕累托前沿。设置选项以在函数空间和参数空间中绘制帕累托集。
fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; lb = [-4,-4]; ub = -lb; options = optimoptions('paretosearch','PlotFcn',{'psplotparetof' 'psplotparetox'}); rng default % For reproducibility [x,fval] = paretosearch(fun,2,[],[],[],[],lb,ub,[],options);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
目标函数空间中的解析解是半径为 18 的四分之一圆弧。在参数空间中，解析解是一条从 [-2,-1] 到 [1,2] 的直线。解点接近解析曲线。
设置选项来监控帕累托集解过程。另外，从 paretosearch 获取更多输出，以便您了解解过程。
options = optimoptions('paretosearch','Display','iter',... 'PlotFcn',{'psplotparetof' 'psplotparetox'}); fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; lb = [-4,-4]; ub = -lb; rng default % For reproducibility [x,fval,exitflag,output] = paretosearch(fun,2,[],[],[],[],lb,ub,[],options);
Iter F-count NumSolutions Spread Volume 0 60 11 - 3.7872e+02 1 386 12 7.6126e-01 3.4654e+02 2 702 27 9.5232e-01 2.9452e+02 3 1029 27 6.6332e-02 2.9904e+02 4 1357 36 1.3874e-01 3.0070e+02 5 1690 37 1.5379e-01 3.0200e+02 6 2014 50 1.7828e-01 3.0252e+02 7 2214 59 1.8536e-01 3.0320e+02 8 2344 60 1.9435e-01 3.0361e+02 9 2464 60 2.1055e-01 3.0388e+02 Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
检查附加输出。
fprintf('Exit flag %d.\n',exitflag)Exit flag 1.
disp(output)
iterations: 10
funccount: 2464
volume: 303.6076
averagedistance: 0.0250
spread: 0.2105
maxconstraint: 0
message: 'Pareto set found that satisfies the constraints. ↵↵Optimization completed because the relative change in the volume of the Pareto set ↵is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within ↵'options.ConstraintTolerance'.'
rngstate: [1×1 struct]
获取并检查帕累托前沿约束残差。创建一个具有线性不等式约束 sum(x) <= -1/2 和非线性不等式约束 norm(x)^2 <= 1.2 的问题。为了提高准确性，在帕累托前沿使用 200 个点，以及 ParetoSetChangeTolerance 的 1e-7，并给出自然边界 -1.2 <= x(i) <= 1.2。
非线性约束函数出现在此示例的末尾，如果您将此示例作为实时脚本运行，则该函数有效。要以其他方式运行此示例，请将非线性约束函数作为文件包含在您的 MATLAB® 路径中。
fun = @(x)[norm(x-[1,2])^2;norm(x+[2,1])^2]; A = [1,1]; b = -1/2; lb = [-1.2,-1.2]; ub = -lb; nonlcon = @circlecons; rng default % For reproducibility options = optimoptions('paretosearch','ParetoSetChangeTolerance',1e-7,... 'PlotFcn',{'psplotparetof' 'psplotparetox'},'ParetoSetSize',200);
使用所有输出调用 paretosearch。
[x,fval,exitflag,output,residuals] = paretosearch(fun,2,A,b,[],[],lb,ub,nonlcon,options);
Pareto set found that satisfies the constraints. Optimization completed because the relative change in the volume of the Pareto set is less than 'options.ParetoSetChangeTolerance' and constraints are satisfied to within 'options.ConstraintTolerance'.
与无约束的集合相比，不等式约束减少了帕累托集的大小。检查返回的残差。
fprintf('The maximum linear inequality constraint residual is %f.\n',max(residuals.ineqlin))The maximum linear inequality constraint residual is 0.000000.
fprintf('The maximum nonlinear inequality constraint residual is %f.\n',max(residuals.ineqnonlin))The maximum nonlinear inequality constraint residual is -0.000537.
返回的最大残差为负，这意味着所有返回的点都是可行的。返回的最大残差接近于零，这意味着每个约束对于某些点都是有效的。
function [c,ceq] = circlecons(x) ceq = []; c = norm(x)^2 - 1.2; end
输入参数
输出参量
详细信息
算法
paretosearch 使用模式搜索来搜索帕累托前沿上的点。有关详细信息，请参阅paretosearch 算法。
替代功能
App
优化实时编辑器任务为 paretosearch 提供了一个可视化界面。
扩展功能
版本历史记录
在 R2018b 中推出
