重み付きlpノルム法によるパレートフロントの計算

その１

p=2の場合

clear; close all;

p = 2;
f1 = @(x) x.^2;
f2 = @(x) (x-2).^2;
w1s = 0:0.1:1;
w2s = 0:0.1:1;
x = -10:0.1:10;
xopts = [];
for idx=1:length(w1s)
    w1 = w1s(idx);
    for jdx=1:length(w2s)
        w2 = w2s(jdx);
        F = ((w1*f1(x)).^p+(w2*f2(x)).^2).^(1/p);
        if length(find(F==min(F)))>3
            continue
        end
        xopt = x(find(F==min(F),1));
        xopts(end+1) = xopt;
    end
end

xopts = unique(sort(xopts));
f1opts = f1(xopts);
f2opts = f2(xopts);
plot(f1opts,f2opts,'o');
hold on;

x1 = 0:0.1:4;
y1 = interp1(f2opts,f1opts,x1);
plot(x1,y1,'-');

big;

その２

clear; close all;

p = 2;
a = 1/sqrt(2);
f1 = @(x) 1-exp(-(x(:,1)-a).^2-(x(:,2)-a).^2);
f2 = @(x) 1-exp(-(x(:,1)+a).^2-(x(:,2)+a).^2);
w1s = 0:0.1:1;
w2s = 0:0.1:1;
x1 = -4:0.1:4;
x2 = -4:0.1:4;
x = [x1;x2]';
x1opts = [];
x2opts = [];
for idx=1:length(w1s)
    w1 = w1s(idx);
    for jdx=1:length(w2s)
        w2 = w2s(jdx);
        F = ((w1*f1(x)).^p+(w2*f2(x)).^2).^(1/p);
        if length(find(F==min(F)))>3
            continue
        end
        xopt = x(find(F==min(F),1),:);
        x1opts(end+1) = xopt(1);
        x2opts(end+1) = xopt(2);        
    end
end

xopts = [x1opts;x2opts]';
xopts = unique(sort(xopts),'rows');
f1opts = f1(xopts);
f2opts = f2(xopts);
plot(f1opts,f2opts,'o');
hold on;

x1 = 0:0.1:1;
y1 = interp1(f2opts,f1opts,x1,'spline');
plot(x1,y1,'-');

big;