Optimizing constants applied to multiple vectors with multiple constraints

Hello, I have 6 vectors given below. I would like to find two constants, a and b, such that:

1) the elements of sqrt( (a*U).^2 + (b*V).^2 ) are as small as possible (hopefully all with values < 0.8)

2) the elements of sqrt( (a*W).^2 + (b*X).^2 ) are as close to 1 as possible

3) the elements of sqrt( (a*Y).^2 + (b*Z).^2 ) are as large as possible (hopefully all with values > 1.2)

How would one go about setting up this problem to find the best a and b using the optimization toolbox?

Thanks for any assistance you can provide.

U = [0.255533929
    0.225391099
    0.309547494
    0.654021159
    0.620657919
    0.702112095
    1.191504738
    1.209453366
    1.211036147
    1.045063574
    1.047682214 ];

V = [0.711221344
    0.480397832
    0.855346308
    0.641913032
    0
    0
    0
    0.450686486
    0.886286133
    0.377284731
    0.715048781 ];

W = [0.703234145
    0.661481123
    0.680938355
    1.348926117
    1.213094284
    1.227020665
    1.373111841
    1.764042991 ];

X = [1.232745135
    1.357287095
    1.28138892
    0
    1.209100227
    1.266457293
    0.505682363
    0 ];

Y = [0.822623864
    0.751947223
    0.717610808
    0.733428167
    0.667499868
    1.328856667
    1.242252969
    1.387983711
    1.070372916
    1.045063574
    1.414452191
    1.373461829
    2.000592955 ];

Z = [2.07106942
    1.593285407
    1.602925001
    1.970190532
    1.732521225
    2.181746415
    1.062587656
    3.489796952
    0.886469414
    0.377284731
    1.234992392
    1.826285175
    0 ];

a = 1 / 1.556484554;
b = 1 / 1.560365817;

figure,
    plot(sqrt( (a*U).^2 + (b*V).^2 ), 'go', 'MarkerFace', 'g')
    hold on
    plot(sqrt( (a*W).^2 + (b*X).^2 ), 'yo', 'MarkerFace', 'y')
    plot(sqrt( (a*Y).^2 + (b*Z).^2 ), 'ro', 'MarkerFace', 'r')
    grid on; grid minor

[Craig]

Hi Alan, Thanks for taking a look. I am only allowed to change "a" and "b". I'm not sure how to write the optimization parameters to make:

sqrt((a*U).^2 + (b*V).^2 ) as small as possible

sqrt((a*Y).^2 + (b*Z).^2 ) as large as possible

and sqrt( (a*W).^2 + (b*X).^2 ) are as close to 1 as possible

all happening at the same time.

Hope that clarifies the issue. I do know how to minimize or maximize a function, or make it equal to a value, just not sure how to do all three at once.

Answer 1

[Alan Weiss]

I do not understand your question. What are you allowed to vary? Are there any constraints? I mean, why not just set all elements of U and V to zero?

Alan Weiss

MATLAB mathematical toolbox documentation