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I am currently trying to solve my own optimality problem.

Let $f :(a,b,c,d) \in \mathbb{R^4} \mapsto 1 + d_1+d_2+d_3+d_4+d_5 \in \mathbb{R}_+$ where:

$d_1 = \sqrt{a^2+b^2}$

$d_2= \sqrt{c^2+d^2}$

$d_3=\sqrt{(a-1)^2+b^2}$

$d_4=\sqrt{(c-1)^2+d^2}$

$d_5=\sqrt{(a-c)^2+(b-d)^2}$

I would like to know what methods can we use to find the minimum value $m$ of $f$ knowing that $\forall i \in \{1,2,3,4,5\}, d_i \geq1$?

For now, I have shown geometrically that $5+\sqrt{3} \geq m >6$. I am not looking necessarily for a full answer but rather for methods to deal with that kind of problem. Any help would be appreciated.

Axel
  • 2,447

1 Answers1

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I have used LocalSolver to tackle your problem:

function model() {
    lowerBound = -100;  //  somewhat arbitrary
    upperBound = +100;

a <- float(lowerBound, upperBound);
b <- float(lowerBound, upperBound);
c <- float(lowerBound, upperBound);
d <- float(lowerBound, upperBound);

d1 <- sqrt2(a, b);
d2 <- sqrt2(c, d);
d3 <- sqrt2(a - 1, b);
d4 <- sqrt2(c - 1, d);
d5 <- sqrt2(a - c, b - d);

constraint d1 >= 1.0;
constraint d2 >= 1.0;
constraint d3 >= 1.0;
constraint d4 >= 1.0;
constraint d5 >= 1.0;

m <- 1 + d1 + d2 + d3 + d4 + d5;

minimize m;



}


function sqrt2(u, v) {
    return pow(uu + vv, 0.5);
}

The result:

m=6.73204901816782

a=0.499997511563293
b=-0.866027201892499
c=1.49999583752018
d=-0.866027201892499


d1=1.00000031299357
d2=1.73204810181347
d3=1.0000028014264
d4=0.999999475977494
d5=0.999998325956891


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Axel Kemper
  • 4,943
  • Thank you for your answer. It is 100% coherent with my intuitive vision of this problem, indeed we end up with $m = 5 +\sqrt{3} \approx 6.73205$. – Axel Jul 31 '20 at 14:40
  • Have you any idea/methods for proof? – Axel Jul 31 '20 at 14:46
  • The constraints are looking like a geometric problem of rectangular triangles. Variables a, b and d1 could be seen as side lengths of one triangle. Similarly, other triangles are described by c, d, d2, and so on. The constraints put on d1, ... d5 could be modelled as circles. Further than that, I do not have a proof to add. – Axel Kemper Jul 31 '20 at 17:02
  • That's right. You can check the original problem here: https://math.stackexchange.com/questions/3775947/geometry-and-optimization-in-a-plane. Thank you for introducing LocalSolver, it looks like a great tool. Have a nice day. – Axel Aug 03 '20 at 11:46