cpack.gms : Packing identical size circles in the unit circle

**Description**

Given the unit circle (of radius 1), find a set of identical size circles with an optimized (maximal) radius r so that all such circles are contained by the unit circle, in a non-overlapping arrangement. A test example from the LGO library

**Reference**

- Pinter, J D, Nonlinear optimization with GAMS/LGO. Journal of Global Optimization 38, 1 (2007), 79-101.

**Small Model of Type :** QCP

**Category :** GAMS Model library

**Main file :** cpack.gms

```
$title Packing identical size circles in the unit circle (CPACK,SEQ=387)
$onText
Given the unit circle (of radius 1), find a set of identical
size circles with an optimized (maximal) radius r so that all
such circles are contained by the unit circle, in a non-overlapping
arrangement.
A test example from the LGO library
Pinter, J D, Nonlinear optimization with GAMS/LGO.
Journal of Global Optimization 38 (2007), 79-101.
Keywords: quadratic constraint programming, circle packing problem, mathematics
$offText
$if not set k $set k 5
Set i / i1*i%k% /;
Alias (i,j);
* Here we define the set ij(i,j) of ordered pairs i,j i < j.
Set ij(i,j); ij(i,j)$(ord(i) < ord(j)) = yes;
Variable
r 'radius of identical sized circles'
x(i) 'x coordinate of circle i'
y(i) 'y coordinate of circle i';
Equation
circumscribe(i) 'enforce circle is enclose in unit circle'
nooverlap(i,j) 'enforce that circles do not overlap';
circumscribe(i).. sqr(1 - r) =g= sqr(x(i)) + sqr(y(i));
nooverlap(ij(i,j)).. sqr(x(i) - x(j)) + sqr(y(i) - y(j)) =g= 4*sqr(r);
x.lo(i) = -1; x.up(i) = 1;
y.lo(i) = -1; y.up(i) = 1;
* starting values for local solvers such that some feasible solution is produced (at least with k=5)
x.l(i) = -0.2 + ord(i)*0.1;
y.l(i) = -0.2 + ord(i)*0.1;
* These bound are valid for k>=5
r.lo = 0.05; r.up = 0.4;
Model m / all /;
solve m using qcp maximizing r;
display r.l;
```