polygon.gms : Largest small polygon COPS 2.0 #1

**Description**

Finds the polygon of maximal area, among polygons with nv sides and diameter d <= 1. This model is from the COPS benchmarking suite. See http://www-unix.mcs.anl.gov/~more/cops/. The number of sides can be specified using the command line parameter --nv. COPS performance tests have been reported for nv = 25, 50, 75, 100

**References**

- Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS. Tech. rep., Mathematics and Computer Science Division, 2000.
- Graham, R L, The Largest Small Hexagon. Journal of Combinatorial Theory, Series A 18, 2 (1975), 165-170.
- Gay, D M, AMPL Models.

**Large Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** polygon.gms

```
$Title Largest small polygon COPS 2.0 #1 (POLYGON,SEQ=229)
$ontext
Finds the polygon of maximal area, among polygons with nv sides and
diameter d <= 1.
This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.
The number of sides can be specified using the command line parameter
--nv. COPS performance tests have been reported for nv = 25, 50, 75,
100
Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.
Graham, R L, The Largest Small Hexagon. J. Combin. Th. 18 (1975),
165-170.
Gay, D, AMPL Models.
$offtext
$if not set nv $set nv 25
set i sides / i1*i%nv% /;
alias (i,j);
scalar pi a famous constant;
positive variables
r(i) polar radius (distance to fixed vertex)
theta(i) polar angle (measured from fixed direction)
variable polygon_area
equations
obj
distance(i,j)
ordered(i) ;
obj.. polygon_area =e= 0.5*sum(j(i+1), r(i+1)*r(i)*sin(theta(i+1)-theta(i)));
ordered(i+1).. theta(i) =l= theta(i+1);
distance(i,j)$(ord(j) > ord(i))..
sqr(r(i)) + sqr(r(j)) - 2*r(i)*r(j)*cos(theta(j)-theta(i)) =l= 1;
pi = 2*arctan(inf);
r.up(i) = 1;
theta.up(i) = pi;
r.fx('i%nv%') = 0;
theta.fx('i%nv%') = pi;
r.l(i) = 4*ord(i)*(card(i)+ 1- ord(i))/sqr(card(i)+1);
theta.l(i) = pi*ord(i)/card(i);
model polygon /all/;
$if set workspace polygon.workspace = %workspace%;
solve polygon using nlp maximizing polygon_area;
```