$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.

Keywords: nonlinear programming, mathematics
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$if not set nv $set nv 25

Set i 'sides' / i1*i%nv% /;

Alias (i,j);

Positive Variable
   r(i)     'polar radius (distance to fixed vertex)'
   theta(i) 'polar angle (measured from fixed direction)';

Variable polygon_area;

Equation
   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;

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;