# Objective: linear
# Constraints: convex quadratic
# Feasible set: convex

# This model finds the shape of a hanging chain
# The solution is known to be y = cosh(a*x) + b
# for appropriate a and b.

param N := 100;	# number of chainlinks
param L := 1;	# difference in x-coords of endlinks

param h := 2*L/N;	# length of each link

var x {0..N};	# x-coordinates of endpoints of chainlinks
var y {0..N};	# y-coordinates of endpoints of chainlinks

minimize pot_energy: sum{j in 1..N} (y[j-1] + y[j])/2;

subject to x_left_anchor: x[0] = 0;
subject to y_left_anchor: y[0] = 0;
subject to x_right_anchor: x[N] = L;
subject to y_right_anchor: y[N] = 0;

subject to link_up {j in 1..N}: (x[j] - x[j-1])^2 + (y[j] - y[j-1])^2 <= h^2;

let {j in 0..N} x[j] := j*L/N;
let {j in 0..N} y[j] := 0;

solve;

printf {j in 0..N}: "%10.5f %10.5f \n", x[j], y[j] > xy.out;