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Catenary - Solving a QCQP

Description
This model finds the shape of a hanging chain by minimizing its potential energy.

Source Files
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xbcatena.cxx

/********************************************************
BCL Example Problems
====================

file xbcatena.cxx

QCQP problem (linear objective, convex quadratic constraints)
Based on AMPL model catenary.mod
(Source: http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/ )

This model finds the shape of a hanging chain by
minimizing its potential energy.

(c) 2008 Fair Isaac Corporation
author: S.Heipcke, June 2008, rev. Mar. 2011
********************************************************/

#include <iostream>
#include "xprb_cpp.h"
#include "xprs.h"

using namespace std;
using namespace ::dashoptimization;

#define N 100	               // Number of chainlinks
#define L 1	               // Difference in x-coordinates of endlinks

double H = 2.0*L/N;	               // Length of each link

int main(int argc, char **argv)
{
int i;
XPRBvar x[N+1],y[N+1];       // x-/y-coordinates of endpoints of chainlinks
XPRBexpr qe;
XPRBctr cobj, c;
XPRBprob prob("catenary");              // Initialize a new problem in BCL

prob.setDictionarySize(XPRB_DICT_NAMES,0);

/**** VARIABLES ****/
for(i=0;i<=N;i++)
x[i] = prob.newVar(XPRBnewname("x(%d)",i), XPRB_PL, -XPRB_INFINITY,
XPRB_INFINITY);
for(i=0;i<=N;i++)
y[i] = prob.newVar(XPRBnewname("y(%d)",i), XPRB_PL, -XPRB_INFINITY,
XPRB_INFINITY);
// Bounds: positions of endpoints
// Left anchor
x[0].fix(0);  y[0].fix(0);
// Right anchor
x[N].fix(L);  y[N].fix(0);

/****OBJECTIVE****/
/* Minimise the potential energy:  sum(j in 1..N) (y(j-1)+y(j))/2  */
qe=0;
for(i=1;i<=N;i++) qe+= y[i-1]+y[i];
cobj = prob.newCtr("Obj", qe*0.5 );
prob.setObj(cobj);                     // Set objective function

/**** CONSTRAINTS ****/
forall(j in 1..N) (x(j)-x(j-1))^2+(y(j)-y(j-1))^2 <= H^2  */
for(i=1;i<=N;i++)
prob.newCtr(XPRBnewname("Link_%d",i), sqr(x[i]-x[i-1]) + sqr(y[i]-y[i-1]) <= H*H);

/****SOLVING + OUTPUT****/
prob.setSense(XPRB_MINIM);             // Choose the sense of optimization

/* Problem printing and matrix output: */
/*
prob.print();
prob.exportProb(XPRB_MPS, "catenary");
prob.exportProb(XPRB_LP, "catenary");
*/

/* Start values:
for(i=0;i<=N;i++) x[j].setInitval(j*L/N)
for(i=0;i<=N;i++) y[j].setInitval(0)
*/
// Disable convexity check
`