*-----------------------------------------------------------
* Date created: February 15, 2001
*
* Distribution of electrons on a sphere  
* (R.J.,Vanderbei)
*          
* Bondarenko, A.S., Bortz, D.M., More, J.J.,
* COPS: Large-Scale Nonlinearly Constrained Optimization
*       Problems.
* Technical Report ANL/MCS-TM-237,
* September 1998  (October 1999 - revision), pp. 5-6      
*
* Dr. Neculai Andrei,
* Research Institute for Informatics - Bucharest
* 8-10, Averescu Avenue,
* 71316 Bucharest - Romania
* E-mail: nandrei@u3.ici.ro
* web: http://www.ici.ro/camo/neculai/nandrei.htm
*-----------------------------------------------------------
*
	subroutine ini(n,m,mb,me,sb,x,icgc,ipgc,icge,ipge,
     1                 nszc,nsze,nb)
*
	double precision x(n)
	integer sb(mb)
	integer icgc(nszc),ipgc(n+1)
	integer icge(nsze),ipge(n+1)
*
* Information about the problem.
*
	write(1,10)
10	format(5x,'Distribution of electrons on a sphere')
	write(1,11)
11	format(5x,'by R.J. Vanderbei')
	write(1,12)
12	format(5x,'COPS, pp.5-6')
*
* Dimension of the problem:
*     
      np=5
	n=3*np
	m=0
	me=np
	mb=0
	nszc=0
	nsze=3*np
*
* Initial point:
*
	do i=1,n
	  x(i)=float(i)
	end do
*
* Index vector for simple bounds: cb >= 0.
*
*
* Rows indices of the nonzeros of the Jacobian of the Equalities
*                  
      k=1
      do i=1,3
        do j=1,np
          icge(k)=j
          k=k+1
        end do
      end do    
*
* Starting address of columns of the Jacobian of the Equalities
*      
      do i=1,n
        ipge(i)=i
      end do
      ipge(n+1)=n+1  
*
	return
	end
*


*--------------------------------------------------------------
* Distribution of electrons on a sphere
* by R.J.,Vanderbei.
* Bondarenko, A.S., Bortz, D.M., More, J.J.,
* COPS: Large-Scale Nonlinearly Constrained Optimization
*       Problems.
* Technical Report ANL/MCS-TM-237,
* September 1998  (October 1999 - revision), pp. 5-6
*--------------------------------------------------------------
*
	subroutine prob(n,m,mb,me,sb,x,objf,gobj,c,gc,cb,e,ge,
     1                  nszc,nsze,nb)
*
* Calculate problem function at iterate x.
*
	double precision x(n),objf,gobj(n),c(m),gc(nszc)
	double precision cb(mb),e(me),ge(nsze)  
	real*8 u, v
*
* Objective function and its gradient:
*
      np=5
*      
	objf=0.d0  
	do i=1,n
	  gobj(i)=0.d0
      end do      
*	
      do i=1,np-1
        do j=i+1,np
      u=(x(i)-x(j))**2 + (x(np+i)-x(np+j))**2 + (x(2*np+i)-x(2*np+j))**2  
      v=2.d0*u*sqrt(u)
*
      objf = objf + 1.d0/sqrt(u)
*      
      gobj(i)    =  -2.d0*(x(i)-x(j))/v           + gobj(i)
      gobj(j)    =   2.d0*(x(i)-x(j))/v           + gobj(j)
      gobj(np+i) =  -2.d0*(x(np+i)-x(np+j))/v     + gobj(np+i)
      gobj(np+j) =   2.d0*(x(np+i)-x(np+j))/v     + gobj(np+j)
      gobj(2*np+i)= -2.d0*(x(2*np+i)-x(2*np+j))/v + gobj(2*np+i)
      gobj(2*np+j)=  2.d0*(x(2*np+i)-x(2*np+j))/v + gobj(2*np+j)
        end do
      end do  
*
*
* Bounds on variables:
*
*
* Constraints. (Equalities):
*                    
      np2=2*np
      do i=1,np
        e(i) = x(i)**2 + x(np+i)**2 + x(np2+i)**2 - 1.d0
      end do  
*
* Jacobian of the Equalities constraints:
*          
      do i=1,n
        ge(i) = 2.d0*x(i)
      end do    
*
	return
	end
*--------------------------------------------------DES.for




* Remark:
* For different values of parameter np, with the initial point
* x0(i)=i, the initial values of objective function, f(x0), and 
* the optimal values of objective, f(x*), are as follows:
*                                        
* -------------------------------------------------------
*       np          f(x0)           f(x*) 
* -------------------------------------------------------
*       50          101.0134        1055.182314691
*       75          168.9337        2454.369689043
*       100         241.7584        4448.350634334
*       150         397.6079        10236.438492618
*       200         563.2665        18439.039580786
* -------------------------------------------------------