$TITLE Example for QAP using penalty and single circle constraint
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 SETS     J         index      / J1 * J4 / ;

 VARIABLES
      X(J)   unknowns
      Z      optimal value  ;

X.L(J) = 10 ;
 X.LO(J) = 0 ;

SCALAR P quadratic penalty parameter /100/;
* SCALAR R log barrier penalty parameter /.0001/;

 EQUATIONS 
    COST   define objective function 
     CONSTR1 single constraint nonlinear  ;
 
 COST ..  Z =E= 12*(X('J1')**2 + X('J2')**2 + X('J3')**2 + X('J4')**2)
                +18*X('J1')*X('J2') + 16*X('J1')*X('J3') + 12*X('J1')*X('J4')
                +12*X('J2')*X('J3') + 16*X('J2')*X('J4') + 18*X('J3')*X('J4')
                - 12*(SUM(J,X(J)))
                +P*((X('J1') + X('J2') -1)**2 
                +(X('J3') + X('J4') -1)**2 
                +(X('J1') + X('J3') -1)**2 
                +(X('J2') + X('J4') -1)**2 ) ;
*                -(R)*(SUM(J,LOG(X(J)))) ;
            
  CONSTR1..  X('J1')**2 + X('J2')**2 + X('J3')**2 + X('J4')**2 =E= 2 ;

MODEL PNLTY1 /ALL/;

 SOLVE PNLTY1  MINIMIZING Z USING NLP;