# outrata4-MPEC.mod	
# Original AMPL coding by Sven Leyffer, Argonne National Laboratory, 2004.
# 
# MPEC formulation of ...
# 
# Multi-Leader-Follower Game (MLF)) derived from outrata31-outrata34, 
# see J. Outrata, SIAM J. Optim. 4(2), pp.340ff, 1994. All Stackelberg 
# players have the same constraints, but different objectives.
#
#########################################################################

# ... parameters to give number of leaders
param nl integer, default 4;       # number of leaders
set L := 1..nl;                    # set of leaders
set I := 1..4;                     # ... useful index set

# ... Stackelberg leaders' variables
var x{L} >= 0, <= 10;
var xa = sum{l in L} x[l] / card(L);

# ... Followers' variables
var y{I} >= 0;
var s{I} >= 0;

# ... multipliers (differ for each leader)
var mF{L,I};                       # multipliers of s = F(x,y) constraints
var my{L,I} >= 0;                  # multipliers of y >= 0
var ms{L,I} >= 0;                  # multipliers of s >= 0
var mC{L,I} >= 0;                  # multipliers of Y s <= 0
var mx{L};                         # multiplier of 0 <= x[l] <= L

# ... definied variables model rhs of KKT conditions
var kktx{l in L}  = mF[l,1]*(0.2*y[1]-1.333) + mF[l,2]*(0.1*y[2]-1) 
                    + mF[l,3]*(-0.1) + mF[l,4]*0.1 + mx[l]*card(L);
var kkty1{l in L} = mF[l,1]*(1+0.2*xa+2*y[4]) + mF[l,3]*0.333 
                    + mF[l,4]*(-2*y[1]) + my[l,1] - mC[l,1]*s[1];
var kkty2{l in L} = mF[l,2]*(1+0.1*xa+2*y[4]) + mF[l,3]*(-1) + mF[l,4]*(-2*y[2]) 
                    + my[l,2] - mC[l,2]*s[2];
var kkty3{l in L} = mF[l,1]*(-0.333) + mF[l,2] + my[l,3] - mC[l,3]*s[3];
var kkty4{l in L} = mF[l,1]*(2*y[1]) + mF[l,2]*(2*y[2]) + my[l,4] - mC[l,4]*s[4];

# ... dummy objective
minimize compl_mult: sum{l in L, i in I} mC[l,i];

subject to 

   # ... FO conditions of leader 1 (gradients in (x[1],y))
   KKT1x:  2*(x[1]-1) = kktx[1] / card(L);
   KKT1y1: 2*(y[1]-3) = kkty1[1];
   KKT1y2: 2*(y[2]-4) = kkty2[1];
   KKT1y3: 2*(y[3]-1) = kkty3[1];
   KKT1y4: 20*y[4]    = kkty4[1];

   # ... FO conditions of leader 2 (gradients in (x[2],y))
   KKT2x:  3*x[2]     = kktx[2] / card(L);
   KKT2y1: 2*(y[1]-3) = kkty1[2];
   KKT2y2: 2*(y[2]-4) = kkty2[2];
   KKT2y3: 2*(y[3]-1) = kkty3[2];
   KKT2y4: 20*y[4]    = kkty4[2];

   # ... FO conditions of leader 3 (gradients in (x[3],y))
   KKT3x:  2*x[3]     = kktx[3] / card(L);
   KKT3y1: 2*(y[1]-3) = kkty1[3];
   KKT3y2: 2*(y[2]-4) = kkty2[3];
   KKT3y3: 2*(y[3]-1) = kkty3[3];
   KKT3y4: 20*y[4]    = kkty4[3];

   # ... FO conditions of leader 4 (gradients in (x[4],y))
   KKT4x:  2*x[4]     = kktx[4] / card(L);
   KKT4y1: 2*(y[1]-3) = kkty1[4];
   KKT4y2: 2*(y[2]-4) = kkty2[4];
   KKT4y3: 2*(y[3]-1) = kkty3[4];
   KKT4y4: 20*y[4]    = kkty4[4];

   # ... first order conditions for the slacks (all players)
   KKTs{l in L, i in I}: 0 = - mF[l,i] + ms[l,i] - mC[l,i]*y[i];

   # ... followers reponse to average of leaders (xa)
   nlcs1{l in L}: mF[l,1]  complements
                  s[1] = (1 + 0.2*xa)*y[1] - (3 + 1.333*xa) - 0.333*y[3] + 2*y[1]*y[4];
   nlcs2{l in L}: mF[l,2]  complements
                  s[2] = (1 + 0.1*xa)*y[2] - xa + y[3] + 2*y[2]*y[4];
   nlcs3{l in L}: mF[l,3]  complements
                  s[3] = 0.333*y[1] - y[2] + 1 - 0.1*xa;
   nlcs4{l in L}: mF[l,4]  complements
                  s[4] = 9 + 0.1*xa - y[1]^2 - y[2]^2;

   # ... complementarity constraints
   complx{l in L}:         0 <= x[l] <= 10 complements  mx[l];
   comply{l in L, i in I}: 0 <= y[i]       complements  my[l,i] >= 0;
   compls{l in L, i in I}: 0 <= s[i]       complements  ms[l,i] >= 0;
   complC{i in I}:         0 >= s[i]*y[i];