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# ex-001-NLP.mod: NLP formulation of a small EPEC with two leaders
# 
# leader i's problem is:
#
#     minimize    (x_i+1)^2
#     subject to  s = x_1 + x_2 + y
#                 0 <= s  _|_  y >= 0
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set I := 1..2;

# ... leader variables
var x{I};

# ... follower variables
var s >= 0;
var y >= 0;

# ... multipliers
var lambda{I};
var sigma{I} >= 0;
var nu{I}    >= 0;
var xi{I}    >= 0;

# ... dummy objective
minimize penalty: sum{i in I}( s*sigma[i] + y*nu[i] + s*y );

subject to 

	# ... first order conditions
	KKTx{i in I}: 2*(x[i]+1) + lambda[i] = 0;
	KKTy{i in I}:   lambda[i] - nu[i]    + xi[i]*s = 0;
	KKTs{i in I}: - lambda[i] - sigma[i] + xi[i]*y = 0;

	slack{i in I}: s = x[1] + x[2] + y;