A growing collection of bilevel problems
Linear-Linear problem from [Savard, 1989]:
Objective value(s) | Solution point(s) |
---|---|
F* = -14.600 | x* = (0.000, 0.650) |
f* = 0.300 | y* = (0.000, 0.300, 0.000) |
Original source:
Other sources:
AMPL
formatvar x{1..2} >= 0, <= 10; # Outer variable
var y{1..3} >= 0, <= 10; # Inner variable
var l{1..9} >= 0, <= 10; # KKT Multipliers
minimize outer_obj: -8*x[1] - 4*x[2] + 4*y[1] - 40*y[2] + 4*y[3]; # Outer objective
subject to
# Outer constraints:
outer_con1: x[1] + 2*x[2] - y[3] - 1.3 <= 0;
# Inner objective:
inner_obj: 2*y[1] + y[2] + 2*y[3] = 0;
# Inner constraints
inner_con1: -y[1] + y[2] + y[3] - 1 <= 0;
inner_con2: 4*x[1] - 2*y[1] + 4*y[2] - y[3] - 2 <= 0;
inner_con3: 4*x[2] + 4*y[1] - 2*y[2] - y[3] - 2 <= 0;
# KKT conditions:
stationarity_1: 2 - l[1] - 2*l[2] + 4*l[3] - l[4] + l[5] = 0;
stationarity_2: 1 + l[1] + 4*l[2] - 2*l[3] - l[6] + l[7] = 0;
stationarity_3: 2 + l[1] - l[2] - l[3] - l[8] + l[9] = 0;
complementarity_1: l[1]*(-y[1] + y[2] + y[3] - 1) = 0;
complementarity_2: l[2]*(4*x[1] - 2*y[1] + 4*y[2] - y[3] - 2) = 0;
complementarity_3: l[3]*(4*x[2] + 4*y[1] - 2*y[2] - y[3] - 2) = 0;
complementarity_4: l[4]*y[1] = 0;
complementarity_5: l[5]*(y[1] - 10) = 0;
complementarity_6: l[6]*y[2] = 0;
complementarity_7: l[7]*(y[2] - 10) = 0;
complementarity_8: l[8]*y[3] = 0;
complementarity_9: l[9]*(y[3] - 10) = 0;