A growing collection of bilevel problems
b_1991_01v : Linear-Linear problem, variation of b_1991_01
This is a variation of b_1991_01 problem. The only difference is the outer objective function, which is changed from -x + 10*y1 - y2
to -x + 10*y1 - 2y2
.
The leader is faced with an ambiguous situation for all choices but one. Only at x = 1
is the follower’s response, y = (0,0)
, unique. Notice, however, that the point x = 0, y = (0,1)
is most preferred by the leader giving F = -2
, but may not be realized despite the fact that it is in the inducible region (bilevel feasible region); that is, the follower might very well pick y = (1,0)
giving F = 10
for the leader, while keeping unchanged value f = -1
for the follower.
This example suggests that without some incentive, the follower has no reason to select the point y = (0,1)
which would be best for the leader.
Objective values | Solution points |
---|---|
F* = -2.000 | x* = 0.000 |
f* = -1.000 | y* = (0.000, 1.000) |
Original source:
Other sources:
AMPL
formatvar x >= 0, <= 10; # Outer variable
var y{1..2} >= 0, <= 10; # Inner variable
var l{1..7} >= 0, <= 10; # KKT Multipliers
minimize outer_obj: -x + 10*y[1] - 2*y[2]; # Outer objective
subject to
# Inner objective:
inner_obj: -y[1] - y[2] = 0;
# Inner constraints
inner_con1: x + y[1] - 1 <= 0;
inner_con2: x + y[2] - 1 <= 0;
inner_con3: y[1] + y[2] - 1 <= 0;
# KKT conditions:
stationarity_1: -1 + l[1] + l[3] - l[4] + l[5] = 0;
stationarity_2: -1 + l[2] + l[3] - l[6] + l[7] = 0;
complementarity_1: l[1]*(x + y[1] - 1) = 0;
complementarity_2: l[2]*(x + y[2] - 1) = 0;
complementarity_3: l[3]*(y[1] + y[2] - 1) = 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;