A growing collection of bilevel problems
Linear-Quadratic type problem from [Aiyoshi & Shimizu, 1984]
Objective values | Solution points |
---|---|
F* = 0.000 | x* = (0.0, 30.0) |
f* = 100.000 | y* = (-10.0, 10.0) |
F* = 0.000 | x* = (0.0, 0.0) |
f* = 200.000 | y* = (-10.0, -10.0) |
The problem has a local solution at x = (25, 30), y = (5, 1O) for which the outer objective function is F = 5. The paper in which this problem is presented originally (Aiyoshi & Shimizu, 1984) locates this local optimum solution. (Shimizu et al., 1997) also report its local minimum as the solution of the problem.
Original source:
Other sources:
AMPL
formatvar x{1..2} >= 0, <= 50; # Outer variable
var y{1..2} >= -10, <= 20; # Inner variable
var l{1..6} >= 0, <= 100; # KKT Multipliers
minimize outer_obj: 2*x[1] + 2*x[2] - 3*y[1] - 3*y[2] - 60; # Outer objective
subject to
# Outer constraints
outer_con: x[1] + x[2] + y[1] - 2*y[2] - 40 <= 0;
# Inner objective:
inner_obj: (y[1] - x[1] + 20)^2 + (y[2] - x[2] + 20)^2 = 0;
# Inner constraints
inner_con1: -x[1] + 2*y[1] + 10 <= 0;
inner_con2: -x[2] + 2*y[2] + 10 <= 0;
# KKT conditions:
stationarity_1: 2*(y[1] - x[1] + 20) + 2*l[1] - l[3] + l[4] = 0;
stationarity_2: 2*(y[2] - x[2] + 20) + 2*l[2] - l[5] + l[6] = 0;
complementarity_1: l[1]*(-x[1] + 2*y[1] + 10) = 0;
complementarity_2: l[2]*(-x[2] + 2*y[2] + 10) = 0;
complementarity_3: l[3]*(-y[1] - 10) = 0;
complementarity_4: l[4]*(y[1] - 20) = 0;
complementarity_5: l[5]*(-y[2] - 10) = 0;
complementarity_6: l[6]*(y[2] - 20) = 0;