nwj_2017_05 : Nonlinear-Nonlinear bilevel problem from [Nie et al., 2017]
| Objective values | Solution points |
|---|---|
| F* = -3.506 | x* = (0.544, 0.468, 0.490, 0.495) |
| f* = -0.834 | y* = (-0.783, -0.501, -0.288, -0.184) |
Sources where this problem occurs
Original source:
- Example 5.8 in (Nie et al., 2017)
Other sources:
Description of the problem in the AMPL format
var x{1..4} >= -1, <= 1; # Outer variables
var y{1..4} >= -10, <= 10; # Inner variables
var l{1..10} >= 0, <= 100; # Multipliers
minimize outer_obj: (x[1] + x[2] + x[3] + x[4])*(y[1] + y[2] + y[3] + y[4]);
subject to
# Outer constraints:
outer_con_1: x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 - 1 <= 0;
outer_con_2: -x[1] + y[2]*y[4] <= 0;
outer_con_3: -x[4] + y[3]^2 <= 0;
# Inner objective
inner_obj: x[1]*y[1] + x[2]*y[2] + 0.1*y[3] + 0.5*y[4] - y[3]*y[4] = 0;
# Inner constraints:
inner_con_1: y[1]^2 + 2*y[2]^2 + 3*y[3]^2 + 4*y[4]^2 - x[1]^2 - x[3]^2 - x[2] - x[4] <= 0;
inner_con_2: y[1]*y[4] - y[2]*y[3] <= 0;
# KKT conditions
stationarity_1: x[1] + 2*l[1]*y[1] + l[2]*y[4] - l[3] + l[4] = 0;
stationarity_2: x[2] + 4*l[1]*y[2] - l[2]*y[3] - l[5] + l[6] = 0;
stationarity_3: 0.1 - y[4] + 6*l[1]*y[3] - l[2]*y[2] - l[7] + l[8] = 0;
stationarity_4: 0.5 - y[3] + 8*l[1]*y[4] + l[2]*y[1] - l[9] + l[10] = 0;
complementarity_1: l[1]*(y[1]^2 + 2*y[2]^2 + 3*y[3]^2 + 4*y[4]^2 - x[1]^2 - x[3]^2 - x[2] - x[4]) = 0;
complementarity_2: l[2]*(y[1]*y[4] - y[2]*y[3]) = 0;
complementarity_3: l[3]*(-y[1] - 10) = 0;
complementarity_4: l[4]*(y[1] - 10) = 0;
complementarity_5: l[5]*(-y[2] - 10) = 0;
complementarity_6: l[6]*(y[2] - 10) = 0;
complementarity_7: l[7]*(-y[3] - 10) = 0;
complementarity_8: l[8]*(y[3] - 10) = 0;
complementarity_9: l[9]*(-y[4] - 10) = 0;
complementarity_10: l[10]*(y[4] - 10) = 0;
References
