nwj_2017_01 : Linear-Nonlinear bilevel problem from [Nie et al., 2017]
Optimal solutions:
| Objective values | Solution points |
|---|---|
| F* = 2.000 | x* = 2.000 |
| f* = 0.000 | y* = (0.000, 0.000) |
Sources where this problem occurs
Original source:
- Example 3.4 in (Nie et al., 2017)
Description of the problem in the AMPL format
var x >= 2, <= 3; # Outer variable
var y{1..2} >= 0, <= 10; # Inner variable
var l{1..5} >= 0, <= 100; # KKT Multipliers
minimize outer_obj: x + y[1] + y[2]; # Outer objective
subject to
# Inner objective:
inner_obj: x*y[1] + x*y[2] = 0;
# Inner constraints
inner_con: -y[1]^2 + y[2]^2 + y[1]^4 + 2*y[1]^2*y[2]^2 + y[2]^4 <= 0;
# KKT conditions:
stationarity_1: x + l[1]*(-2*y[1] + 4*y[1]^3 + 4*y[1]*y[2]^2) - l[2] + l[3] = 0;
stationarity_2: x + l[1]*(2*y[2] + 4*y[1]^2*y[2] + 4*y[2]^3) - l[4] + l[5] = 0;
complementarity_1: l[1]*(-y[1]^2 + y[2]^2 + y[1]^4 + 2*y[1]^2*y[2]^2 + y[2]^4) = 0;
complementarity_4: l[2]*y[1] = 0;
complementarity_5: l[3]*(y[1] - 10) = 0;
complementarity_6: l[4]*y[2] = 0;
complementarity_7: l[5]*(y[2] - 10) = 0;
References
