A growing collection of bilevel problems
Quadratic-Quadratic bilevel problem from [De Silva, 1978]
Objective values | Solution points |
---|---|
F* = -1.0 | x* = (0.5, 0.5) |
f* = 0.0 | y* = (0.5, 0.5) |
Original source:
Other sources:
AMPL
formatvar x{1..2} >= 0, <= 10; # Outer variable
var y{1..2} >= 0.5, <= 1.5; # Inner variable
var l{1..4} >= 0, <= 100; # KKT Multipliers
minimize outer_obj: x[1]^2 - 2*x[1] + x[2]^2 - 2*x[2] + y[1]^2 + y[2]^2; # Outer objective
subject to
# Inner objective:
inner_obj: (y[1] - x[1])^2 + (y[2] - x[2])^2 = 0;
# KKT conditions:
stationarity_1: 2*(y[1] - x[1]) - l[1] + l[2] = 0;
stationarity_2: 2*(y[2] - x[2]) - l[3] + l[4] = 0;
complementarity_1: l[1]*(-y[1] + 0.5) = 0;
complementarity_2: l[2]*(y[1] - 1.5) = 0;
complementarity_3: l[3]*(-y[2] + 0.5) = 0;
complementarity_4: l[4]*(y[2] - 1.5) = 0;