BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

nwj_2017_02 : Nonlinear-Nonlinear bilevel problem from [Nie et al., 2017]

Optimal solution

Objective values Solution points
F* = -1.71 x* = (-1.0, -1.0)
f* = -2.23 y* = (1.11, 0.31, -0.82)

Sources where this problem occurs

Original source:

Description of the problem in the AMPL format

var x{1..2} >= -1, <= 1;    # Outer variables
var y{1..3} >= -10, <= 10;  # Inner variables
var l{1..8} >= 0, <= 100;   # Multipliers

minimize outer_obj: x[1]*y[1] + x[2]*y[2] + x[1]*x[2]*y[1]*y[2]*y[3];

subject to
# Outer constraints:
    outer_con: -x[1]^2 + y[1]*y[2] <= 0;
# Inner objective
    inner_obj: x[1]*y[1]^2 + x[2]^2*y[2]*y[3] - y[1]*y[3]^2 = 0;
# Inner constraints:
    inner_con_1:  -y[1]^2 - y[2]^2 - y[3]^2 + 1 <= 0;
    inner_con_2:  y[1]^2 + y[2]^2 + y[3]^2 - 2 <= 0;
# KKT conditions
    stationarity_1: 2*x[1]*y[1] - y[3]^2 - 2*l[1]*y[1] + 2*l[2]*y[1] - l[3] + l[4] = 0;
    stationarity_2: x[2]^2*y[3] - 2*l[1]*y[2] + 2*l[2]*y[2] - l[5] + l[6] = 0;
    stationarity_3: x[2]^2*y[2] - 2*y[1]*y[3] -2*l[1]*y[3] + 2*l[2]*y[3] - l[7] + l[8] = 0;
    complementarity_1: l[1]*(-y[1]^2 - y[2]^2 - y[3]^2 + 1) = 0;
    complementarity_2: l[2]*(y[1]^2 + y[2]^2 + y[3]^2 - 2) = 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;

References