BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

nwj_2017_04 : Nonlinear-Nonlinear bilevel problem from [Nie et al., 2017]

Optimal solution

Objective values Solution points
F* = -2.000 x* = (1.000, 1.000)
f* = -1.000 y* = (0.000, 0.000, 1.000)

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: 0.5*x[1]^2*y[1] + x[2]*y[2]^2 - (x[1] + x[2]^2)*y[3];

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