BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

mb_2007_24 : Nonlinear-Nonlinear bilevel problem from [Mitsos & Barton, 2007]


Comments on the problem

Note, that two global optimal solutions exist, while in (Mitsos & Barton, 2007) only one solution was reported. Moreover, the third outer constraint 2.5 + y^2_1 + y^2_2 + y^2_3 <= 0 is always infeasible, as y^2_1 + y^2_2 + y^2_3 >= 0. To make it feasible, we negated the constant term. After this modification, optimal solution point is the same, as was reported in the original source. Note, that the authors in (Nie et al., 2017) fixed this problem in the same way, but have not mentioned in a text.

Sources where this problem occurs

Original source:

Other sources:

Optimal solution

Objective values Solution point(s)
F* = -2.350 x* = (-1.0,-1.0) x* = (-1.0,-1.0)
f* = -2.000 y* = (1.0,1.0,-0.707) y* = (1.0,-1.0,-0.707)

Description in the AMPL format

var x{1..2} >= -1, <= 1;   # Outer variables
var y{1..3} >= -1, <= 1;   # Inner variables
var l{1..6} >= 0, <= 100;  # Multipliers

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

subject to
# Outer constraints:
    outer_con_1: -x[1]^2 <= -0.1;
    outer_con_2: -(y[1]^2 + y[2]^2 + y[3]^2) <= -1.5;
    outer_con_3: y[1]^2 + y[2]^2 + y[3]^2 <= 2.5;
# Inner objective
    inner_obj: x[1]*y[1]^2 + x[2]*y[2]^2 + (x[1]-x[2])*y[3]^2 = 0;
# KKT conditions
    stationarity_1: 2*x[1]*y[1] -l[1] + l[2] = 0;
    stationarity_2: 2*x[2]*y[2] -l[3] + l[4] = 0;
    stationarity_3: 2*(x[1]- x[2])*y[3] -l[5] + l[6] = 0;
    complementarity_1: -l[1] -l[1]*y[1] = 0;
    complementarity_2: -l[2] + l[2]*y[1] = 0;
    complementarity_3: -l[3] -l[3]*y[2] = 0;
    complementarity_4: -l[4] + l[4]*y[2] = 0;
    complementarity_5: -l[5] -l[5]*y[3] = 0;
    complementarity_6: -l[6] + l[6]*y[3] = 0;

References