BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

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

Optimal solution

Objective values Solution point(s)
F* = 0.210 x* = -0.555
f* = -0.069 y* = 0.455

Illustration of the problem

Outer Problem Inner Problem

Description in the AMPL format

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

minimize outer_obj: (x + 0.6)^2 + y^2;

subject to
# Inner objective
    inner_obj: y^4 + (4/30)*(-x+1)*y^3 + (-0.02*x^2+0.16*x-0.4)*y^2 + (0.004*x^3-0.036*x^2+0.08*x)*y = 0;
# KKT conditions
    stationarity:      0.004*x^3-0.036*x^2+0.08*x + (-0.04*x^2+0.32*x-0.8)*y + (-0.4*x+0.4)*y^2 + 4*y^3 - l[1] + l[2] = 0;
    complementarity_1: l[1]*(-1 - y) = 0;
    complementarity_2: l[2]*(y - 1) = 0;

References