BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

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

Optimal solution

Objective values Solution point(s)
F* = -2.000 x* = -1.000 x* = -0.500
f* = 0.000 y* = 0.000 y* = -1.000

Illustration of the problem

Outer Problem Inner Problem

Description in the AMPL format

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

minimize outer_obj: 2*x + y;

subject to
# Inner objective
    inner_obj: -0.5*x*y^2 - y^4/4 = 0;
# KKT conditions
    stationarity:      -x*y - y^3 - l[1] + l[2] = 0;
    complementarity_1: l[1]*(-1 - y) = 0;
    complementarity_2: l[2]*(y - 1) = 0;

References