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* = 0.1875 x* = -0.250 x* = -0.250
f* = -0.016 y* = 0.500 y* = -0.500

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: (x + 0.5)^2 + 0.5*y^2;

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