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* = -1.755 x* = 0.211
f* = 0.009 y* = 1.799

Illustration of the problem

Outer Problem Inner Problem

Description in the AMPL format

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

minimize outer_obj: x^2 - y;

subject to
# Inner objective
    inner_obj: ( (y-1-0.1*x)^2 - 0.5 - 0.5*x )^2 = 0;
# KKT conditions
    stationarity:      4*((y-1-0.1*x)^2-0.5-0.5*x)*(y-1-0.1*x) - l[1] + l[2] = 0;
    complementarity_1: -l[1]*y = 0;
    complementarity_2: l[2]*(y - 3) = 0;

References