BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

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

Optimal solution

Objective values Solution point(s)
F* = 1.000 y* = 1.000
f* = -1.000  

The inner program has the unique optimal solution y* = 1 and therefore the bilevel problem has the unique feasible point y = 1.

Illustration of the problem

Outer Problem Inner Problem

Description in the AMPL format

var y >= -0.5, <= 1;      # Inner variables
var l{1..2} >= 0, <= 2;   # KKT Multipliers

minimize outer_obj: y;    # Outer objective

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

References