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* = 0.500 y* = 0.500
f* = -1.000  

The inner program has the unique optimal solution y = 0.5 and, therefore, the bilevel problem has the unique optimal solution y* = 1.

Illustration of the problem

Outer Problem Inner Problem

Description in the AMPL format

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

minimize outer_obj: y;    # Outer objective

subject to
# Inner objective:
    inner_obj: 16*y^4 + 2*y^3 - 8*y^2 - 1.5*y + 0.5 = 0;
# KKT conditions
    stationarity:      64*y^3 + 6*y^2 - 16*y - 1.5 - l[1] + l[2] = 0;
    complementarity_2: l[1]*(-1 - y) = 0;
    complementarity_3: l[2]*(y - 1) = 0;

References