BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

mb_2007_19 : Quadratic-Nonlinear bilevel problem from [Mitsos & Barton, 2007]

Comments on the problem

In the original source (Mitsos & Barton, 2007) the reported incorrect optimal solution achieved at (x,y) = (0.189,−0.768) with F* = -0.258. The correct solution provided in the table below.

Optimal solution

Objective values Solution point(s)
F* = -0.258 x* = 0.189
f* = -0.018 y* = 0.439

Illustration of the problem

Outer Problem Inner Problem

Sources where this problem occurs

Original source:

Other sources:

Description in the AMPL format

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

minimize outer_obj: x*y - y + 0.5*y^2;

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

References