BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Nonlinear-Nonlinear bilevel problem from (Colson, 2002)


Optimal solution

Objective values Solution point
F* = 2.000 x* = 4.000
f* = 24.018 y* = 0.000

Sources where this problem occurs

Original source:

Other sources:

Description of the problem in the AMPL format

var x >= 0, <= 10;          # Outer variable
var y >= 0, <= 10;          # Inner variable
var l{1..3} >= 0, <= 100;   # KKT Multipliers

minimize outer_obj: (x - 5)^4 + (2*y + 1)^4;  # Outer objective

subject to
# Outer constraints
    outer_con:  x + y - 4 <= 0;
# Inner objective:
    inner_obj: exp(-x+y) + x^2 + 2*x*y + y^2 + 2*x + 6*y = 0;
# Inner constraints
    inner_con:    -x + y - 2 <= 0;
# KKT conditions:
    stationarity:      exp(-x+y) + 2*x + 2*y + 6 + l[1] - l[2] + l[3] = 0;
    complementarity_1: l[1]*(-x + y - 2) = 0;
    complementarity_2: l[2]*y = 0;
    complementarity_3: l[3]*(y - 10) = 0;

References