BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Quadratic-Nonlinear bilevel problem from [Colson, 2002]

Optimal solution

Objective values Solution point
F* = 17.000 x* = 1.000
f* = 2.000 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..5} >= 0, <= 10;   # KKT Multipliers

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

subject to
# Inner objective:
    inner_obj: (y - 1)^2 - 1.5*x*y + x^3 = 0;
# Inner constraints
    inner_con1:    -3*x + y + 3 <= 0;
    inner_con2:     x - 0.5*y - 4 <= 0;
    inner_con3:     x + y - 7 <= 0;
# KKT conditions:
    stationarity:      2*(y - 1) - 1.5*x + l[1] - 0.5*l[2] + l[3] - l[4] + l[5] = 0;
    complementarity_1: l[1]*(-3*x + y + 3) = 0;
    complementarity_2: l[2]*(x - 0.5*y -4) = 0;
    complementarity_3: l[3]*(x + y -7) = 0;
    complementarity_4: l[4]*y = 0;
    complementarity_5: l[5]*(y - 10) = 0;

References