BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Linear-Quadratic bilevel problem from [Bard, 1991]


Optimal solution

Objective values Solution points
F* = 2.000 x* = 2.000
f* = 12.000 y* = (6.000, 0.000)

Sources where this problem occurs

Original source:

Other sources:

Description of the problem in the AMPL format

var x >= 2, <= 4;          # Outer variable
var y{1..2} >= 0, <= 10;   # Inner variable
var l{1..5} >= 0, <= 100;  # KKT Multipliers

minimize outer_obj: x + y[2];  # Outer objective

subject to
# Inner objective:
    inner_obj:      2*y[1] + x*y[2] = 0;
# Inner constraints
    inner_con:      x - y[1] - y[2] + 4 <= 0;
# KKT conditions:
    stationarity_1:    2 - l[1] - l[2] + l[3] = 0;
    stationarity_2:    x - l[1] - l[4] + l[5] = 0;
    complementarity_1: l[1]*(x - y[1] - y[2] + 4) = 0;
    complementarity_4: l[2]*y[1] = 0;
    complementarity_5: l[3]*(y[1] - 10) = 0;
    complementarity_6: l[4]*y[2] = 0;
    complementarity_7: l[5]*(y[2] - 10) = 0;

References