BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

lh_1994_01 : Linear-Linear problem from [Liu & Hart, 1994]:

Optimal solution

Objective values Solution points
F* = -16.000 x* = 4.000
f* = 4.000 y* = 4.000

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 - 3*y;  # Outer objective

subject to
# Inner objective:
    inner_obj: y = 0;
# Inner constraints
    inner_con1:  -x + y - 3 <= 0;
    inner_con2:   x + 2*y - 12 <= 0;
    inner_con3:   4*x - y - 12 <= 0;
# KKT conditions:
    stationarity:      1 + l[1] + 2*l[2] - l[3] - l[4] + l[5] = 0;
    complementarity_1: l[1]*(-x + y - 3) = 0;
    complementarity_2: l[2]*(x + 2*y - 12) = 0;
    complementarity_3: l[3]*(4*x - y - 12) = 0;
    complementarity_4: l[4]*y = 0;
    complementarity_5: l[5]*(y - 10) = 0;

References