BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Linear-Linear problem from [Candler & Townsley, 1982]


Optimal solution

Objective values Solution point
F* = -29.200 x* = (0.0, 0.9)
f* = 3.200 y* = (0.0, 0.6, 0.4, 0.0, 0.0, 0.0)

Original source:

Other sources:

Description in the AMPL format

var x{1..2} >= 0, <= 10;   # Outer variable
var y{1..6} >= 0, <= 10;   # Inner variable
var l{1..15} >= 0, <= 10;  # KKT Multipliers

minimize outer_obj: -8*x[1] - 4*x[2] + 4*y[1] - 40*y[2] - 4*y[3];  # Outer objective

subject to
# Inner objective:
    inner_obj: x[1] + 2*x[2] + y[1] + y[2] + 2*y[3] = 0;
# Inner constraints
    inner_con1: -y[1]  + y[2] + y[3] + y[4] - 1 = 0;
    inner_con2: 2*x[1] - y[1] + 2*y[2] - 0.5*y[3] + y[5] - 1 = 0;
    inner_con3: 2*x[2] + 2*y[1] - y[2] - 0.5*y[3] + y[6] - 1 = 0;
# KKT conditions:
    stationarity_1:    1 - l[1] - l[2] + 2*l[3] - l[4] + l[5] = 0;
    stationarity_2:    1 + l[1] + 2*l[2] - l[3] - l[6] + l[7] = 0;
    stationarity_3:    2 + l[1] - 0.5*l[2] - 0.5*l[3] - l[8] + l[9] = 0;
    stationarity_4:    l[1] - l[10] + l[11] = 0;
    stationarity_5:    l[2] - l[12] + l[13] = 0;
    stationarity_6:    l[3] - l[14] + l[15] = 0;
    complementarity_1: l[1]*(-y[1]  + y[2] + y[3] + y[4] - 1) = 0;
    complementarity_2: l[2]*(2*x[1] - y[1] + 2*y[2] - 0.5*y[3] + y[5] - 1) = 0;
    complementarity_3: l[3]*(2*x[2] + 2*y[1] - y[2] - 0.5*y[3] + y[6] - 1) = 0;
    complementarity_4: l[4]*y[1] = 0;
    complementarity_5: l[5]*(y[1] - 10) = 0;
    complementarity_6: l[6]*y[2] = 0;
    complementarity_7: l[7]*(y[2] - 10) = 0;
    complementarity_8: l[8]*y[3] = 0;
    complementarity_9: l[9]*(y[3] - 10) = 0;
    complementarity_10: l[10]*y[4] = 0;
    complementarity_11: l[11]*(y[4] - 10) = 0;
    complementarity_12: l[12]*y[5] = 0;
    complementarity_13: l[13]*(y[5] - 10) = 0;
    complementarity_14: l[14]*y[6] = 0;
    complementarity_15: l[15]*(y[6] - 10) = 0;

References