BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

b_1984_02 : Quadratic-Quadratic bilevel problem from (Bard, 1984)

Optimal solution

Objective values Solution point
F* = b_1984_02 x* = (0.0, 2.0)
f* = -6.473 y* = (1.875, 0.9063)

Sources where this problem occurs

Original source:

Other sources:

Note, that Problem TP3 from www.bilevel.org (and others sources used by the same authors) is based on this problem, but the outer objective is changed from -x[1]^2 - 3*x[2] - 4*y[1] + y[2]^2 to -x[1]^2 - 3*x[2]^2 - 4*y[1] + y[2]^2 keeping the rest unchanged.

Description of the problem in the AMPL format

var x{1..2} >= 0, <= 10;   # Outer variable
var y{1..2} >= 0, <= 10;   # Inner variable
var l{1..6} >= 0, <= 100;  # KKT Multipliers

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

subject to
# Outer constraints
    outer_con: x[1]^2 + 2*x[2] - 4 <= 0;
# Inner objective:
    inner_obj:  2*x[1]^2 + y[1]^2 - 5*y[2] = 0;
# Inner constraints
    inner_con1: -x[1]^2 + 2*x[1] - x[2]^2 + 2*y[1] - y[2] - 3 <= 0;
    inner_con2: -x[2] - 3*y[1] + 4*y[2] + 4 <= 0;
# KKT conditions:
    stationarity_1:    2*y[1] + 2*l[1] -3*l[2] - l[3] + l[4] = 0;
    stationarity_2:    -5 - l[1] + 4*l[2] - l[5] + l[6] = 0;
    complementarity_1: l[1]*(-x[1]^2 + 2*x[1] - x[2]^2 + 2*y[1] - y[2] -3) = 0;
    complementarity_2: l[2]*(-x[2] - 3*y[1] + 4*y[2] + 4) = 0;
    complementarity_3: l[3]*y[1] = 0;
    complementarity_4: l[4]*(y[1] - 10) = 0;
    complementarity_5: l[5]*y[2] = 0;
    complementarity_6: l[6]*(y[2] - 10) = 0;

References