BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

dd_2012_02 : Quadratic-Quadratic bilevel problem from (Dempe & Dutta, 2012)

Optimal solution

Objective values Solution point
F* = -1.000 x* = (0.707, 0.707)
f* = 4.000 y* = (0.000, 1.000)

Sources where this problem occurs

Original source:

Other sources:

Description of the problem in the AMPL format

var x{1..2} >= 0,   <= 10;    # Outer variable
var y{1..2} >= -10, <= 10;    # Inner variable
var l{1..6} >= 0,   <= 200;   # KKT Multipliers

minimize outer_obj:    -y[2];  # Outer objective

subject to
# Outer constraints
    outer_con_1:       y[1]*y[2] = 0;
# Inner objective:
    inner_obj:         y[1]^2 + (y[2] + 1)^2 = 0;
# Inner constraints
    inner_con_1:       (y[1] - x[1])^2 + (y[2] - 1 - x[1])^2 - 1 <= 0;
    inner_con_2:       (y[1] + x[2])^2 + (y[2] - 1 - x[2])^2 - 1 <= 0;
# KKT condition
    stationarity_1:    2*y[1] + 2*l[1]*(y[1] - x[1]) + 2*l[2]*(y[1] + x[2]) - l[3] + l[4] = 0;
    stationarity_2:    2*(y[2] + 1) + 2*l[1]*(y[2] - 1 - x[1]) + 2*l[2]*(y[2] - 1 - x[2]) - l[5] + l[6] = 0;
    complementarity_1: l[1]*((y[1] - x[1])^2 + (y[2] - 1 - x[1])^2 - 1) = 0;
    complementarity_2: l[2]*((y[1] + x[2])^2 + (y[2] - 1 - x[2])^2 - 1) = 0;
    complementarity_3: l[3]*(-y[1] - 10) = 0;
    complementarity_4: l[4]*(y[1] - 10) = 0;
    complementarity_5: l[5]*(-y[2] - 10) = 0;
    complementarity_6: l[6]*(y[2] - 10) = 0;

References