BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Quadratic-Quadratic bilevel problem from (Sahin & Ciric, 1998)

Comments on the problem

This problem has the same objective functions as cw_1990_02, but the constraints of the inner problem have been moved to the outer problem.

Optimal solution

Objective values Solution point
F* = 9.0 x* = 3.0
f* = 0.0 y* = 5.0

Illustration of the problem

Outer Problem Inner Problem

Sources where this problem occurs

Original source:

Other sources:

Description of the problem in the AMPL format

var x >= 0, <= 8;         # Outer variables
var y >= 0, <= 8;         # Inner variables
var l{1..2} >= 0, <= 10;  # KKT Multipliers

minimize outer_obj: (x-3)^2 + (y-2)^2;   # Outer objective

subject to
# Outer constraints:
    outer_con1:  -2*x + y - 1 <= 0;
    outer_con2:  x - 2*y + 2 <= 0;
    outer_con3:  x + 2*y -14 <= 0;
# Inner objective: To avoid AMPL variable reordering, added redundant (x-x)^2
    inner_obj: (y - 5)^2 + (x-x)^2 = 0;
# KKT conditions:
    stationarity:      2*(y - 5) - l[1] + l[2] = 0;
    complementarity_1: l[1]*y  = 0;
    complementarity_2: l[2]*(y - 10) = 0;

References