BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

cw_1990_02 : Quadratic-Quadratic bilevel problem from [Clark & Westerberg, 1990]

Comments on the problem

Also has two locally optimal solutions at (x, y) = (3, 5), and (x, y) = (4.4, 4.8).

Optimal solution

Objective values Solution points
F* = 5.0 x* = 1.0
f* = 4.0 y* = 3.0

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 variable
var y >= 0, <= 8;           # Inner variable
var l{1..5} >= 0, <= 10;    # KKT Multipliers

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

subject to
# Inner objective:
    inner_obj: (y - 5)^2 = 0;
# Inner constraints
    inner_con1:  -2*x + y - 1 <= 0;
    inner_con2:   x - 2*y + 2 <= 0;
    inner_con3:   x + 2*y - 14 <= 0;
# KKT conditions:
    stationarity:      2*(y - 5) + l[1] - 2*l[2] + 2*l[3] - l[4] + l[5] = 0;
    complementarity_1: l[1]*(-2*x + y - 1) = 0;
    complementarity_2: l[2]*(x - 2*y + 2) = 0;
    complementarity_3: l[3]*(x + 2*y - 14) = 0;
    complementarity_4: l[4]*y = 0;
    complementarity_5: l[5]*(y - 8) = 0;

References