BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Quadratic-Quadratic bilevel problem from (Shimizu & Aiyoshi, 1981)

Optimal solution

Objective values Solution point
F* = 100.0 x* = 10.0
f* = 0.0 y* = 10.0

Sources where this problem occurs

Original source:

Other sources:

Description of the problem in the AMPL format

var x >= 0, <= 15;        # Outer variable
var y >= 0, <= 20;        # Inner variable
var l{1..3} >= 0, <= 10;  # KKT Multipliers

minimize outer_obj: x^2 + (y-10)^2;  # Outer objective

subject to
# Outer constraints
    outer_con:  -x + y <= 0;
# Inner objective:
    inner_obj: (x + 2*y -30)^2 = 0;
# Inner constraints
    inner_con:   x + y - 20 <= 0;
# KKT conditions:
    stationarity:      4*(x + 2*y -30) + l[1] - l[2] + l[3] = 0;
    complementarity_1: l[1]*(x + y - 20) = 0;
    complementarity_2: l[2]*y = 0;
    complementarity_3: l[3]*(y - 20) = 0;

References