BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Quadratic-Quadratic bilevel problem from [De Silva, 1978]

Optimal solution

Objective values Solution points
F* = -1.0 x* = (0.5, 0.5)
f* = 0.0 y* = (0.5, 0.5)

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} >= 0.5, <= 1.5;  # Inner variable
var l{1..4} >= 0, <= 100;    # KKT Multipliers

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

subject to
# Inner objective:
    inner_obj:  (y[1] - x[1])^2 + (y[2] - x[2])^2 = 0;
# KKT conditions:
    stationarity_1:    2*(y[1] - x[1]) - l[1] + l[2] = 0;
    stationarity_2:    2*(y[2] - x[2]) - l[3] + l[4] = 0;
    complementarity_1: l[1]*(-y[1] + 0.5) = 0;
    complementarity_2: l[2]*(y[1] - 1.5) = 0;
    complementarity_3: l[3]*(-y[2] + 0.5) = 0;
    complementarity_4: l[4]*(y[2] - 1.5) = 0;

References