BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

Quadratic-Nonlinear bilevel problem from (Shimizu et al., 1997)

Optimal solution

Objective values Solution points
F* = 2250.0 x* = 11.25
f* = 194.80 y* = 5.000

This problem has two local optima, (7.2, 12.8) and (11.25, 5), where (11.25, 5) is the global solution. Note that all the inner level functions are convex, and therefore the inner problem KKT conditions are both necessary and sufficient.

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, <= 12.5;       # Outer variables
var y >= 0, <= 50;         # Inner variables
var l{1..3} >= 0, <= 200;  # KKT Multipliers

minimize outer_obj: 16*x^2 + 9*y^2;   # Outer objective

subject to
# Outer constraints:
    outer_con1:  -4*x + y <= 0;
# Inner objective:
    inner_obj: (x + y -20)^4 = 0;
# Inner constraints
    inner_con1: 4*x + y -50 <= 0;
# KKT conditions:
    stationarity:      3*(x + y -20)^3 + l[1] - l[2] + l[3] = 0;
    complementarity_1: l[1]*(4*x + y -50) = 0;
    complementarity_2: l[2]*y = 0;
    complementarity_3: l[3]*(y - 50) = 0;

References