BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

mb_2007_01 : Linear-Linear problem from (Mitsos & Barton, 2007)

Optimal solution

Objective values Solution point
F* = 1.000 y* = 1.000
f* = -1.000  

The inner program has the unique optimal solution (y = 1) and therefore the bilevel program has the unique feasible point (see illustration on the right hand side) and therefore unique global minimum (see illustration on the left hand side). The inner program is linear and therefore the KKT conditions are both necessary and sufficient for a minimum.

Illustration of the problem

Outer Problem Inner Problem

Sources where this problem occur

Original source:

Other sources:

Description in the AMPL format

var y >= -1, <= 1;        # Inner variable
var l{1..2} >= 0, <= 2;   # KKT Multipliers

minimize outer_obj: y;    # Outer objective

subject to
# Inner objective:
    inner_obj: -y = 0;
# KKT conditions:
    stationarity:      -1 - l[1] + l[2] = 0;
    complementarity_1: (y + 1)*l[1]  = 0;
    complementarity_2: (-y + 1)*l[2] = 0;

References