BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

aw_1990_01 : Linear-Linear problem from [Anandalingam & White, 1990]:

Optimal solution

Objective value(s) Solution point(s)
F* = -49.000 x* = 16.000
f* = 17.000 y* = 11.000

Sources where this problem occurs

Original source:

Other sources:

Description of the problem in the AMPL format

var x >= 0, <= 50;         # Outer variable
var y >= 0, <= 50;         # Inner variable
var l{1..7} >= 0, <= 10;   # KKT Multipliers

minimize outer_obj: -x - 3*y;  # Outer objective

subject to
# Inner objective:
    inner_obj: -x + 3*y = 0;
# Inner constraints
    inner_con1:    -x - 2*y + 10 <= 0;
    inner_con2:    x - 2*y - 6 <= 0;
    inner_con3:    2*x - y - 21 <= 0;
    inner_con4:    x + 2*y - 38 <= 0;
    inner_con5:    -x + 2*y - 18 <= 0;
# KKT conditions:
    stationarity:      3 - 2*l[1] - 2*l[2] - l[3] + 2*l[4] + 2*l[5] - l[6] + l[7]= 0;
    complementarity_1: l[1]*(-x - 2*y + 10) = 0;
    complementarity_2: l[2]*(x - 2*y - 6) = 0;
    complementarity_3: l[3]*(2*x - y - 21) = 0;
    complementarity_4: l[4]*(x + 2*y - 38) = 0;
    complementarity_5: l[5]*(-x + 2*y - 18) = 0;
    complementarity_6: l[6]*y = 0;
    complementarity_7: l[7]*(y - 50) = 0;

References