BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

sib_1997_02v : Linear-Linear problem, variation of sib_1997_02

Comments on the problem

This is a variation of sib_1997_02 problem. The only difference is the fourth inner constraint, which is changed from -3x + 2y + 4 <= 0 into 3x - 2y - 4 <= 0. After this modification, the optimal solution occurs at (x, y) = (4.0, 4.0) with F* = -12.0 and f* = 4.0, i.e., at the same point which was reported in the literature for sib_1997_02 problem.

Optimal solution

Objective values Solution point
F* = -12.000 x* = 4.000
f* = 4.000 y* = 4.000

Illustration of the problem

Outer Problem Inner Problem

Description of the problem in the AMPL format

var x >= 0, <= 10;         # Outer variable
var y >= 0, <= 10;         # Inner variable
var l{1..6} >= 0, <= 10;   # KKT Multipliers

minimize outer_obj: x - 4*y;  # Outer objective

subject to
# Inner objective:
    inner_obj: y = 0;
# Inner constraints
    inner_con1: -x - y + 3 <= 0;
    inner_con2: -2*x + y <= 0;
    inner_con3: 2*x + y - 12 <= 0;
    inner_con4: 3*x - 2*y - 4 <= 0;
# KKT conditions:
    stationarity:      1 - l[1] + l[2] + l[3] - 2*l[4] - l[5] + l[6] = 0;
    complementarity_1: l[1]*(-x - y + 3) = 0;
    complementarity_2: l[2]*(-2*x + y) = 0;
    complementarity_3: l[3]*(2*x + y - 12) = 0;
    complementarity_4: l[4]*(3*x - 2*y - 4) = 0;
    complementarity_5: l[5]*y = 0;
    complementarity_6: l[6]*(y - 10) = 0;

References