BASBLib - A Library of Bilevel Test Problems

A growing collection of bilevel problems

cw_1990_01 : Linear-Linear problem from [Clark & Westerberg, 1990]

Optimal solution

Objective value(s) Solution point(s)
F* = -13.000 x* = 5.000
f* = -4.000 y* = (4.000, 2.000)

Original source:

Other sources:

Description of the problem in the AMPL format

var x >= 0, <= 8;           # Outer variable
var y{1..2} >= 0, <= 4;     # Inner variable
var l{1..7} >= 0, <= 10;    # KKT Multipliers

minimize outer_obj: -x - 3*y[1] + 2*y[2];  # Outer objective

subject to
# Inner objective:
    inner_obj: -y[1] = 0;
# Inner constraints
    inner_con1: -2*x + y[1] + 4*y[2] - 16 <= 0;
    inner_con2:  8*x + 3*y[1] - 2*y[2] - 48 <= 0;
    inner_con3: -2*x + y[1] - 3*y[2] + 12 <= 0;
# KKT conditions:
    stationarity_1:    -1 + l[1] + 3*l[2] + l[3] - l[4] + l[5] = 0;
    stationarity_2:    4*l[1] - 2*l[2] - 3*l[3] - l[6] + l[7] = 0;
    complementarity_1: l[1]*(-2*x + y[1] + 4*y[2] - 16) = 0;
    complementarity_2: l[2]*(8*x + 3*y[1] - 2*y[2] - 48) = 0;
    complementarity_3: l[3]*(-2*x + y[1] - 3*y[2] + 12) = 0;
    complementarity_4: l[4]*y[1] = 0;
    complementarity_5: l[5]*(y[1] - 4) = 0;
    complementarity_6: l[6]*y[2] = 0;
    complementarity_7: l[7]*(y[2] - 8) = 0;

References