BASBLib - A Library of Bilevel Test Problems | A growing collection of bilevel problems

cw_1988_01 : Linear-Linear problem from [Clark & Westerberg, 1988]:

Optimal solution

Objective value(s)	Solution point(s)
F* = -37.000	x* = 19.000
f* = 14.000	y* = 14.000

Illustration of the problem

Outer Problem	Inner Problem

Sources where this problem occurs

Original source:

Example from Section 2 in (Clark & Westerberg, 1988)

Other sources:

Example 5.1 in (Chen & Florian, 1995)
Example 3.2 in (Yezza, 1996)
Test Problem 4 from Section 9.2.5 in (Floudas et al., 1999)

Description of the problem in the `AMPL` format

var x >= 0, <= 30;         # Outer variable
var y >= 0, <= 30;         # Inner variable
var l{1..5} >= 0, <= 10;   # KKT Multipliers

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

subject to
# Inner objective:
    inner_obj: y = 0;
# Inner constraints
    inner_con1:  -2*x + y <= 0;
    inner_con2:   2*x + 5*y - 108 <= 0;
    inner_con3:   2*x - 3*y + 4 <= 0;
# KKT conditions:
    stationarity:      1 + l[1] + 5*l[2] - 3*l[3] - l[4] + l[5] = 0;
    complementarity_1: l[1]*(-2*x + y) = 0;
    complementarity_2: l[2]*(2*x + 5*y - 108) = 0;
    complementarity_3: l[3]*(2*x - 3*y + 4) = 0;
    complementarity_4: l[4]*y = 0;
    complementarity_5: l[5]*(y - 30) = 0;

Optimal solution

Illustration of the problem

Sources where this problem occurs

Description of the problem in the AMPL format

References

Description of the problem in the `AMPL` format