as_2013_01 : Linear-Linear problem from [Allende & Still, 2013]

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Optimal solution

Objective value(s)	Solution point(s)
F* = 0.000	x* = 0.000
f* = 0.000	y* = 0.000

Sources where this problem occurs

Original source:

• Example 3.1 in (Allende & Still, 2013)

Other sources:

• Example 5.9, Problem No. 1 in (Nie et al., 2017)

Description in the AMPL format

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

minimize outer_obj: -x - y;  # Outer objective

subject to
# Inner objective:
    inner_obj: y = 0;
# Inner constraints
    inner_con_1:  x - y <= 0;
    inner_con_2:  y <= 0;
# KKT conditions:
    stationarity:      1 - l[1] + l[2] - l[3] - l[4] = 0;
    complementarity_1: l[1]*(x - y) = 0;
    complementarity_2: l[2]*y = 0;
    complementarity_3: l[3]*(-y - 10) = 0;
    complementarity_4: l[4]*(y - 10) = 0;

References

• G. B. Allende and G. Still, Solving bilevel programs with the KKT-approach, Mathematical programming, 138 (2013), pp. 309–332
• J. Nie, L. Wang, and J. Ye, Bilevel polynomial programs and semidefinite relaxation methods, SIAM Journal on Optimization, 27 (2017), pp. 1728–1757