Quadratic-Quadratic bilevel problem from (Dempe, 2000)

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Optimal solution

Objective values	Solution points
F* = 0.0	x* = 0.5
f* = -0.25	y* = -0.5

Sources where this problem occurs

Original source:

• Example 5.6 in (Dempe, 2000)

Other sources:

Description of the problem in the AMPL format

var x >= -0.5, <= 10;     # Outer variables
var y >= -10, <= 11;      # Inner variables
var l{1..4} >= 0, <= 10;  # KKT Multipliers

minimize outer_obj: (-x + y + 1)^2;  # d > 0 fixed, so we use d = 1

subject to
# Inner objective
    inner_obj: x*y = 0;
# Inner constraints
    inner_con1: -x - y <= 0;
    inner_con2: -x + y -1 <= 0;
# KKT conditions
    stationarity:      x - l[1] + l[2] - l[3] + l[4] = 0;
    complementarity_1: l[1]*(-x - y) = 0;
    complementarity_2: l[2]*(-x + y - 1) = 0;
    complementarity_3: l[3]*(-10 - y) = 0;
    complementarity_4: l[4]*(y - 11) = 0;

References

• S. Dempe, A bundle algorithm applied to bilevel programming problems with non-unique lower level solutions, Computational Optimization and Applications, 15 (2000), pp. 145–166