Quadratic-Quadratic bilevel problem from (Dempe, 1992)

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Optimal solution

Objective values	Solution points
F* = 31.25	x* = 1.0
f* = 4.00	y* = 1.0

Sources where this problem occurs

Original source:

• Example 4.1 in (Dempe, 1992)

Other sources:

• Dempe92 problem in (Colson, 2002) and in (Colson et al., 2005)

Description of the problem in the AMPL format

var x >= 1, <= 10;         # Outer variables
var y >= 1, <= 10;         # Inner variables
var l{1..3} >= 0, <= 100;  # KKT Multipliers

minimize outer_obj: (x - 3.5)^2 + (y + 4)^2;  # d > 0 fixed, so we use d = 1

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

References

• B. Colson, P. Marcotte, and G. Savard, A trust-region method for nonlinear bilevel programming: algorithm and computational experience, Computational Optimization and Applications, 30 (2005), pp. 211–227
• B. Colson, BIPA(BIlevel Programming with Approximation Methods)(software guide and test problems), Cahiers du GERAD, (2002)
• S. Dempe, A necessary and a sufficient optimality condition for bilevel programming problems, Optimization, 25 (1992), pp. 341–354