Quadratic-Quadratic bilevel problem from [Bard, 1998]

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Optimal solution

Objective values	Solution points
F* = -1.41	x* = 1.89
f* = 7.62	y* = (0.89, 0.0)

Description of the problem in the AMPL format

var x >= 0, <= 10;         # Outer variable
var y{1..2} >= 0, <= 10;   # Inner variable
var l{1..8} >= 0, <= 100;  # KKT Multipliers

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

subject to
# Inner objective:
    inner_obj: (2*y[1] - 4)^2 + (2*y[2] - 1)^2 + x*y[1] = 0;
# Inner constraints
    inner_con1:  4*x + 5*y[1] + 4*y[2] - 12 <= 0;
    inner_con2:  4*x - 4*y[1] + 5*y[2] - 4 <= 0;
    inner_con3: -4*x - 5*y[1] + 4*y[2] + 4 <= 0;
    inner_con4: -4*x + 4*y[1] + 5*y[2] - 4 <= 0;
# KKT condition
    stationarity_1:   4*(2*y[1] - 4) + x + 5*l[1] - 4*l[2] - 5*l[3] + 4*l[4] - l[5] + l[6] = 0;
    stationarity_2:   4*(2*y[2] - 4) + 4*l[1] + 5*l[2] + 4*l[3] + 5*l[4] - l[7] + l[8] = 0;
    complementarity_1: l[1]*(4*x + 5*y[1] + 4*y[2] - 12) = 0;
    complementarity_2: l[2]*(4*x - 4*y[1] + 5*y[2] - 4) = 0;
    complementarity_3: l[3]*(-4*x - 5*y[1] + 4*y[2] + 4) = 0;
    complementarity_4: l[4]*(-4*x + 4*y[1] + 5*y[2] - 4) = 0;
    complementarity_5: l[5]*y[1] = 0;
    complementarity_6: l[6]*(y[1] - 10) = 0;
    complementarity_7: l[7]*y[2] = 0;
    complementarity_8: l[8]*(y[2] - 10) = 0;

References

• J. F. Bard, Practical Bilevel Optimization, vol. 30 of Nonconvex Optimization and Its Applications, Springer US, 1998