Quadratic-Quadratic bilevel problem from [Bard, 1998]

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Optimal solution

Objective values	Solution points
F* = 0.000	x* = (0.800, 0.200)
f* = -0.900	y* = 1.000

Description of the problem in the AMPL format

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

minimize outer_obj: 0.5*(x[1] - 0.8)^2 + 0.5*(x[2] - 0.2)^2 + 0.5*(y - 1)^2;  # Outer objective

subject to
# Inner objective:
    inner_obj: 0.5*y^2 - y + (-1)*x[1]*y + 2*x[2]*y = 0;
# KKT condition
    stationarity:      y - 1 - x[1] + 2*x[2] - l[1] + l[2] = 0;
    complementarity_1: l[1]*y = 0;
    complementarity_2: l[2]*(y - 1) = 0;

References

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