Quadratic-Nonlinear bilevel problem from (Ye & Zhu, 2010)

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Optimal solution

Objective values	Solution points
F* = 1.000	x* = 1.000
f* = -2.000	y* = 1.000

Sources where this problem occurs

Original source:

• Example 4.2 in (Ye & Zhu, 2010)

Other sources:

• Example 5.9, Problem No. 2 in (Nie et al., 2017)

Description of the problem in the AMPL format

var x >= -3, <= 2;        # Outer variables
var y >= -10, <= 10;      # Inner variables
var l{1..3} >= 0, <= 10;  # KKT Multipliers

minimize outer_obj: (x - 1)^2 + y^2;

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

References

• Ye, J. J., & Zhu, D. (2010). New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM Journal on Optimization, 20(4), 1885-1905
• J. Nie, L. Wang, and J. Ye, Bilevel polynomial programs and semidefinite relaxation methods, SIAM Journal on Optimization, 27 (2017), pp. 1728–1757