dd_2012_01 : Quadratic-Nonlinear bilevel problem from (Dempe & Dutta, 2012)

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Optimal solution

Objective values	Solution point
F* = 1.0	x* = 0.000
f* = 0.0	y* = 0.000

Sources where this problem occurs

Original source:

• Example 2.4 in (Dempe & Dutta, 2012)

Other sources:

• Example 3.3 in (Nie et al., 2017)

Description of the problem in the AMPL format

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

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

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

References

• S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical Programming, 131 (2012), pp. 37–48
• J. Nie, L. Wang, and J. Ye, Bilevel polynomial programs and semidefinite relaxation methods, SIAM Journal on Optimization, 27 (2017), pp. 1728–1757