Linear-Linear problem from [Bard & Falk, 1982]:

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Optimal solution

Objective values	Solution points
F* = -26.000	x* = (0.000, 0.900)
f* = 3.200	y* = (0.000, 0.600, 0.400)

Description of the problem in the AMPL format

var x{1..2} >= 0, <= 10;   # Outer variable
var y{1..3} >= 0, <= 10;   # Inner variable
var l{1..9} >= 0, <= 10;   # KKT Multipliers

minimize outer_obj: -8*x[1] - 4*x[2] + 4*y[1] - 40*y[2] + 4*y[3];  # Outer objective

subject to
# Inner objective:
    inner_obj: x[1] + 2*x[2] + y[1] + y[2] + 2*y[3] = 0;
# Inner constraints
    inner_con1: -y[1]  + y[2] + y[3] - 1 <= 0;
    inner_con2: 2*x[1] - y[1] + 2*y[2] - 0.5*y[3] - 1 <= 0;
    inner_con3: 2*x[2] + 2*y[1] - y[2] - 0.5*y[3] - 1 <= 0;
# KKT conditions:
    stationarity_1:    1 - l[1] - l[2] + 2*l[3] - l[4] + l[5] = 0;
    stationarity_2:    1 + l[1] + 2*l[2] - l[3] - l[6] + l[7] = 0;
    stationarity_3:    2 + l[1] - 0.5*l[2] - 0.5*l[3] - l[8] + l[9] = 0;
    complementarity_1: l[1]*(-y[1]  + y[2] + y[3] - 1) = 0;
    complementarity_2: l[2]*(2*x[1] - y[1] + 2*y[2] - 0.5*y[3] - 1) = 0;
    complementarity_3: l[3]*(2*x[2] + 2*y[1] - y[2] - 0.5*y[3] - 1) = 0;
    complementarity_4: l[4]*y[1] = 0;
    complementarity_5: l[5]*(y[1] - 10) = 0;
    complementarity_6: l[6]*y[2] = 0;
    complementarity_7: l[7]*(y[2] - 10) = 0;
    complementarity_8: l[8]*y[3] = 0;
    complementarity_9: l[9]*(y[3] - 10) = 0;

References

• J. F. Bard and J. E. Falk, An explicit solution to the multi-level programming problem, Computers & Operations Research, 9 (1982), pp. 77–100