Abstract
In the paper, an algorithm is presented for solving two-level programming problems. This algorithm combines a direction finding problem with a regularization of the lower level problem. The upper level objective function is included in the regularzation to yield uniqueness of the follower's solution set. This is possible if the problem functions are convex and the upper level objective function has a positive definite Hessian. The computation of a direction of descent and of the step size is discussed in more detail. Afterwards the convergence proof is given.
Last but not least some remarks and examples describing the difficulty of the inclusion of upper-level constraints also depending on the variables of the lower level are added.
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References
B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Non-Linear Parametric Optimization, Akademie-Verlag: Berlin, 1982.
J.F. Bard, “Geometric and algorithmic developments for a hierarchical planning problem,” European J. Oper. Res., vol. 19, pp. 372–383, 1985.
J.F. Bard, “Convex two-level optimization,” Math. Programming, vol. 40, no. 1, pp. 15–27, 1988.
S. Dempe, “Directional differentiability of optimal solutions under Slater's condition,” Math. Programming, vol. 59, no. 1, pp. 49–69, 1993.
S. Dempe, “A necessary and a sufficient optimality condition for bilevel programming problems,” Optimization, vol. 25, pp. 341–354, 1992.
A.V. Fiacco, Introduction to Sensitivity and Stability analysis in Nonlinear Programming, Academic Press: New York, 1983.
J. Gauvin, “A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming,” Math. Programming, vol. 12, no. 1, pp. 136–139, 1977.
J. Guddat, F.G. Vasquez, and H.Th. Jongen, Parametric Optimization: Singularities, Pathfollowing and Jumps, B.G. Teubner: Stuttgart, 1990.
P.T. Harker and J.-S. Pang, “Existence of optimal solutions to mathematical programs with equilibrium constraints,” Oper. Res. Lett., vol. 7, no. 2, pp. 61–64, 1988.
W.W. Hogan, “Directional derivatives for extremal-value functions with applications to the complelety convex case,” Oper. Res., vol. 21, no. 1, pp. 188–209, 1973.
Y. Ishizuka and E. Aiyoshi, “Double penalty method for bilevel optimization problems,” Ann. of Oper. Res., vol. 34, pp. 73–88, 1992.
K. Jittorntrum, “Solution point differentiability without strict complementarity in nonlinear programming,” Math. Programming Stud., no. 21, pp. 127–138, 1984.
I. Kaneko, “On some recent engineering applications of complementarity problems,” Math. Programming Stud., vol. 17, pp. 111–125, 1982.
P. Loridan and J. Morgan, “ε-regularized two-level optimization problems: Approximation and existence results,” in Optimization—Fifth French-German Conference, Varetz, 1988, Lect. Notes in Mathematics no. 1405, pp. 99–113, 1989.
R. Lucchetti, F. Mignanego, and G. Pieri, “Existence theorem of equilibrium points in Stackelberg games with constraints,” Optimization, vol. 18, no. 6, pp. 857–866, 1987.
B. Luderer, “Über die Äquivalenz nichtlinearer Optimierungsaufgaben,” Working Paper, Techn. Univ. Karl-Marx-Stadt, 1983.
P. Marcotte, “Network design problem with congestion effects: A case of bilevel programming,” Math. Programming, vol. 34, no. 2, pp. 142–162, 1986.
W. Oeder, Ein Verfahren zur Lösung von Zwei-Ebenen-Optimierungsaufgaben in Verbindung mit der Untersuchung von chemischen Gleichgewichten, Ph.D.-Thesis, Techn. Univ. Karl-Marx-Stadt, 1988.
J.S. Pang, S.P. Han, and N. Rangaraj, “Minimization of locally Lipschitzian functions,” SIAM J. Optim. vol. 1, pp. 57–82, 1991.
T. Petersen, Optimale Anreizsysteme, Gabler Verlag: Wiesbaden, 1989.
S.M. Robinson, “Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity,” Math. Programming Stud., vol. 30, pp. 45–66, 1987.
A. Shapiro, “Sensitivity analysis of nonlinear programs and differentiability properties of metric projections,” SIAM J. Control Optim., vol. 26, pp. 628–645, 1988.
W.R. Smith and R.W. Missen, Chemical Reaction Equilibrium Analysis: Theory and Algorithms, J. Wiley & Sons: New York, 1982.
L.N. Vicente and P.H. Calamai, “Bilevel and Multilevel Programming: A Bibliography Review,” Report, Department of Systems Design Engineering, University of Waterloo, Waterloo, N2L 3G1 Ontario, Canada, 1993.
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Dempe, S., Schmidt, H. On an algorithm solving two-level programming problems with nonunique lower level solutions. Comput Optim Applic 6, 227–249 (1996). https://doi.org/10.1007/BF00247793
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DOI: https://doi.org/10.1007/BF00247793