Necessary Optimality Conditions for Semi-vectorial Bi-level Optimization with Convex Lower Level: Theoretical Results and Applications to the Quadratic Case
Tóm tắt
This paper explores related aspects to post-Pareto analysis arising from the multi-criteria optimization problem. It consists of two main parts. In the first one, we give first-order necessary optimality conditions for a semi-vectorial bi-level optimization problem: the upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multi-objective optimization problem to be solved by several followers acting in a cooperative way (greatest coalition multi-players game). For the lower level, we deal with weakly or properly Pareto (efficient) solutions and we consider the so-called optimistic problem, i.e. when followers choose amongst Pareto solutions one which is the most favourable for the leader. In order to handle real-life applications, in the second part of the paper, we consider the case where each follower objective is expressed in a quadratic form. In this setting, we give explicit first-order necessary optimality conditions. Finally, some computational results are given to illustrate the paper.
Tài liệu tham khảo
Philip, J.: Algorithms for the vector maximization problem. Math. Program. 2, 207–229 (1972)
An, L.T.H., Muu, L.D., Tao, P.D.: Numerical solution for optimization over the efficient set by d.c. optimization algorithms. Oper. Res. Lett. 19, 117–128 (1996)
Benson, H.P., LEE, D.: Outcome-based algorithm for optimizing over the efficient set of a bi-criteria linear programming problem. J. Optim. Theory Appl. 88, 77–105 (1996)
Benson, H.P.: Generating the efficient outcome set in multiple objective linear programs: the bi-criteria case. Acta Math. Vietnam 22, 29–51 (1997)
Benson, H.P.: Further analysis of an outcome set-based algorithm for multiple objective linear programming. J. Optim. Theory Appl. 97, 1–10 (1998)
Benson, H.P.: Hybrid approach for solving multiple objective linear programs in outcome space. J. Optim. Theory Appl. 98, 17–35 (1998)
Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optim. 13, 1–24 (1998)
Bolintinéanu, S.: Optimality conditions for minimization over the (weakly or properly) efficient set. J. Math. Anal. Appl. 173(2), 523–541 (1993)
Bolintinéanu, S.: Necessary conditions for nonlinear suboptimization over the weakly-efficient set. J. Optim. Theory Appl. 78, 579–598 (1993)
Bonnel, H., Kaya, C.Y.: Optimization over the efficient set of multi-objective control problems. J. Optim. Theory Appl. 147(1), 93–112 (2010)
Craven, B.D.: Aspects of multicriteria optimization. In: Recent Developments in Mathematical Programming, pp. 93–100. Gordon and Breach Science Publishers (1991)
Dauer, J.P.: Optimization over the efficient set using an active constraint approach. J. Oper. Res. 35, 185–195 (1991)
Dauer, J.P., Fosnaugh, T.A.: Optimization over the efficient set. J. Global Optim. 7, 261–277 (1995)
Fülöp, J.: A cutting plane algorithm for linear optimization over the efficient set. In: Generalized Convexity. Lecture notes in Economics and Mathematical System. vol. 405, pp. 374–385. Springer, Berlin (1994)
Horst, R., Thoai, N.V.: Maximizing a concave function over the efficient or weakly-efficient set. Eur. J. Oper. Res. 117, 239–252 (1999)
Horst, R., Thoai, N.V., Yamamoto, Y., Zenke, D.: On optimization over the efficient set in linear multicriteria programming. J. Optim. Theory Appl. 134, 433–443 (2007)
Kim, N.T.B., Thang, T.N.: Optimization over the efficient set of a bi-criteria convex programming problem. Pac. J. Optim. 9, 103–115 (2013)
Bonnel, H., Collonge, J.: Stochastic optimization over a pareto set associated with a stochastic multi-objective optimization problem. J. Optim. Theory Appl. 162, 405–427 (2014)
Bonnel, H., Collonge, J.: Optimization over the pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case. J. Global Optim. 62, 481–505 (2015)
Yamamoto, Y.: Optimization over the efficient set : an overview. J. Global Optim. 22, 285–317 (2002)
Konno, H., Thach, P.T., Yokota, D.: Dual approach to minimization on the set of pareto-optimal solutions. J. Optim. Theory Appl. 88, 689–707 (1996)
Konno, H., Inori, H.M.: Bond portfolio optimization by bilinear fractional programming. J Oper Res Soc Jpn 32, 143–158 (1989)
Benson, H.P.: Optimization over the efficient set. J. Math. Anal. Appl. 98, 562–580 (1984)
Benson, H.P.: A finite, non-adjacent extreme point search algorithm for optimization over the efficient set. J. Optim. Theory Appl. 73, 47–64 (1992)
Bolintinéanu, S.: Minimization of a quasi-concave function over an efficient set. Math. Program. 61, 89–110 (1993)
Bonnel, H., Morgan, J.: Semivectorial bilevel optimization problem: penalty approach. J. Optim. Theory Appl. 131, 365–382 (2006)
Bonnel, H., TodjihoundÉ, L., Udrişte, C.: Semivectorial bilevel optimization on Riemannian manifolds. J. Optim. Theory Appl. 167, 464–486 (2015)
Bonnel, H.: Optimality conditions for the semivectorial bilevel optimization problem. Pac. J. Optim. 2, 447–468 (2006)
Bonnel, H., Morgan, J.: Semivectorial bilevel convex optimal control problems. SIAM J. Control Optim. 50(6), 3224–3241 (2012)
Bonnel, H., Morgan, J.: Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems Computational and analytical mathematics. Springer, New York (2013)
Ren, A., Wang, Y.: A novel penalty function method for semivectorial bilevel programming problem Appl. Math. Model. 40(1), 135–149 (2016)
Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht (2002)
Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56(5–6), 577–604 (2007)
Dempe, S., Gadhi, N., Zemkoho, A.B.: New optimality conditions for the semivectorial bilevel optimization problem. J. Optim. Theory Appl. 157(1), 54–74 (2013)
Dempe, S., Mehlitz, P.: Semivectorial bilevel programming versus bilevel programming. Optimization (2019). https://doi.org/10.1080/02331934.2019.1625900
Liu, B., Wan, Z., Chen, J., et al.: Optimality conditions for pessimistic semivectorial bilevel programming problem. J. Inequal. Appl. 41(1), 1–26 (2014)
Zemkoho, A.B.: Solving ill-posed bilevel programs. Set-valued Var. Anal. 24(3), 423–448 (2016)
Zheng, Y., Chen, J., Cao, X.: A global solution method for semivectorial bilevel programming problem. Filomat 28(8), 1619–1627 (2014)
Ankhili, Z., Mansouri, A.: An exact penalty on bilevel programs with linear vector optimization lower level. Eur. J. Oper. Res. 197, 36–41 (2009)
Zheng, Y., Wan, Z.: A solution method for semivectorial bilevel programming problem via penalty method. J. Appl. Math. Comput. 37, 207–219 (2011)
Eichfelder, G.: Multiobjective bilevel optimization. Math. Program. Ser. 123, 419–449 (2010)
Morgan, J.: Constrained well-posed two-level optimization problems. In: Clarke, F., Demyanov, V., Giannessi, F. (eds.) Nonsmooth Optimization and Related Topics, pp. 307–326. Plenum Press, New York (1989)
Loridan, P., Morgan, J.: A theoretical approximation scheme for stackelberg problems. J. Optim. Theory Appl. 61, 95–110 (1989)
Loridan, P., Morgan, J.: Weak via strong stackelberg problem: new results. J. Global Optim. 8, 263–287 (1996)
Lignola, M.B., Morgan, J.: Stability of regularized bilevel programming problems. J. Optim. Theory Appl. 93, 575–596 (1997)
Calvete, H., Gale, C.: On linear bilevel problems with multiple objectives at the lower level. Omega 39, 33–40 (2011)
Bonnel, H., Morgan, J.: Optimality conditions for semivectorial bilevel convex optimal control problems. Computational and analytical mathematics. In: Bailey, D.H., Bauschke, H.H., Borwein, P., Garvan, F., Théra, M., J.D. Vanderwer, H. Wolkowicz (eds.) Honor of Jonathan Borwein’s 60th Birthday, pp. 43–74. Springer (2013)
Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)
Ehrgott, M.: Multicriteria Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 491. Springer, Berlin (2000)
Göpfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Chen, G., Huang, X., Yang, X.: Vector Optimization: Set Valued and Variational Analysis. Springer, Berlin (2005)
Jahn, J.: Vector Optimization. Springer, Berlin (2004)
Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems 319. Springer, Berlin (1989)
Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Dordrecht (1998)
Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)
Phelps, R.R.: Convex Functions Monotone Operators and Differentiability. Lecture Notes in Mathematics. Springer, Berlin (1993)
Yu, P.L.: Multiple-Criteria Decision Making. Plenum Press, New York (1985)
Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)
Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)