Abstract
It is not accidental that we unify the exposition of these two areas of optimization theory in one chapter.
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References
L. Adam, R. Henrion and J. V. Outrara (2017), On M-stationarity conditions in MPECs and the associated qualification conditions, Math. Program., DOI 10.1007/s10107-017-1146-3.
R. Andreani, G. Haeser, M. L. Schuverdt and P. J. S. Silva (2012), Two new weak constraint qualifications and applications, SIAM J. Optim. 22, 1109–1135.
T. Q. Bao, P. Gupta and B. S. Mordukhovich (2007), Necessary conditions for multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl. 135, 179–203.
M. Bardi (1989), A boundary value problem for the minimal time function, SIAM J. Control Optim. 27, 776–785.
R. E. Bellman (1957), Dynamic Programming, Princeton University Press, Princeton, New Jersey.
M. Benko and H. Gfrerer (2017), New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints, to appear in Optimization, arXiv https://arxiv.org/pdf/1611.08206.pdf.
D. P. Bertsekas and A. E. Ozdaglar (2002), Pseudonormality and a Lagrange multiplier theory for constrained optimization, J. Optim. Theory Appl. 114, 287–343.
J. M. Borwein and D. Preiss (1987), A smooth variational principle with applications to subdifferentiability and differentiability of convex functions, Trans. Amer. Math. Soc. 303, 517–527.
J. V. Burke and S. Deng (2005), Weak sharp minima revisited, II: applications to linear regularity and error bounds, Math. Program. 104, 235–261.
J. V. Burke and M. C. Ferris (1993), Weak sharp minima in mathematical programming, SIAM J. Control Optim. 31, 1340–1359.
P. Cannarsa and C. Sinestrari (2004), Semiconvex Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhäuser, Boston, Massachusetts.
F. H. Clarke (1976), A new approach to Lagrange multipliers, Math. Oper. Res. 2, 165–174.
F. H. Clarke (1983), Optimization and Nonsmooth Analysis, Wiley-Interscience, New York.
F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski (1998), Nonsmooth Analysis and Control Theory, Springer, New York.
B. Colson, P. Marcotte and G. Savard (2007), An overview of bilevel optimization, Ann. Oper. Res. 153, 235–256.
S. Dempe (2003), Foundations of Bilevel Programming, Kluwer, Dordrecht, The Netherlands.
S. Dempe and J. Dutta (2012), Is bilevel programming a special case of mathematical programming with complementarity constraints?, Math. Program. 131, 37–48.
S. Dempe, J. Dutta and B. S. Mordukhovich (2007), New necessary optimality conditions in optimistic bilevel programming, Optimization 56, 577–604.
S. Dempe, N. A. Gadhi and L. Lafhim (2010), Fuzzy and exact optimality conditions for a bilevel set-valued problem via extremal principles, Numer. Funct. Anal. Optim. 31, 907–920.
S. Dempe, V. Kalashnikov, G. A. Pérez-Valdés and N. Kalashnykova (2015), Bilevel Programming Problems, Springer, New York.
S. Dempe, B. S. Mordukhovich and A. B. Zemkoho (2012), Sensitivity analysis for two-level value functions with applications to bilevel programming, SIAM J. Optimization 22, 1309–1343.
S. Dempe, B. S. Mordukhovich and A. B. Zemkoho (2014), Necessary optimality conditions in pessimistic bilevel programming, Optimization 63, 505–533.
S. Dempe and A. B. Zemkoho (2012), On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem, Nonlinear Anal. 75, 1202–1218.
S. Dempe and A. B. Zemkoho (2013), The bilevel programming problem: reformulations, constraint qualifications and optimality conditions, Math. Program. 138, 447–473.
S. Dempe and A. B. Zemkoho (2014), KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization, SIAM J. Optim. 24, 1639–1669.
V. F. Demyanov and A. M. Rubinov (2000), Quasidifferentiable and Related Topics, Kluwer, Dordrecht, The Netherlands.
M. Durea and R. Strugariu (2010), Necessary optimality conditions for weak sharp minima in set-valued optimization, Nonlinear Anal. 73, 2148–2157.
M. C. Ferris (1988), Weak Sharp Minima and Penalty Functions in Mathematical Programming, Ph.D. dissertation, University of Cambridge, Cambridge, United Kingdom.
M. L. Flegel and C. Kanzow (2005), On M-stationarity for mathematical programs with equilibrium constraints, J. Math. Anal. Appl. 310, 286–302.
W. H. Fleming and H. M. Soner (1993), Controlled Markov Processes and Viscosity Solutions, Springer, New York.
F. Flores-Bazán and G. Mastroeni (2015), Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications, SIAM J. Optim. 25 (2015), 647–676.
H. Gfrerer and J. J. Ye (2017), New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis, SIAM J. Optim. 27, 842–865.
V. V. Gorokhovik and M. Trafimovich (2016), Positively homogeneous functions revisited, J. Optim. Theory Appl. 171, 481–503.
J. Grzyzbowski, D. Pallaschke and R. Urbanski (2017), Characterization of differences of sublinear functions, to appear in Pure Appl. Funct. Anal.
L. Guo and J. J. Ye (2016), Necessary optimality conditions for optimal control problems with equilibrium constraints, SIAM J. Control Optim. 54, 2710–2733.
N. T. V. Hang (2014), The penalty functions method and multiplier rules based on the Mordukhovich subdifferential, Set-Valued Var. Anal. 22, 299–312.
N. T. V. Hang and J.-C. Yao (2016), Sufficient conditions for error bounds of difference functions and applications, J. Global Optim. 66, 439–456.
N. T. V. Hang and N. D. Yen (2015), Optimality conditions and stability analysis via the Mordukhovich subdifferential, Numer. Funct. Anal. Optim. 36, 364–386.
N. T. V. Hang and N. D. Yen (2016), On the problem of minimizing a difference of polyhedral convex functions under linear constraints, J. Optim. Theory Appl. 171, 617–642.
R. Henrion, A. Jourani and J. V. Outrata (2002), On the calmness of a class of multifunctions, SIAM J. Optim. 13, 603–618.
R. Henrion and J. V. Outrata (2005), Calmness of constraint systems with applications, Math. Program. 104, 437–464.
R. Henrion and T. Surowiec (2011), On calmness conditions in convex bilevel programming, Applic. Anal. 90, 951–970.
M. Hintermüller, B. S. Mordukhovich and T. Surowiec (2014), Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints, Math. Program. 146, 555–582.
J.-B. Hiriart-Urruty (1989), From convex optimization to nonconvex optimization: necessary and sufficient conditions for global optimality, in Nonconvex Optimization and Related Topics, pp. 219–239, Plenum Press, New York.
R. Horst, P. D. Pardalos and N. V. Thoai (2000), Introduction to Global Optimization, 2nd edition, Springer, New York.
A. D. Ioffe (1981), Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), 1–56.
A. Y. Kruger (1981), Generalized Differentials of Nonsmooth Functions and Necessary Conditions for an Extremum, Ph.D. dissertation, Department of Applied Mathematics, Belarus State University, Minsk, Belarus (in Russian).
A. Y. Kruger (1985), Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian Math. J. 26, 370–379.
A. Y. Kruger and B. S. Mordukhovich (1978), Minimization of nonsmooth functionals in optimal control problems, Eng. Cybernetics 16, 126–133.
A. Y. Kruger and B. S. Mordukhovich (1980), Generalized normals and derivatives, and necessary optimality conditions in nondifferential programming, I&II, Depon. VINITI: I# 408-80, II# 494-80, Moscow (in Russian).
A. Y. Kruger and B. S. Mordukhovich (1980), Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24, 684–687 (in Russian).
C. Li, B. S. Mordukhovich, J. Wang and J.-C. Yao (2011), Weak sharp minima on Riamennian manifolds, SIAM J. Optim. 21, 1523–1560.
M. B. Lignola and J. Morgan (2017), Inner regularizations and viscosity solutions for pessimistic bilevel optimization problems, J. Optim. Theory Appl. 173, 183–202.
S. Lu (2011), Implications of the constant rank constraint qualification, Math. Program. 126, 365–392.
Z. Q. Luo, J.-S. Pang and D. Ralph (1996), Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom.
J.-E. Martínez-Legaz and M. Volle (1999). Duality in DC programming: the case of several DC constraints, J. Math. Anal. Appl. 237, 657–671.
K. Meng and X. Yang (2016), Variational analysis of weak sharp minima via exact penalization, Set-Valued Var. Anal. 24, 619–635.
L. Minchenko and S. Stakhovski (2011), Parametric nonlinear programming problems under the relaxed constant rank condition, SIAM J. Optim. 21, 314–332.
B. S. Mordukhovich (1976), Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40, 960–969.
B. S. Mordukhovich (1977), Approximation and maximum principle for nonsmooth problems of optimal control, Russian Math. Surveys 196, 263–264.
B. S. Mordukhovich (1980), Metric approximations and necessary optimality conditions for general classes of extremal problems, Soviet Math. Dokl. 22, 526–530.
B. S. Mordukhovich (1988), Approximation Methods in Problems of Optimization and Control, Nauka, Moscow (in Russian).
B. S. Mordukhovich (2001), The extremal principle and its applications to optimization and economics, in Optimization and Related Topics, edited by A. Rubinov and B. Glover, pp. 343–369, Kluwer, Dordrecht, The Netherlands.
B. S. Mordukhovich (2004), Necessary conditions in nonsmooth minimization via lower and upper subgradients, Set-Valued Anal. 12, 163–193.
B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin.
B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin.
B. S. Mordukhovich and A. Y. Kruger (1976), Necessary optimality conditions for a terminal control problem with nonfunctional constraints, Dokl. Akad. Nauk BSSR 20, 1064–1067 (in Russian).
B. S. Mordukhovich and N. M. Nam (2014), An Easy Path to Convex Analysis and Applications, Morgan & Claypool Publishers, San Rafael, California.
B. S. Mordukhovich, N. M. Nam and H. M. Phan (2011), Variational analysis of marginal functions with applications to bilevel programming, J. Optim. Theory Appl. 152, 557–586.
B. S. Mordukhovich, N. M. Nam and N. D. Yen (2006), Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization 55, 685–396.
B. S. Mordukhovich and B. Wang (2002), Necessary optimality and suboptimality conditions in nondifferentiable programming via variational principles, SIAM J. Control Optim. 41, 623–640.
K. F. Ng and X. Y. Zheng (2003), Global weak sharp minima on Banach spaces, SIAM J. Control Optim. 41, 1868–1885.
J. V. Outrata (1990), On the numerical solution of a class of Stackelberg problems, ZOR–Methods Models Oper. Res. 34, 255–277.
J. V. Outrata (1999), Optimality conditions for a class of mathematical programs with equilibrium constraints, Math. Oper. Res. 24, 627–644.
J. V. Outrata, J. Jarušek and J. Stará (2011), On optimality conditions in control of elliptic variational inequalities, Set-Valued Var. Anal. 19, 23–42.
J. V. Outrata, M. Kočvara and J. Zowe (1998), Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer, Dordrecht, The Netherlands.
B. T. Polyak (1979), Sharp Minima, Institute of Control Sciences Lecture Notes, Moscow; presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria.
B. T. Polyak (1987), Introduction to Optimization, Optimization Software, New York.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko (1962), The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.
R. T. Rockafellar and R. J-B. Wets (1998), Variational Analysis, Springer, Berlin.
H. Scheel and S. Scholtes (2000), Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity, Math. Oper. Res. 25, 1–22.
W. Schirotzek (2007), Nonsmooth Analysis, Springer, Berlin.
M. Studniarski and D. E. Ward (1999), Weak sharp minima: characterizations and sufficient conditions, SIAM J. Control Optim. 38, 219–236.
A. I. Subbotin (1995), Generalized Solutions of First-Order PDEs, Birkhäuser, Boston, Massachusetts.
T. Tanino and T. Ogawa (1984), An algorithm for solving two-level convex optimization problems, Inter. J. Syst. Sci. 15, 163–174.
J. Warga (1976), Derivate containers, inverse functions, and controllability, in Calculus of Variations and Control Theory, edited by D. L. Russel, pp. 13–46, Academic Press, New York.
J. Warga (1978), Controllability and a multiplier rule for nondifferentiable optimization problems, SIAM J. Control Optim. 16, 803–812.
J. J. Ye (1998), New uniform parametric error bounds, J. Optim. Theory Appl. 98, 197–219.
J. J. Ye (2000), Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints, SIAM J. Optim. 10, 943–962.
J. J. Ye and X. Y. Ye (1997), Necessary optimality conditions for optimization problems with variational inequality constraints, Math. Oper. Res. 22, 977–997.
J. J. Ye and D. L. Zhu (1995), Optimality conditions for bilevel programming problems, Optimization 33, 9–27.
J. J. Ye and D. L. Zhu (1997), A note on optimality conditions for bilevel programming problems, Optimization 39, 361–366.
J. J. Ye and D. L. Zhu (2010), New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach, SIAM J. Optim. 20, 1885–1905.
A. J. Zaslavski (2012), Necessary optimality conditions for bilevel minimization problems, Nonlinear Anal. 75, 1655–1678.
A. Zemkoho (2016), Solving ill-posed bilevel programs, Set-Valued Var. Anal. 24, 423–448.
R. Zhang (2003), Multistage bilevel programming problems, Optimization 52, 605–616.
X. Y. Zheng and K. F. Ng (2015), Hölder stable minimizers, tilt stability and Hölder metric regularity of subdifferentials, SIAM J. Optim. 25, 416–438.
X. Y. Zheng and X. Q. Yang (2007), Weak sharp minima for semi-infinite optimization problems with applications, SIAM J. Optim. 18, 573–588.
J. Zhou, B. S. Mordukhovich and N. Xiu (2012), Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity, Nonlinear Anal. 75, 1700–1718.
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Mordukhovich, B.S. (2018). Nondifferentiable and Bilevel Optimization. In: Variational Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-92775-6_6
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