Abstract
We discuss new applications of advanced tools of variational analysis and generalized differentiation to a number of important problems in optimization theory, equilibria, optimal control, and feedback control design. The presented results are largely based on the recent work by the author and his collaborators. Among the main topics considered and briefly surveyed in this paper are new calculus rules for generalized differentiation of nonsmooth and set-valued mappings; necessary and sufficient conditions for new notions of linear subextremality and suboptimality in constrained problems; optimality conditions for mathematical problems with equilibrium constraints; necessary optimality conditions for optimistic bilevel programming with smooth and nonsmooth data; existence theorems and optimality conditions for various notions of Pareto-type optimality in problems of multiobjective optimization with vector-valued and set-valued cost mappings; Lipschitzian stability and metric regularity aspects for constrained and variational systems.
Research was partially supported by the US National Science Foundation under grants DMS-0304989 and DMS-0603846 and also by the Australian Research Council under grant DP-04511668.
Chapter PDF
Similar content being viewed by others
Keywords
- Multiobjective Optimization
- Generalize Differential Construction
- Bilevel Program
- Multiobjective Optimization Problem
- Equilibrium Constraint
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Attouch, A., Buttazzo, G., Michaelle, G.: Variational Analysis in Sobolev and BV Spaces. In: Applications to PDEs and Optimization. MPS-SIAM Series on Optimization. SIAM Publications, Philadelphia (2006)
Bao, T.Q., Gupta, P., Mordukhovich, B.S.: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135, 179–203 (2007)
Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 36, 531–562 (2007)
Bao, T.Q., Mordukhovich, B.S.: Existence of minimizers and necessary conditions in multiobjective optimization with equilibrium constraints. Appl. Math. 26, 453–472 (2007)
Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Global Optim. (to appear)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers to multiobjective problems: existence and optimality conditions: Tech. Rep. 11, Department of Mathematics, Wayne State University (2007)
Borwein, J.M., Zhu, Q.J.: Techniques in Variational Analysis. CMS Books in Mathematics Series, vol. 20. Springer, New York (2005)
Borwein, J.M., Zhuang, D.M.: Super efficiency in vector optimization. Trans. Amer. Math. Soc. 338, 105–122 (1993)
Clarke, F.H.: Optimization and Nonsmooth Analysis. CMS Books in Mathematics Series, vol. 1. Wiley, New York (1983)
Crandall, M.G., Lions, P.-L.: Viscosity solutions to Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42 (1983)
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, 577–604 (2007)
Donchev, T., Farkhi, E., Mordukhovich, B.S.: Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces. J. Diff. Eq. 243, 301–328 (2007)
Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Amer. Math. Soc. 355, 493–517 (2003)
Eberhard, A., Mordukhovich, B.S., Pearce, C.E.M.: On differentiability properties of prox-regular functions, preprint (2008)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Facchinei, F., Pang, J.-P.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Series in Operations Research. Springer, New York (2003)
Geremev, W., Mordukhovich, B.S., Nam, N.M.: Coderivative calculus and metric regularity for constraint and variational systems. Nonlinear Anal. (to appear)
Göpfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics, vol. 17. Springer, New York (2003)
Ioffe, A.D.: Metric regularity and subdifferential calculus. Russian Math. Surv. 55, 501–558 (2000)
Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Series in Operations Research. Springer, New York (2004)
Jeyakumar, V., Luc, D.T.: Nonsmooth Vector Functions and Continuous Optimization. Optimization and Its Applications Series, vol. 10. Springer, New York (2008)
Jourani, A., Thibault, L.: Coderivative of multivalued mappings, locally compact cones and metric regularity. Nonlinear Anal. 35, 925–945 (1999)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods, and Applications. Kluwer Academic Publishers, Boston (2002)
Kruger, A.Y.: Weak stationarity: eliminating the gap between necessary and sufficient conditions. Optimization 53, 147–164 (2004)
Levy, A.B., Poliquin, R.A.: Characterizing the single-valuedness of multifunctions. Set-Valued Anal. 5, 351–364 (1997)
Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)
Mordukhovich, B.S.: Metric approximations and necesary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22, 526–530 (1980)
Mordukhovich, B.S.: Complete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340, 1–35 (1993)
Mordukhovich, B.S.: Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM Journal on Control and Optimization 33, 815–882 (1995)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 331. Springer, Berlin (2006)
Mordukhovich, B.S.: Coderivative calculus and robust Lipschitzian stability for variational systems. J. Convex Anal. 13, 799–822 (2006)
Mordukhovich, B.S.: Variational analysis of evolution inclusions. SIAM J. Optim. 18, 752–777 (2007)
Mordukhovich, B.S.: Variational analysis in nonsmooth optimization and discrete optimal control. Math. Oper. Res. 26, 840–856 (2007)
Mordukhovich, B.S.: Necessary and sufficient conditions for linear suboptimality in constrained optimization. J. Global Optim. 40, 225–244 (2008)
Mordukhovich, B.S.: Methods of variational analysis in multiobjective optimization. Optimization (to appear)
Mordukhovich, B.S.: Characterizations of linear suboptimality for mathematical programs with equilibrium constraints. Math. Prog. (to appear)
Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Prog. (to appear)
Mordukhovich, B.S.: Optimization and equilibrium problems with equilibrium constraints in infinite-dimensional spaces. Optimization (to appear)
Mordukhovich, B.S.: Suboptimal minimax design of constrained parabolic systems with mixed boundary control. Appl. Math. Comp. (to appear)
Mordukhovich, B.S.: Optimization and feedback design of state-constrained parabolic systems. Pacific J. Math. (to appear)
Mordukhovich, B.S.: Failure of metric regularity for major classes of variational systems. Nonlinear Anal. (to appear)
Mordukhovich, B.S.: Suboptimal feedback control of constrained parabolic systems in uncertainty conditions: Tech. Rep. 15, Department of Mathematics, Wayne State University (2006)
Mordukhovich, B.S., Nam, N.M.: Exact calculus for proximal subgradients with applications to optimization. ESAIM Proc. 17, 80–95 (2007)
Mordukhovich, B.S., Nam, N.M.: Variational analysis of extended generalized equations via coderivative calculus in Asplund spaces. J. Math. Anal. Appl. (to appear)
Mordukhovich, B.S., Nam, N.M., Phan, H.: Generalized differentiation of value functions in parametric optimization with applications to bilevel programming, preprint (2008)
Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55, 685–708 (2006)
Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Prog. (to appear)
Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasivariational inequalities with applications to stability and optimization. SIAM J. Optim. 18, 389–412 (2007)
Mordukhovich, B.S., Outrata, J.V., Červinka, M.: Equilibrium problem with complementarity constraints: case study with applications to oligopolistic markets. Optimization 56, 479–494 (2007)
Mordukhovich, B.S., Pennanen, T.: Epiconvergent discretization of the generalized Bolza problem in dynamic optimization. Optimization Letters 1, 379–390 (2007)
Mordukhovich, B.S., Seidman, T.I.: Asymmetric games for convolution systems with applications to feedback control of constrained parabolic equations. J. Math. Anal. Appl. 333, 401–415 (2007)
Mordukhovich, B.S., Wang, B.: Generalized differentiation of parameter-dependent sets and mappings. Optimization 57, 17–40 (2008)
Mordukhovich, B.S., Wang, D., Wang, L.: Optimal control of delay-differential inclusions with multivalued initial conditions in infinite-dimensional spaces. Control Cybern. (to appear)
Outrata, J.V.: Mathematical programs with equilibrium constraints: theory and numerical methods. In: Haslinger, J., Stavroulakis, G.E. (eds.) Nonsmooth Mechanics of Solids. CISM Courses and Lecture Notes, vol. 485, pp. 221–274. Springer, New York (2006)
Robinson, S.M.: Generalized equations and their solutions. I: Basic theory. Math. Prog. Study 10, 128–141 (1979)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 317. Springer, Berlin (1998)
Schirotzek, W.: Nonsmooth Analysis. Series Universitext. Springer, Berlin (2007)
Vinter, R.B.: Optimal Control. Series Systems & Control: Foundations & Applications. Birkhäuser, Boston (2000)
Yao, J.C., Chadi, O.: Pseudomonotone complementarity problems and variational inequalities. In: Handbook of Generalized Convexity and Generalized Monotonicity, pp. 501–558. Springer, New York (2005)
Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)
Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 IFIP International Federation for Information Processing
About this paper
Cite this paper
Mordukhovich, B.S. (2009). New Applications of Variational Analysis to Optimization and Control. In: Korytowski, A., Malanowski, K., Mitkowski, W., Szymkat, M. (eds) System Modeling and Optimization. CSMO 2007. IFIP Advances in Information and Communication Technology, vol 312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04802-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-04802-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04801-2
Online ISBN: 978-3-642-04802-9
eBook Packages: Computer ScienceComputer Science (R0)