dontchev [at] umich [dot] edu
This webpage provides information about my research. I am currently employed by Mathematical Reviews, a division of the American Mathematical Society. I am also affiliated with the University of Michigan and the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. For more details, see my vita. More about me and my work at Mathematical Reviews you can find here.
My research interests are centered around estimating the effect of perturbations and approximations on solutions of variational problems, in particular problems in the general areas of optimization, optimal control, differential equations, calculus of variations and approximation theory. The perturbations may be represented by changes of various parameters in the problems, e.g., singular perturbations in differential equations/control problems, while approximations are understood broadly as any changes in the original problem that may help in solving it; for example, discrete approximations to infinite-dimensional problems or sequences of approximate solutions that are generated by iterative methods. I am mostly interested in variational problems with constraints where tools from classical analysis can not be applied. For more specific information about my current research interests please see my recent book coauthored with R. T. Rockafellar, "Implicit Functions and Solution Mappings. A View from Variational Analysis," Second edition, Springer 2014, see Errata and Addenda.
- Strong metric subregularity of mappings in variational analysis and optimization (with R. Cibulka and A. Kruger), J. Math. Anal. Appl., accepted.
- Kantorovich-type theorems for generalized equations (with R. Cibulka, J. Preininger, T. Roubal and V. Veliov) J. Convex Analysis, accepted.
- Lyusternik-Graves theorems for the sum of a Lipschitz function and a set-valued mapping (with R. Cibulka and V. Veliov), SIAM J. Control and Optim 54 (2016) 3273–3296.
- Omega limit sets of differential inclusions (with M. Krastanov and V. Veliov), Analysis and geometry in control theory and its applications,
159–169, Springer INdAM Ser. 11, 2015.
- Regularity properties of mappings in optimal control (with V. Veliov), Proceedings of SDCP (2014) 35-41.
- Inexact Newton methods and Dennis-More theorems for nonsmooth generalized equations (with R. Cibulka and M. Geoffroy), SIAM J. Control and Optim. 53 (2015) 1003-1019.
- A nonsmooth Robinson's inverse function theorem in Banach spaces (with R. Cibulka), Math. Program. A, electronically published April 2015.
- A proof of the Lyusternik-Graves theorem, Optimization 64 (2015) 41-48.
- The weak Ekeland variational principle and fixed points (with G. Beer) Nonlinear Anal. 102 (2014)
- On one-sided Lipschitz stability of set-valued contractions (with S. Adly and M. Thera), Numer. Funct. Anal.Optim. 35 (2014)
- Local convergence of quasi-Newton methods under metric regularity (with F. Aragon Artacho, A. Belyakov and M. Lopez), Comp. Optim. Appl. 58 (2014) 225–247.
- An Euler-Newton continuation method for tracking solution trajectories of parametric variational inequalities (with M. I. Krastanov, R. T. Rockafellar and V. M. Veliov) SIAM J. Control Optim. 53 (2013) 1823-1840.
- Generalizations of the Dennis-More theorem, SIAM J. Optim. 22 (2012) 821-830.
- Parametric stability of solutions in models of economic equilibrium (with R. T. Rockafellar) J. Convex Analysis 19 (2012) 975–997.
- On derivative criteria for metric regularity (with H. Frankowska), JonFest Proceedings, Springer, accepted Febr. 9, 2012.
- Convergence of inexact Newton methods for generalized equations (with R. T. Rockafellar), Math. Programming, 139 (2013) 115-137.
- Lyusternik-Graves Theorem and Fixed Points II (with H. Frankowska), J. Convex Analysis 19 (2012) 955-973.
- Metric regularity of Newton's iteration (with F. Aragon, M. Geoffroy, M.Gaydu and V. Veliov), SIAM J. Control Optim. 49 (2011) 339-362.
- Lyusternik-Graves Theorem and Fixed Points (with H. Frankowska), Proc. AMS 139 (2011) 521–534.
- Metric Regularity under Approximations (with V. M. Veliov), Control and Cybernetics 38 (2009) 1283--1303.
- Newton's method for generalized equations: a sequential implicit function theorem (with R. T. Rockafellar), Math. Progr. 123 (2010) 139-159.
- Robinson’s implicit function theorem and its extensions (with R. T. Rockafellar) Math. Progr. 117 (2009) 120-14.
- A local selection theorem for metrically regular mappings, J. Convex Anal. 11 (2004) 81-94.
- Aubin criterion for metric regularity (with M. Quincampoix, N. Zlateva) J. Convex Anal. 13 (2006) 281-297.
- Regularity and conditioning of solution mappings in variational analysis (with R. T. Rockafellar) Set-Valued Anal. 12 (2004) 79-109.
- Perturbations and metric regularity (with A. Lewis) Set-Valued Anal. 13 (2005) 417-438.
- On the Bartle-Graves theorem (with J. M. Borwein) Proc. Amer. Math. Soc. 131 (2003) 2553-2560.
- Singular Perturbations in Infinite-Dimensional Control Systems (with Ts. Donchev) SIAM J. Control Optim. 42 (2003) 1795-1812.
- The radius of metric regularity, (with A. S. Lewis and R. T. Rockafellar) Trans. Amer. Math. Soc. 355 (2003) 493-517.
- A Newton method for shape-preserving spline interpolation (with Qi, H., Qi, L., Yin, H.) SIAM J. Optim. 13 (2002) 588-602.
- The Euler approximation in state constrained optimal control (with W. W. Hager), Math. of Computations 70 (2001) 173-203.
- Ample parameterization of variational inclusions (with R. T. Rockafellar) SIAM J. Optim. 12 (2001) 170-187.
- Second-order Runge-Kutta approximations in control constrained optimal control (with W. W. Hager and V. M. Veliov) SIAM J. Numer. Anal. 38 (2000) 202-226.
- On the effect of neglecting sensor dynamics in parameter identification problems (with M. Polis and V. M. Veliov) SIAM J. Control Optim. 38 (2000), 1309-1321.
- Error bounds for Euler approximation of a state and control constrained optimal control problem (with W. W. Hager and K. Malanowski) Numer. Funct. Anal. Optim. 21 (2000) 653-682.
- Lipschitzian stability for state constrained nonlinear optimal control (with W. W. Hager) SIAM J. Control Optim. 36 (1998) 698-718.
- Characterizations of strong regularity for variational inequalities over polyhedral convex sets (with R. T. Rockafellar) SIAM J. Optim. 6 (1996) 1087-1105.
- An a priori estimate for discrete approximations in nonlinear optimal control, SIAM J. Control Optim. 34 (1996) 1315-1328.
- The Graves theorem revisited, J. Convex Anal. 3 (1996) 45-53.
- An inverse mapping theorem for set-valued maps (with W. W. Hager) Proc. Amer. Math. Soc. 121 (1994) 481-489.
- Optimality, stability and convergence in nonlinear control (with W. W. Hager, A. B. Poore, and B. Yang) Appl. Math. and Optim. 31 (1995) 297-326.
- Well-Posed Optimization Problems (with T. Zolezzi) Lecture Notes in Mathematics, vol. 1543, Springer 1993.
- Best interpolation in a strip, J. Approx. Theory 73 (1993). 334--342.
- Time scale decomposition of the reachable set of linear systems, Mathematics of Control, Signals, and Systems 5 (1992) 327-34.
- Difference methods for differential inclusions - a survey (with F. Lempio) SIAM Review 34 (1992) 263-294.
Error estimates for discretized differential inclusions (with E. Farhi), Computing 41 (1989) 349-358.
- Continuity of the family of trajectories of linear control systems with respect to singular perturbations (with V. Veliov) Soviet Math. Dokl. 35 (1987) 283-286.
- On the regularity of Kuhn-Tucker curve (with H. Th. Jongen) SIAM J. Control and Optim. (1986) 169-176.
- Singular perturbations in Mayer's problem for linear systems (with V. Veliov) SIAM J. Control Optim. 21 (1983) 566-581.
- Error estimates for a discrete approximation to constrained control problems, SIAM J. Num. Anal. 18 (1981) 500-514.
- Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems, Lecture Notes in Control and Inf. Sc. 52, Springer 1983 (translation in Russian by Mir, Moscow 1987).