Research Interest

Variational techniques and their applications in mathematical finance, optimization, nonsmooth analysis and control problems.

Book

Techniques of Variational Analysis, J. M. Borwein and Q. J. Zhu, Springer, New York, 2005

Research papers by areas

Financial Mathematics

• Vector majorization and a robust option replacement trading strategy, Set-Valued Analysis (2008).
• Mathematical analaysis of investment systems, Journal of Mathematical Analysis and Applications, 326 (2007) 708-720.

Nonsmooth analysis on smooth manifolds

• Nonsmooth analysis on smooth manifold, (With Yu. Ledyaev) Transactions of AMS, 359 (2007) 3687--3732.
• Helly's Intersection Theorem on Manifolds of Nonpositive Curvature, (with Yu. Ledyaev and J. Treiman)  Journal of Convex Analysis, 13 (2006) 785-798.
• Techniques for nonsmooth analysis on smooth manifolds: Part I:The local theory, (With Yu. Ledyaev) Optimal control, stabilization and nonsmooth analysis, 283-297, Lecture Notes in Control and Inform. Sci. 301, Springer, Berlin, 2004.
• Techniques for nonsmooth analysis on smooth manifolds: Part II:Using deformations and flows, (With Yu. Ledyaev) Optimal control, stabilization and nonsmooth analysis, 299-311, Lecture Notes in Control and Inform. Sci. 301, Springer, Berlin, 2004.

Nonsmooth and variational analysis

• Variational methods in convex analysis, (with J. Borwein) J. Global Optimization 35 (2006) 197-213.
• Multidirectional mean value inequality and weak monotonicity, (With Yu. Ledyaev) J. London Math. Soc. 71 (2005), 187-202.
• Nonconvex separation theorem for multifunctions, Set-Valued Analysis, 12 (2004), 275-290.
• A variational proof of Birkhoff's theorem on doubly stochastic matrices, Math. Ineq. Appl. 7 (2004), 309-313.
• An extended extremal principle with applications to multi-objective optimization, (With B. Mordukhovich and J. Treiman) SIAM J. Optimization, 14 (2003), 359-379.
• Lower semicontinuous Lyapunov functions and stability, J. Nonlinear and Convex Analysis, 4 (2003), 325-332.
• Necessary conditions for constrained optimization problems in smooth Banach spaces and applications, SIAM Optimization, 12 (2002), 1032-1047.
• Convex spectral functions of compact operators, Part II: Lower semicontinuity and rearragement invariance, (With Borwein, J. M. and Lewis, A. S.) Proceedings of the Optimization Miniconference VI, Kluwer Academic Publ. 2000.
• Viscosity coderivatives and their limiting behavior in smooth spaces, (With Mordukhovich, B. S. and Shao, Y.) Positivity 4 (2000) 1-39.
• Convex spectral functions of compact operators, (With Borwein, J. M., Read, J. and Lewis, A. S.) Journal of Nonlinear and Convex Analysis, 1 (2000) 17-35.
• Implicit multifunctions theorems, (With Ledyaev, Yu. S.), Set-valued Analysis, 7 (1999) 209-238.
• A survey of subdifferential calculus with applications, (With Borwein, J.M.),  Nonlinear Analysis, TMA. 38 (1999) 687-773.
• Partially smooth variational principles and applications. (With Borwein, J. M. and Treiman, J. S.) Nonlinear Analysis: TMA, 35. (1999), 1031-1059.
• Multi-Functional and Functional Analytic Techniques in Nonsmooth Analysis, (With Borwein, J.M.),  Proceedings of the 37th consecutive edition of the SMS on Nonlinear Analysis and Control,  Montreal, July-August,1998.
• Limiting convex examples for nonconvex subdifferential calculus,  (With Borwein, J.M.), Convex Analysis. 5. (1998),  221-235.
• The equivalence of several basic theorems for subdifferentials. Set-valued Analysis, 6. (1998), 171-185.
• A limiting example for the local ``fuzzy'' sum rule in nonsmooth analysis, (joint with J. Vanderwerff)  Proceedings of the American Mathematics Society.  126, (1998), 2691-2697
• Clarke-Ledyaev mean value inequality in smooth Banach spaces,  Nonlinear Analysis, Theory, Methods and Applications, 32. (1998), 315-324.
• Perturbed differential inclusion problem with nonadditive $L^1$ perturbations and applications, (joint with J. J. Ye) Journal of Optimization Theory and Applications, 92 no. 1 (1997), 189-208.
• Variational analysis in smooth Banach spaces and control problems with $L^1$ perturbations, ( joint with J. M. Borwein) Nonlinear Analysis, Theory, Methods and Applications, 28. (1997), 889-915.

Optimization and optimal control

• Multiobjective optimization problems with variational inequality constraints, (With J. J. Ye) Math. Program. 96 (2003), 139-160.
• Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints, SIAM J. Contr. & Optim. 39 (2000), no. 1, 97-112.
• Necessary conditions for constrained optimization problems with semicontinuous and continuous data, (joint with J. M. Borwein and J. S. Treiman) Trans. Amer. Math. Soc., 350, (1998), 2409-2429.
• Hamilton-Jacobi theory for a generalized optimal stopping time problem, (joint with J. J. Ye) to appear in Nonlinear Analysis: Theory, Methods and Applications, 34, (1998),1029-1053.
• On the characterization of properly relaxed delayed controls (joint with J. F. Rosenblueth, J. Warga), J. Math. Anal. Appl. 209 (1997) 274-290.
• Exact penalization and necessary optimality conditions for generalized bilevel programming problem, (joint with J. J. Ye and D. L. Zhu) SIAM J. Optimization, 7, (1997), 481-507.
• Necessary optimality conditions for nonconvex differential inclusion with endpoint constraints, Journal of Differential Equations, 124, (1996) 186-204.
• The equivalence of extremals in different representations of unbounded control problems, (joint with J. Warga) SIAM Journal on Control and Optimization, 32 (1994), 1151-1169.

Control of flexible manufacturing systems

• Production set of failure prone flexible manufacturing systems, (With Boukas, E. K.) Stochastic Anal. Appl. 18 (2000), no. 1, 39--61.
• Studying flexible manufacturing systems via deterministic control problems, "Recent Advances in Control and Optimization of Manufacturing Systems," Q. Zhang and G. Yin eds. Lecture Notes in Control and Information Sciences Vol. 214, Springer-Verlag, New York, (1996) 181-196.
• Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, (joint with J. M. Borwein) SIAM Journal on Control and Optimization, 34. (1996), 1568-1591.
• Optimal production and maintenance planning for flexible manufacturing systems,  (joint with E. K. Boukas and Q. Zhang) Control-Theorey and Advanced Technology, 10, (1995) 1421-1430.
• A piecewise deterministic Markov process model for flexible manufacturing systems with preventive maintenance, (joint with E. K. Boukas and Q. Zhang) Journal Optimization Thoery and Applications, 81 (1994), 259-275.