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

Recent research papers

• Investment system specific European option pricing intervals, Submitted..
• Mathematical analaysis of investment systems, Journal of Mathematical Analysis and Applications, 326 (2007) 708-720.
• Helly's Intersection Theorem on Manifolds of Nonpositive Curvature, (with Yu. Ledyaev and J. Treiman) To appear in Journal of Convex Analysis.
• Variational methods in convex analysis, (with J. Borwein) to appear in J. Global Optimization.
• Multidirectional mean value inequality and weak monotonicity, (With Yu. Ledyaev) J. London Math. Soc. 71 (2005), 187-202.
• A variational proof of Birkhoff's theorem on doubly stochastic matrices, Math. Ineq. Appl. 7 (2004), 309-313.
• 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.
• Nonconvex separation theorem for multifunctions, Set-Valued Analysis, 12 (2004), 275-290.
• Nonsmooth analysis on smooth manifold, (With Yu. Ledyaev) to appear in Transactions of AMS.
• Multiobjective optimization problems with variational inequality constraints, (With J. J. Ye) Math. Program. 96 (2003), 139-160.
• 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.
• Convergence in evolutionary programs with self-adaptation in evolution strategies, (With Greenwood, G.) Evolutionary Computation, 9 (2001), no. 2 147-157.
• 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.
• Production set of failure prone flexible manufacturing systems, (With Boukas, E. K.) Stochastic Anal. Appl. 18 (2000), no. 1, 39--61.
• Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints, SIAM J. Contr. & Optim. 39 (2000), no. 1, 97-112.
• 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.) CECM Research Report 99-105,  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.), CECM Research Report 98-105,  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.
• 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.
• 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.
• 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.
• Necessary optimality conditions for nonconvex differential inclusion with endpoint constraints, Journal of Differential Equations, 124, (1996) 186-204.
• 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.
• 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.
• Integral inclusions with delays in Banach spaces, Journal of Mathematical Analysis and Applications, 188 (1994), 1-24.
• A proper relaxation of shifted and delayed controls, Journal of Mathematical Analysis and Applications, 168 (1992), 546-561.
• An algorithm for the reachability set of a linear control systems, (joint with N. Zhang and Y. He) Journal of Optimization Theory and Applications, 72 (1992), 333-353.
• On the solution set of differential inclusions in Banach spaces, Journal of Differential Equations, 93 (1991), 213-237.