• Key Note Lectures and Plenary Talks at Conferences
    1. Duality Theory in Nonsmooth/Nonconvex Dynamical Systems: A Survey and New Results, Key Note Lecture presented at the  International Conference on Nonsmooth/Nonconvex Mechanics,  Aristotle University of Thessaloniki (A.U.Th.),July 5-6, 2002.
    2. Complete Solutions and Canonical Dual Transformation in Global Optimization, Plenary Talk presented at  The 2nd International Conference on  Optimization and Control with Application (OCA2002), August 18-22, 2002.   Yellow Mountains International Hotel, Anhui, China, ABSTRACT
    3. Canonica l duality and solutions to a class of nonconvex nonsmooth problems.  J.J. Moreau's 80th birthday  colloquium , November, 2003,  Montpellier , France.

    Invited Lectures and Colloquium Talks at Institutions
    1. Promoting Research in Engineering, Science and Computing in a University Environment, Division of Engineering and Science, Curtin University, Australia, Feb. 13, 2003
    2. Framework in Natural Science and Philosophy, Department of Ocean Engineering, Shanghai Jiao-Tong University, Shanghai, China, August 13, 2002. 
    3. Duality and Triality in Mathematics and Scientific Computations, Department of Mathematics, University of Auckland, New Zealand, January 22, 2002.
    4. Duality and Triality: Unifying Mathematics and Natural Sciences, Colloquium talk at Department of Math., University of Glasgow, January 18, 2001.
    5. Closed solution for large deformation mechanics and applications
      6. Two-hours lecture at the Department of Engineering Mechanics and Technology,
      Tongji University , January 11, 2000.
    6. Canonical dual transformation method and applications in nonlinear boundary value problems and nonconvex Hamiltonian systems
      7. Two-hours lecture at the Department of Mathematics,
      University of Science and Technology of China, January 9, 2000.
    7. Complementary and dual variational methods in large deformation nonsmooth and nonconvex mechanics
      8. Two-hours lecture at the Department of Modern Mechanics,
      University of Science and Technology of China, January 8, 2000.
    8. Generalized complementarity methodology in science and technology
      9. One-hour honoring lecture presented for the Mayer of Huainan, Presidents and department heads at Huainan Institute of Technology, January 5, 2000.
    9. Philosophical Principles of Natural Systems
      10. Two-hours Institute Address at Huainan Institute of Technology, January 4, 2000.
    10. General Analytic Solutions and Duality Theory for Fully Nonlinear Variational/Boundary Value Problems Governed by Nonsmooth Constitutive Laws.
      11. Colloquium talk at the Department of Mathematics,
      George Mason University, March 26, 1999.
    11. Hellinger-Reissner's Open Problem and Analytic Solutions for Finite Deformation Theory
      12. Colloquium Talk at the Dept. of Engineering Science and Mechanics Virginia Tech., November, 1998.
    12. Department of Mathematics, The George Washington University, April 25, 1997
    13. Department of Engineering Science and Mechanics, Virginia Tech. October 2, 1996.
    14. Department of Mathematics, Virginia Tech., Oct. 11, 1996.
    15. Department of Mathematics, Univ. Michigan, April, 1992.
    16. Department of Mathematics, Univ. California, Irvin, Feb. 1992.
    17. Institute of Applied Mathematics, Indiana University, Sep. 1991.
    18. Department of Mathematics, West Virginia University, March, 1991.
    19. Dept. of Mathematics, West Chester University, March, 1991.
    20. Department of Mathematics, Beijing University, December, 1988.

    Invited/Contributed Talks at Professional Conferences

    1. Generalized Min-Max Theory and Sequential Canonical Dual Transformation Methods in Nonconvex Analysis
    2. Triality and Primal-Dual Algorithm for Ginzburg-Landau Equation
    3. Triality and new phenomena in non-smooth, non-convex and non-conservative Hamilton systems, Symposium on Hamiltonian systems,  Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, May 24-27,2002
    4. Duality reformulation and extended min-max theory for nonsmooth global optimization with applications. The 17th International Symposium on Mathematical Programming, Georgia Tech., Atlanta, August 7-14, 2000.
    5. Canonical dual transformation and generalized Caratheodory condition in global optimization, INTERNATIONAL CONFERENCE ON ADVANCES IN CONVEX ANALYSIS AND GLOBAL OPTIMIZATION, Honoring the memory of C. Caratheodory (1873-1950), June 5-9, 2000 - Pythagorion, Samos, Greece.
    6. Dual feedback control against chaotic vibrations, Year 2000 International Conference on Dynamical Systems and Differential Equations, Kennesaw State University, Atlanta, Georgia, USA, May 18 to 21, 2000.
    7. Triality theory and general solutions for nonconvex/nonsmooth variational problems with applications in nonconvex dynamics and differential geometry, Sixteenth Southeastern Analysis Meeting. University of Virginia, March 17-18, 2000
    8. General Analytic Solutions and Duality Theory for Nonconvex Variational/Boundary Value Problems with Applications in Dynamical Post-Bifurcation of Nonlinear Structures
      9. NSF-CBMS Regional Research Conference on  Mathematical Control Theory of Coupled Systems of Partial Differential Equations.  Lincoln, Nebraska, August 5-9, 1999.
    9. Analytic solutions for 3-D nonconvex-nonsmooth finite deformation elastoplasticity with applications,
      10.  Nonsmooth-Nonconvex Mechanics: Internaltional Symposium in Honor of P.D. Panagiotopoulos.
      Virginia Tech, June 27-29, 1999 (Symposium organizer).
    10. Minimax and triality theory in nonconvex, nonsmooth optimization with applications
      11. Sixth SIAM Conference on Optimization, Sheraton Atlanta Hotel, Atlanta, Georgia, May 10-12, 1999
    11. Analytic solutions and duality theory in fully nonlinear partial differential systems with applications
      12. 1999 SIAM Annual Meeting (PDE session Chair) Sheraton Atlanta Hotel, Atlanta, Georgia, May 12-15, 1999
    12. Post-buckling analysis of nonlinear dynamical thick beam model and dual variational principles,
      13. The 7-th Int. Conf. on Nonlinear Virbrations and Control,
      Virginia Tech, June, 1998.
    13. On the Hellinger-Reissner's Open Problem and the General Analytic Solution in Finite Deformation Theory.
      14. Thirteenth US Congress of Appl. Mechanics,
      Gainesville, Florida, June 21-26, 1988.
    14. International Conference on Dynamics and Control of Partial Differential Equations,
      15. CIMAT, Gunajuato, Mexico, November 29-30, 1997.
    15. SIAM 45th Anniversary meeting,
      16. Stanford University, July 14-19, 1997, Organizer for minisymposium on ``Complementary-Dual Variational Principles and Numerical Methods''.
    16. Modern Mechanics and Advancement of Science and Technology in Society,
      17. A Conference Celebrating the 40th Anniversary of the Chinese Society of Theoretical and Applied Mechanics, Beijing, China, August, 1997,
    17. The 16th International Symposium on Mathematical Programming,
      18. Lausanne, Switzerland, August 24-29, 1997.
    18. The International ISAAC'97 Congress,
      19. June 3-7, 1997, Delaware, USA.
    19. Fifth Pan American Congress on Applied Mechanics,
      20. January, 2-4, 1997, San Juan, Puerto Rico.
    20. 14th U.S. Army Symposium on Solid Mechanics,
      21. 16-18 October, 1996, Myrtle Beach, South Carolina.
    21. U.S.-China Workshop on Mechanics and Manufacturing Science,
      22. Beijing, August, 1996, sponsored by NSF and NNSF of China (50 min invited talk).
    22. Symposium on Nonlinear Elasticity in Honor of J.L. Eriksen,
      23. ASME Mechanics and Materials Conference, The Johns Hopkins University, Baltimore, MD, June 12-14, 1996.
    23. 1996 AMS-IMS-SIAM Summer Research Conference on Optimization Methods in Partial Differential Equations,
      24. June 16-20, 1996, Mount Holyoke, sponsored by AMS (30 min. invited talk).
    24. The 18th Southeastern Conference on Theoretical and Applied Mechanics,
      25. April 14-16, 1996, Tuscaloosa, AL.
    25. Int. Conf. Complementarity Problems,
      26. Baltimore, MD, Nov. 1-4, 1995. ( 30 min. invited talk).
    26. SIAM Annual Meeting,
      27. Three contributed talks, Charlotte, NC, Oct. 23-26, 1995
    27. Int. Sym. Methods and Appl. of Analysis,
      28. Hong Kong, Dec. 16-19, 1994
    28. Int. Conf. Computational Methods in Structural and Geotechnical Engineering,
      29. Hong Kong, Dec. 12-14, 1994.
    29. The 15th International Symposium in Math. Programming,
      30. Ann Arbor, MI, 1994 (30 min. invited talk and Chair for Variational Inequality and Complementarity Session).
    30. SIAM Annual Meeting,
      31. San Diago, CA, July 25-29, 1994.
    31. The 17th Southeastern Conference on Theoretical and Applied Mechanics,
      32. Hot Springs National Park, Arkansas, April 10-12, 1994.
    32. The 3rd SIAM Conference on Geometric Design,
      33. Nov. 1993, Tempe, Arizona. Chair for Surface Modeling
      session.
    33. The 2nd Int. Conf. Indust. Appl. Math.,
      34. Washington, D.C. USA, July, 1991.
    34. The 4th Int. Conf. of System Research, Informatics and Cybernetics,
      35. Baden-Baden, W. Germany, Aug. 1988 (30 min. invited talk).
    35. The 17th Int. Union of Theoretical and Applied Mechanics,
      36. Grenoble, France, Aug. 1988.
    36. The 20th Midwestern Mechanics Conference,
      37. Purdue University, 1987.
    37. Int. Conf. Computational Mechanics,
      38. Tokyo, Japan, August, 1986.
    38. The 2nd National Conf. on Engineering Computational Mechanics,
      39. July, 1986, Shanghai, China.
    39. The 1st National Conference on Modern Mathematics and Mechanics,
      40. Beijing University, Aug. 1986 (45 minute invited lecture).
    40. National Congress of Natural Philosophy,
      41. July, 1986, Yellow Mountain, Anhui (1 hour invited lecture, excellent paper award).
    41. International Conference on Nonlinear Mechanics,
      42. Shanghai, May, 1985.
    42. The 2nd National Conference on Plastic Forming,
      43. October 10-15, 1981, Jinan, Shandong.
    43. Duality Method for Parametric Controlling of Chaotic Systems: An Potentially Powerful New Approach and Applications. SIAM Annual Meeting, Westin Rio Mar Resort, Puerto Rico, July 10-14, 2000.
    44. Prima-Dual Method and Triality Theory for Nonsmooth Global Optimization. SIAM Annual Meeting, Westin Rio Mar Resort, Puerto Rico, July 10-14, 2000.
    45. General Analytic Solutions and Duality Methods for Fully Nonlinear, Nonsmooth Variational/Boundary Value Problems, Chair for the session: Differential Equations and Applications II. SIAM Annual Meeting, Westin Rio Mar Resort, Puerto Rico, July 10-14, 2000.
    46. Duality Theory in Parametric Controlling of Chaotic Beam Vibrations. The 8th Conference on Nonlinear Vibrations, Stability and Dynamics of Structures. Virginia Tech. July 23-27, 2000.
    47. New phenomena in chaotic systems and dual feedback control, Southeastern Atlantic Regional Conference on Differential Equations, Virginia Tech. Oct. 21-22, 2000.
      Duality and Triality in Nonconvex-Nonsmooth Systems: A Powerful Approach for Nonlinear Phenomena with Applications
      Perspectives in Applied Mathematics: A Scientific Conference in Honor of Gil Strang on the Occasion of his 65th Birthday.
      Cambridge, MA, USA December 3-4, 1999.

     

        Abstract: This talk will present a brief survey and some new developments on the duality and triality theories in general
        nonsmooth/nonconvex, nonconservative Hamilton systems. By using the canonical dual transformation method developed by the
        speaker recently, many nonsmooth problems can be transformed into smooth dual problems, and certain nonconvex partial
        differential systems can be converted into the so-called Differential-Algebraic equations (DAEs). The triality theory reveals
        some interesting extremality properties and intrinsic symmetry in nonconvex dynamical systems. Based on this theory, an
        alternative algorithms is proposed for solving nonsmooth/nonconvex dynamical problems.

        Application is illustrated by a semi-linear, nonconvex parametric variational problem. Some interesting phenomena, i.e.
        meta-chaos and post-chaos are discovered in nonconvex/nonconservative Hamilton systems with double-well potential. Dual
        feed-back control against chaos is discussed.

       
    GAO, David
    Complete Solutions and Triality Theory for Global Optimization with Applications

    Abstract : Duality is a fundamental concept that underlies almost all natural phenomena.  Triality, a newly discovered theory, reveals the intrinsic pattern of dualities in complex systems.  Based on the powerful canonical dual transformation method developed recently [1], the speaker will show that many very difficult nonsmooth/ nonconvex optimization problems in n-dimensional space can be converted into certain smooth, picewise convex dual problems in lower dimensional space.  Therefore, complete solutions, including all extremum points (both minimizers and maximizers) and all saddle points, are obtained for a large class of nonconvex optimization problems.  The extremality conditions for all these solutions are controlled by the triality theory.  Applications are illustrated by examples from nonsmooth d.c. programming, chaotic dynamical systems and phase transitions in physics and material science.



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