David Yang Gao

Welcome to the Web Page of

DAVID YANG GAO

Department of Mathematics

Virginia Tech
Blacksburg, VA 24061
Office: McBryde Hall 524
Tel: (540) 231-2768

Duality: The Natural Beauty

Triality : The Natural Mystery

Teaching	Research	Books	Publications
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Announcements:

Foundations of Computer-Aided Process Operations (FOCAPO) 2008, June 29-July 2, 2009, Cambridge, MA

World Congress on Global Optimization, June 1-5, 2009, Changsha-Zhangjiajie, Hunan, China.

The 3^rd International Conference on Complementarity, Duality, and Global Optimization with Applications in Engineering Science (CDGO-09), May 1-3, 2009, MIT, Cambridge, MA

9^th Conference on Dynamical Systems, Theory and Applications, Lodz, Poland, December 17-20

The 2^nd International Conference on Complementarity, Duality, and Global Optimization, Gainesville, Florida, February 28 - March 2, 2007

The Fifth International Conference on Nonlinear Analysis and Convex Analysis (NACA2007)

The 7th International Conference on Optimization: Techniques and Applications (ICOTA7), Kobe, Japan, Dec. 12-15, 2007.

International Workshop on Duality and Conic Optimization, Tsinghua, Beijing, December 16-18, 2006.

The 11^th Asian Technology Conference in Mathematics, 12 - 16 December 2006 The Hong Kong Polytechnic University Hong Kong SAR, China

International Conference on Complementarity, Duality, and Global Optimization,
August 15-17, 2005, Virginia Tech, Organizers: D. Gao and H. Shirali

Book Reviews on Gao’s book “Duality Principles in Nonconvex Systems”

Canonical Duality Theory is a newly developed, potentially powerful methodology which can be used to solve a large class of nonconvex/nonsmooth problems in nonlinear analysis, global optimization, and complex systems. This theory is composed mainly of a canonical dual transformation and an associated triality theory, whose components comprise a saddle min-max duality and two pairs of double-min, double-max dualities. The canonical dual transformation can be used to formulate perfect dual problems (with zero duality gap), while the triality theory can be used to identify both global and local extrema. In analysis and optimization, traditional Lagrangian duality can be used only for solving convex minimization problems. In nonconvex systems, the well-developed Fenchel-Moreau-Rockafelar duality theory will produce a so-called duality gap if the primal problem is nonconvex. Traditional sufficient conditions in convex analysis and programming can not be used to identify global extrema. Therefore, direct methods for solving nonconvex problems may produce the so-called Chaotic solutions in dynamical systems. In global optimization, many problems are NP-hard. However, by using the canonical dual transformation, a large class of nonconvex/nonsmooth problems can be reformulated as canonical dual problems, i.e., either concave maximization or convex minimization, many nonlinear PDEs can be converted into certain algebraic equations, and nonsmooth/discrete problems can be transformed into smooth/continuous dual problems. Both global and local extrema can be identified by the triality theory. Therefore, complete solutions to a class of nonconvex boundary value problems and global optimization problems have been obtained recently (see papers #1, 2, 3, 4, 5 given below).

In finite deformation theory, the existence of purely stress based complementary variational principle has been a well-known debate existing for more than 50 years since E. Reissner (1953). In a series of publications (see Gao’s book and the papers #6, #7 listed below), this open problem has been solved and a pure complementary energy principle was proposed as a perfect duality theory to the minimal potential variational principle. By using this principle, many nonlinear partial differential equations can be converted into certain algebraic (tensor) equations. Large scale nonconvex/nonsmooth optimization problems can be reformulated as canonical dual (i.e., either convex minimization or concave maximization) problems. Therefore, analytic solutions can be obtained for a class of nonconvex/nonsmooth variational problems. Both global and local extremality conditions can be identified by the triality theory. For detailed discussion on this pure complementary energy principle in nonlinear elasticity, please check Here.

Canonical Duality and Triality in Global Optimization

Recent Results and publications:

Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation, with Ray Ogden

This paper presents a complete set of analytical solutions to general variational/boundary value problems with either mixed or Dirichlet boundary conditions. It shows that the global minimizer may not be a smooth function and can’t be obtained by traditional methods. Criteria for the existence, uniqueness, smoothness and multiplicity of solutions are presented and discussed. The iterative finite-difference method (FDM) is used to illustrate the difficulty of capturing non-smooth solutions with traditional FDMs.

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem, with Ray Ogden,

This paper solved nonlinear variational/boundary value problems in nonlinear elasticity with both convex and nonconvex strain energy densities. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.

CANONICAL DUAL APPROACH TO SOLVING 0-1 QUADRATIC PROGRAMMING PROBLEMS This paper presents an application of the canonical duality theory for 0-1 programming problems. It shows that by the canonical dual transformation, discrete integer minimization problems can be converted into canonical dual problems in continuous space, which can be solved easily under certain conditions. Both global and local minimizers can be identified by Triality theory. Multi-integer programming problems can be solved too, results will be posed soon.

Multi-scale modelling and canonical dual finite element method in phase transitions of solids Int. J. Solids and Structure. This paper presents a multi-scale model in phase transitions of solid materials with both macro and micro effects. This model is governed by a semi-linear nonconvex partial differential equation which can be converted into a coupled quadratic mixed variational problem by the canonical dual transformation method. The extremality conditions of this variational problem are controlled by a triality theory, which reveals the multi-scale effects in phase transitions. Therefore, a potentially useful canonical dual finite element method is proposed for the first time to solve the nonconvex variational problems in multi-scale phase transitions of solids

Solutions and Optimality Criteria to Box Constrained Nonconvex Minimization Problems, J. Industrial and Management Optimization, 3(2), 293-304, 2007. This paper solved a class of box constrained nonconvex minimization problems, including quadratic minimization, integer programming, and Boolean least squares problems. This paper shows that some “NP-hard problems” can be solved by polynomial algorithms.

Complete solutions and extremality criteria to polynomial optimization problems, J. Global Optimization, 35 : 131-143, 2006

Canonical duality theory and solutions to constrained nonconvex quadratic programming. J. Global Optimization. This paper presents a set of complete solutions to quadratic minimization over a sphere, as well as a canonical dual form for quadratic minimization with linear inequality constraints.

Perfect duality theory and complete solutions to a class of global optimization problems, Optimization. This paper presents a potentially useful methodology for solving a class of nonconvex variational/optimization problems. A set of complete solutions is obtained for Landau-Ginzburg equation, nonlinear Schrödinger, equation and Cahn-Hilliard theory in finite dimensional space.

Sufficient conditions and perfect duality in nonconvexs minimization with inequality, J. Ind. Management Optimization. This paper solved quadratic minimization problem with a quadratic inequality constraint.

Complete solutions to a class of polynomial minimization. J. Global Optimization

Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications, Nonlinear Analysis. This paper present a complete set of solutions to a class of nonconvex/nonsmooth variational/boundary value problems. The global minimizer and local extrema can be identified by the triality theory.

General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics, Meccanica. This paper solved an open problem in finite deformation theory and proposed a pure complementary energy principle (i.e. the Gao Principle) which can be used to solve a large class of nonconvex variational/boundary value problems.

Pure complementary energy principle and triality theory in finite elasticity. Mechanics Research Communication.

Nonconvex Semi-Linear Problems and Canonical Duality Solutions, Advances in Mechanics and Mathematics, Vol II, 2003, Springer, 261-311.

Breaking News: Nature exists in duality pairs!!!

Researchers Get First Look into Antimatter Atom (NSF report)
First Production of Cold Antihydrogen (see picture)

New Journals:

Journal of Industrial and Management Optimization

Optimization Letters

Book Series:

Advances in Mechanics and Mathematics (AMMA) by Springer

Modern Mechanics& Mathematics (MMM), Taylor & Francis CRC Press

New books published:

Proceedings of IUTAM Symposium on Complementarity, Duality, and Symmetry in Mechanics, Kluwer, 404pp. 2004.

AMMA 2003 by David Y. Gao and Ray W. Ogden, Kluwer, 2003, 332pp

AMMA 2002 by David Y. Gao and Ray W. Ogden, Kluwer, 2002, 302pp

Nonconvex and Nonsmooth Mechanics: Modeling,Analysis and Numerical Methods By David Y. Gao, Ray W. Ogden and G.E.Stavroulakis, 517pp

Duality in Nonconvex Systems: Theory, Methods and Applications, by David Y. Gao, Kluwer Academic Publishers,Dordrecht/Boston xviii + 454pp

New web sites

Global Optimization & Control: Bi-Duality and Tri-Duality

Nonconvex and Nonsmooth Mechanics

Duality in nature is amazingly beautiful, for it is the way nature was created.
Duality in nature is simply mysterious, for it is the way that nature exists.
It is beautiful because all things were originally created in a splendid harmonious world.
It is mysterious because different creatures have different patterns of duality.
If we are not confused very often about the duality of natural phenomena,
we do not really understand what it is.
This may be the way that we exist.

David Y. Gao, 1999
Duality Principle