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Department of Mathematics

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

E-mail: gao at vt edu

Duality:  The Natural Beauty

http://www.math.vt.edu/people/ gao

Triality : The Natural Mystery













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The 2nd World Congress of Global Optimization and Applications, Chania, Greece, July 3-7, 2011.

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

The Forth International Conference on Optimization and Control with Applications (OCA2009)  June 6-11, 2009, Harbin and Wudalianchi, China

The 3rd International Conference on Modeling of Complex Systems, CMCS’09, Doha,  Qatar, May 5-7, 2009. 
International Conference on Optimization, Simulation and Control  Ulaanbaatar, Mongolia   July 25 - 28, 2010
International Conference on Mathematical Control Theory 
In honor of David L. Russell on the occasion of his 70th Birthday May 15—17, 2009, Chinese Academy of Sciences (CAS), Beijing, China

The 10th Conference on Dynamical Systems - Theory and Applications, DSTA 2009, December 7-10, 2009,  Łódź, Poland

International School of Mathematics "G. Stampacchia" Workshop on "Nonlinear Optimization, Variational Inequalities and Equilibrium Problems",    July 2 to 10, 2010,   "Ettore Majorana Centre for Scientific Culture"   Erice, Sicily, Italy.

Summer Workshop on Duality Theory and Application, May 23-24, 2009, Tsinghua University. 
Winter Workshop on Duality Theory and Application, Dec. 27-31, 2009, Hyatt Orlando Airport, Florida. 

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

Book Reviews on the book “Duality Principles in Nonconvex Systems”

Canonical Duality Theory is a unified methodology which is composed mainly of a canonical dual transformation, a complementary-dual principle, and an associated triality theory. The canonical dual transformation can be used for modeling complex systems and to formulate perfect dual problems without duality gap; the complementary-dual principle presents a unified analytic solution form for general problems in continuous and discrete systems; the triality theory is comprised by a saddle min-max duality and two pairs of double-min, double-max dualities. This theory reveals an intrinsic duality pattern in complex phenomena and can be used to identify both global and local extrema. In analysis and optimization, the traditional Lagrangian duality can be used only for solving convex minimization problems. In nonconvex systems, the well-developed Fenchel-Moreau-Rockafelar duality theory produces a so-called duality gap if the primal problem is nonconvex. Traditional sufficient conditions in convex analysis and programming cannot 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, 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 given below). The original idea of the canonical duality theory is   from the joint work by Gao and Strang in 1989. It is now understood that the popular semi-definite programming method is actually a special application of the complementary-dual variational principle proposed in this joined work. The canonical duality theory can be used to solve a large class of nonconvex/nonsmooth/discrete problems in nonlinear analysis, global optimization, and complex systems. 

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). The canonical duality theory solved this open problem 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:

Advances in canonical duality theory with  applications to  global optimization, invited lecture presented at Foundations of Computer-Aided Process Operations (FOCAPO) 2008, June 29-July 2, 2009, Cambridge, MA     This paper presents a brief review and  recent developments of this theory with applications to some well-know problems, including polynomial minimization,  mixed integer and fractional  programming,  nonconvex minimization with nonconvex quadratic constraints, etc. Results shown that under certain conditions, these difficult problems can be solved by deterministic methods within polynomial times, and   NP-hard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Concluding remarks and  open problems are presented in the end.

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.


Canonical dual least squares method for solving general nonlinear systems of quadratic equations. This paper presents a canonical dual approach for solving general nonlinear algebraic systems. By using least square method, the nonlinear system of  m-quadratic equations in n-dimensional space is first formulated as a nonconvex optimization problem. We then proved that, by the canonical duality theory developed by the second author, this nonconvex problem is equivalent to a concave maximization  problem in Rm, which can be solved easily by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.



 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 minimizationJ. 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 (click here for details) 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)

Journal of Industrial and Management Optimization, According to the Journal Citation Report 2008impact factor of this journal is 1.181 and it is now ranked the top 20th out of 64 in the category of Management Science and Operations Research.

Optimization Letters

Book Series:

Advances  in Mechanics and Mathematics (AMMA)  by Springer

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

Optimization and Control of Complex Systems, Taylor & Francis

Computational Mechanics and Mathematics, (New) Springer

New books published:

Advances in Applied Mathematics and Global Optimization, Springer, 2009

Complementarity, Duality, and Symmetry in Mechanics, Kluwer, 404pp. 2004.

Advances in Mechanics and Mathematics,  2003 by David Y. Gao and Ray W. Ogden, Kluwer, 2003, 332pp

Advances in Mechanics and Mathematics,  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 

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

Movie: Click  here for a finite element simulation

of large deformation nonconvex  problem (and here for details) 




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

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

The 2nd 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.

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

The International Conference on Nonlinear Programming with Applications
(NPA2006) Date : 29 May - 1 June, 2006, Fudan University, Shanghai, China

International Conference on Nonsmooth/Nonconvex Mechanics, ARISTOTLE UNIVERSITY OF THESSALONIKI, Greece.
7 & 8 July, 2006

Modern Mechanics and Mathematics ---
An International Conference in Honour of Ray Ogden's 60th Birthday.

August 26-28, 2003, Keele University, UK. Organisers:
M.A. Hayes, Y.B. Fu, and D.Y. Gao

The Third International Conferenceon
Optimization and Control with Applications(OCA2004),


Mathematical Theory in Fine Art and Design