David Yang Gao

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School of Science, Information Technology and Engineering

Federation University Australia Mt Helen, Ballarat, Victoria 3350 Office: T137 Tel: (+61) 3 5327 9791

Australian National University https://researchers.anu.edu.au/researchers/gao-d

E-mail: d.gao at federation.edu.au or david.gao at anu.edu.au

Duality:  The Natural Beauty

Triality : The Natural Mystery


  Announcements:Complete set of Analytic Solutions to Nonlinear Elasticity paper1 (for 3-D fully nonlinear PDEs), paper2 (for 2-D nonconvex PDEs)  Canonical Duality-Triality Theory: Breaking-Through from Challenges to Victory!  Four Lectures on Canonical Duality-Triality Theory

Int’l Conf on Computational Mathematics and Sciences June 6-8 2015

International Symposium on Interdisciplinary Computation and Optimization The 3rd World Congress of Global Optimization, July 7-12, 2013, The Yellow Mountains, China.

New Book on Canonical Duality theory and applications

New papers on the canonical duality theory for solving NP-Hard Problems. These papers shows that many nonconvex problems are not NP-Hard unless their canonical dual problems have no solution in positive domain!

  1. Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization,  
  2. Global optimal solutions to Sensor Network Localization problems:
  3. Max-Cut,  By using perturbation method, some of this type NP-hard problems can be solved in polynomial time.
  4. Integer Programming. Quadratic 0-1 integer programming problem is identical to continuous unconstrained Lipschitzian global optimization problem which can be solved deterministically (but not in polynomial times).
  5. Complete solution to the hard case in trust-region sub-problem
  6. Unified method for solving nonconvex constrained problems.

The 8th International Conference on Bio-Inspired Computing: Theories and Application. July 12-14, 2013. The Yellow Mountains. China

Triality Theory: An open problem left in 2003 has now been solved completely. Please read the paper (http://arxiv.org/abs/1104.2970 or http://arxiv.org/abs/1110.0293 ) to see the intrinsic duality, potential applications, as well as the “super triality” of this theory. International Society of Global Optimization (iSoGO)  

Book Reviews on the book “Duality Principles in Nonconvex Systems” Canonical Duality Theory is a unified methodological theory 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 not only to model complex systems within a unified framework, but also 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. This theory has been challenged recently and won a great victory. See here for details 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) 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