David Yang Gao
Welcome to the Web Page of 



Announcements:Complete set of Analytic Solutions to Nonlinear Elasticity paper1 (for 3D fully nonlinear PDEs), paper2 (for 2D nonconvex PDEs) Canonical DualityTriality Theory: BreakingThrough from Challenges to Victory! Four Lectures on Canonical DualityTriality Theory
Int’l Conf on Computational Mathematics and Sciences June 68 2015International Symposium on Interdisciplinary Computation and Optimization The 3^{rd} World Congress of Global Optimization, July 712, 2013, The Yellow Mountains, China. New Book on Canonical Duality theory and applications New papers on the canonical duality theory for solving NPHard Problems. These papers shows that many nonconvex problems are not NPHard unless their canonical dual problems have no solution in positive domain!
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 complementarydual 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 complementarydual principle presents a unified analytic solution form for general problems in continuous and discrete systems; the triality theory is comprised by a saddle minmax duality and two pairs of doublemin, doublemax 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 welldeveloped FenchelMoreauRockafelar duality theory produces a socalled 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 socalled chaotic solutions in dynamical systems. In global optimization, many problems are NPhard. 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 semidefinite programming method is actually a special application of the complementarydual 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 wellknown 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 ComputerAided Process Operations (FOCAPO) 2008, June 29July 2, 2009, Cambridge, MA This paper presents a brief review and recent developments of this theory with applications to some wellknow 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 NPhard 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 nonconvex 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 finitedifference method (FDM) is used to illustrate the difficulty of capturing nonsmooth 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 mquadratic equations in ndimensional 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 R^{m}, which can be solved easily by welldeveloped convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented. Closedform 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 nonconvex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundaryvalue 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 01 QUADRATIC PROGRAMMING PROBLEMS This paper presents an application of the canonical duality theory for 01 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. Multiinteger programming problems can be solved too, results will be posed soon. Multiscale modelling and canonical dual finite element method in phase transitions of solids Int. J. Solids and Structure. This paper presents a multiscale model in phase transitions of solid materials with both macro and micro effects. This model is governed by a semilinear 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 multiscale 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 multiscale phase transitions of solids Solutions and Optimality Criteria to Box Constrained Nonconvex Minimization Problems, J. Industrial and Management Optimization, 3(2), 293304, 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 “NPhard problems” can be solved by polynomial algorithms. Complete solutions and extremality criteria to polynomial optimization problems, J. Global Optimization, 35 : 131143, 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 LandauGinzburg equation, nonlinear Schrödinger, equation and CahnHilliard 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 (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 SemiLinear Problems and Canonical Duality Solutions, Advances in Mechanics and Mathematics, Vol II, 2003, Springer, 261311. 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 