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Announcements: 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
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,
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 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
Semi-Linear Problems and Canonical Duality Solutions, Advances in
Mechanics and Mathematics, Vol II, 2003, Springer, 261-311.
Researchers
Get First Look into Antimatter Atom
(NSF report)
Movie: Click here for a finite element simulation of large deformation nonconvex problem
(and here
for details) |
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Conferences: Foundations of Computer-Aided
Process Operations (FOCAPO) 2008, June 29-July 2, 2009, 9th Conference
on Dynamical Systems, Theory and Applications, The 2nd International
Conference on Complementarity, Duality, and Global Optimization, The Fifth
International Conference on Nonlinear Analysis and Convex Analysis (NACA2007) International
Workshop on Duality and Conic Optimization, Tsinghua, Beijing, December
16-18, 2006. International Conference on
Complementarity, Duality, and Global Optimization, International Conference on
Nonsmooth/Nonconvex Mechanics, ARISTOTLE UNIVERSITY OF THESSALONIKI, Modern Mechanics and
Mathematics --- The
Third International Conferenceon |
