EXAMS:
There will have only one take-home project.

REFERENCES:
There are some books on this topic which may serve well as a references for the course:

• D.Y. Gao, Duality principles in nonconvex systems, Theory, methods and applications. Kluwer Academic Publishers, 1999.
• W. Kaplan, Maxima and Minima with Applications, Practical optimization and duality, Wiley-Interscience, 1999.
• I. Ekeland and R. Temam, Convex analysis and Variational Problems, North-Holland, 1976
• D. Luenberger, Optimization by Vector Space Methods John Wiley and Sons, Inc., New York, 1969.
• D. Luenberger, Investment Science, Oxford University Press, New York, 1997.

PREREQUISITES AND COREQUISITES:
Advanced calculus and linear algebra. Some previous experience with functional analysis and partial differential equations would be helpful but is not essential.

OBJECTIVES:
Global optimization is concerned with the computation and characterization of global minima (or maxima) of nonlinear, nonconvex and even nonsmooth functions. Global optimization problems are widespred in the mathematical modelling of real world systems for a very broad range of applications. Such applications include structural engineering, civil and aerospace mechanical design, finite element methods, network, economics of scale, fixed charges, finance, allocation and location problems, operations research, statistics, transportation problems, chip design and database problems, chamincal engineering design and control, molecular biology and a number of other combinatorial optimization problems such as integer programming and related graph theory.

With the development and implementation of practical global optimization algorithms, more and more scientists in diverse disciplines have been using global optimization techniques to solve problems. Global optimization seems to be playing a significant role in many sciences, and therefore there is a need to have a special course designed for ALL graduate students at Virginia Tech. The ``Applied Global Optimization''is an attempt to fulfill this need, and the instructor himself is one of editors for the Journal of Global Optimization, an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management, and engineering (Website: http://www.wkap.nl/journalhome.htm/0925-5001).

Motivated by many practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines and natural philosophy, this course will discuss, within a unified framework, a self-contained comprehensive mathematical theory, methods, algorithms and applications.

COURSE CONTENT:
The course will discuss the following themes (these can be adjusted to suit the interests of the students):

• 1. Motivations and Framework in engineering science, general systems and mathematical modelings.
• 2. Fundamentals of Convex Analysis:
  Vector space, convex sets, convex functions, extended differential of functions (i.e. Gateaux derivative, sub-differentials, etc).
• 3. Convex optimization with equality/inequality constraints.
  Classical Legendre conjugate transformation and geometry, saddle-Lagrange duality theory and min-max methods, primal-dual interior and exterior point methods, algorithms and applications. linear and quadratic programming, Simplex method and Karmarkar's method.
• 4. Non-convex optimization:
  Quasiconvexity, Polyconvexity, and Rank-1 convexity, concave minimizations and d.c. programming. Relaxation methods for solving nonconvex global optimizations.
• 5. Nonsmooth analysis:
  Subdifferentials and Fenchel conjugate transformation. Applications in economics and solid mechanics
• 6. Bi-Duality Theory in Non-Convex Optimization Problems.
  Classification of the critical points of Lagrangian, Applications in d.c. programming, Hamilton systems, Phase transition and buckling analysis of mechanics.
• 7. Tri-duality and Triality Theories in Global Optimization.
  Canonical dual transformation methods for solving general nonconvex global minimization problems.
• 8. Applications in engineering
  Finite element analysis, structural mechanics, optimal design, Post-buckling analysis of large deformed structures. Contact problems in nonlinear mechanics.
• 9. Network flows and combinatorial optimization.
  Game theory, Transportation and Location Decisions, computational methods and algorithms.

General:
I reserve the option of modifying these policies where appropriate as the course develops.

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