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Optimization Theory: Course Outline (MAT 455)

In mathematics, computer science and operations research, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

Course Contents

Introduction to Optimization, variables and Objective functions, Stationary values and Extrema,  Relative  and  Absolute  extrema,  Equivalence  of  minimum  and  maximum, Convex,  Concave  and  uni-model  functions,  Constraints,  Mathematical  programming problems. Optimization of  one-dimensional  functions.  Optimization  of  two  dimensional functions  and  derivatives  of  sufficient  conditions  for  existence  of  optima  for  them Exercises.  Optimization of function of several variables and necessary and sufficient conditions for it Exercises. Optimization by equality constraints: direct substitution method and Lagrange multiplier method.  Behavior of  the  Lagrangian  functions.  Exercises. Necessary and sufficient conditions for an equality constrained optimum with bounded independent variables. Inequality constraints and Lagrange multipliers. Multidimensional  optimization  by  Gradient  method.  Exercises.  Convex and Concave programming Linearization. Exercises. Calculus of variation, Euler-Lagrange equations. Functional of several variable. Functionals depending on higher derivatives. Functionals depending on several independent variables. Variational  problems in parametric form. Some applications, Constraints   variational   problem.  A minimum   path   problem.   Dynamic programming fundamentals. Generalized mathematical formulation of dynamic programming, Problems  and exercises. Dynamic programming and variational calculus. Control theory.

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