Zahra_Mohaghegh_April 08th_NLP

Zahra_Mohaghegh_April 08th_NLP - Engineering Management...

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1 Engineering Management & Systems Engineering The George Washington University EMSE 102 EMSE 102 - - 202 202 Survey of Operations Research Methods Survey of Operations Research Methods Zahra Mohaghegh mohagheg@umd.edu Post-Doctoral Research Associate Center For Risk and Reliability University of Maryland April 08, 2008
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2 Engineering Management & Systems Engineering The George Washington University Nonlinear Nonlinear Programming Programming
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3 Engineering Management & Systems Engineering The George Washington University Outline Outline ± Introductory Concepts of Nonlinear Programming (NLP) ± NLPs: Formulation ± NLPs : Theory & Algorithms ± Convex & Concave Functions ± Solving NLPs with one variables ± Golden Section Search ± Steepest Ascent Method ± Quadratic Problems
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4 Engineering Management & Systems Engineering The George Washington University Linear Programming Model Linear Programming Model 11 22 11 1 12 2 1n n 1 21 1 2 2n n 2 m1 1 2 2 mn n m Maximize ..... subject to a x + a x + . .. +a x b b b nn m cx cx cx x + ++ MM M 12 , ,..., 0 n xx ASSUMPTIONS : ± Proportionality Assumption ± Objective function ± Constraints ± Additivity Assumption ± Objective function ± Constraints
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5 Engineering Management & Systems Engineering The George Washington University What is a non What is a non - - linear program? linear program? ± maximize 3 sin x + xy + y 3 - 3z + log z Subject to x 2 + y 2 = 1 x + 4z 2 z 0 ± A non-linear program is permitted to have non-linear constraints and/or objectives. ± NLPs have non-proportional and non- additive relationships .
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6 Engineering Management & Systems Engineering The George Washington University Nonlinear Programs (NLP) Nonlinear Programs (NLP) ± Nonlinear objective function f(x) and/or Nonlinear constraints g i (x) . ± An NLP with no constraints is an unconstrained NLP . Common operations such as multiplication (x 1 x 2 ) and power (x 2 ) make a model nonlinear even if only one of them appears just once anywhere in the model. ( ) 12 ,,, ( ) ( ) , 1,2, , n ii Let x x x x Max f x gx b i m =… ≤∀ =
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7 Engineering Management & Systems Engineering The George Washington University NLPs NLPs ± NLPs are closer to general and realistic (and possibly unsolvable) models than the LPs. ± The most general mathematical model is likely to have nonlinear terms with random (and possibly dependent) coefficients. These difficulties are some of the reasons why a deterministic LP is, often, used as an approximation to a stochastic NLP . ± NLP is much more diverse as an area than LP. There is no one special algorithm such as simplex.
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8 Engineering Management & Systems Engineering The George Washington University NLP vs. LP NLP vs. LP If possible, the analyst should strive to model a decision process as a LP. Many management and production type problems have long been solved as LPs.
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Zahra_Mohaghegh_April 08th_NLP - Engineering Management...

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