Documents about Mathematical Programming

  • 10 Pages

    Lecture1

    Lehigh, IE 406

    Excerpt: ... Introduction to Mathematical Programming IE406 Lecture1 Dr. Ted Ralphs IE406 Lecture 1 1 Reading for This Lecture Primary Reading Bertsimas 1.1-1.2, 1.4-1.5 Supplementary Reading Bertsimas 1.3 Operations Research Methods and Models Model Building in Mathematical Programming IE406 Lecture 1 2 Systems Engineering A system is a functionally related group of elements, such as Raw Material B - manufacturing systems, distribution systems, financial systems, computer systems, biological systems, and political systems. M at er ia lA Ra w cost availability Such systems can be modeled and analyzed using techniques we'll learn about in this class. This type of analysis is used in every industry and every sector of the economy. Ra w M at ia er lC Product 1 Product Mix Problem Product 2 selling price demand IE406 Lecture 1 3 What is the purpose of a model? The exercise of building a model can provide insight. It's possible to do thi ...

  • 10 Pages

    Lecture1

    Lehigh, IE 316

    Excerpt: ... ically complementary organs or parts: the nervous system; the skeletal system. A group of interacting mechanical or electrical components. A network of structures and channels, as for communication, travel, or distribution. A network of related computer software, hardware, and data transmission devices. IE316 Lecture 1 4 Why do we model systems? The exercise of building a model can provide insight. It's possible to do things with models that we can't do with "the real thing." Analyzing models can help us decide on a course of action. IE316 Lecture 1 5 Examples of Models Physical Models Simulation Models Probability Models Economic Models Biological Models Mathematical Programming Models IE316 Lecture 1 6 Mathematical Programming Models What does mathematical programming mean? Programming here means "planning." Literally, these are "mathematical models for planning." Also called optimization models. Essential elements Decision variables ...

  • 1 Pages

    exinfo

    Wisconsin, ECE 556

    Excerpt: ... ECE 556 Design Automation of Digital Systems Midterm Examination Information Date: October 14, 2004 Time: 1:00 - 2:15 PM Location: Class room Rules: a. Closed book examination b. one single-side, letter-size "fact sheet" is allowed during exam. Coverage: Combinatorial algorithms, bin-packing problem (Appendix A, lecture notes) Graph theory, shortest path algorithm (lecture notes) Circuit paritioning: Kernighan-Lin, Fiduccia Mattheyses, Simulated Annealing (Section 2.1-2.4) Floor planning, slicing structure, Polish expression, bounding curve, Simulated annealing, mathematical programming method (Section 3.1-3.2, 3.3.2, 3.3.3) ...

  • 22 Pages

    Lecture1

    Lehigh, IE 418

    Excerpt: ... he skeletal system. A group of interacting mechanical or electrical components. A network of structures and channels, as for communication, travel, or distribution. A network of related computer software, hardware, and data transmission devices. IE418 Lecture 1 4 Why do we model systems? The exercise of building a model can provide insight. It's possible to do things with models that we can't do with "the real thing." Analyzing models can help us decide on a course of action. IE418 Lecture 1 5 Examples of Standard Model Types Simulation Models Probability Models Economic Models Biological Models Mathematical Programming Models IE418 Lecture 1 6 Mathematical Programming Models What does mathematical programming mean? Programming here means "planning." Literally, these are "mathematical models for planning." Also called optimization models. The essential element is the existence of an objective. Some categories of mathematical programs (see the ...

  • 1 Pages

    Modified-Spring-2005-Outline-864

    E. Kentucky, EECS 864

    Excerpt: ... Course Outline s s s s s Overview of Enabling Technologies-Physical Layer Issues in WDM Networking Wavelength Routed Networks Material from Chapter 2 and Chapter 5 IP over WDM Network Survivability s Optical Control Plane s Link Management Protocol (LMP) MPLS MPS GMLPS Optical Link Layer s s s Same basics of Graph Theory Gigbit and 10 Gigbit Ethernet Digital Wrapper Generic Framing Procedure Some Basics of Mathematical Programming Student Lectures Introduction 1 ...

  • 1 Pages

    ps8

    Lehigh, IE 406

    Excerpt: ... Problem Set 8 IE406 Introduction to Mathematical Programming Dr. Ralphs Due November 19, 2006 1. Bertsimas 7.3 2. Bertsimas 7.10 3. Bertsimas 7.11 4. Bertsimas 7.20 1 ...

  • 1 Pages

    ps5

    Lehigh, IE 406

    Excerpt: ... Problem Set 5 IE406 Introduction to Mathematical Programming Dr. Ralphs Due Oct 17, 2007 1. Bertsimas 4.3 2. Bertimas 4.5 3. Bertsimas 4.8 4. AMPL Chapter 1, problem 3 1 ...

  • 1 Pages

    ps3

    Lehigh, IE 406

    Excerpt: ... Problem Set 3 IE406 Introduction to Mathematical Programming Dr. Ralphs Due Sept 19, 2007 1. 2.2 2. 2.9 3. 3.2 4. 3.4 1 ...

  • 1 Pages

    ps9

    Lehigh, IE 406

    Excerpt: ... Problem Set 9 IE406 Introduction to Mathematical Programming Dr. Ralphs Due November 28, 2007 1. Bertsimas 10.4 2. Bertsimas 11.1 3. Bertsimas 11.4 4. Bertsimas 11.8 1 ...

  • 1 Pages

    ps2

    Lehigh, IE 406

    Excerpt: ... Problem Set 2 IE406 Introduction to Mathematical Programming Dr. Ralphs Due Sept 12, 2007 1. Bertsimas 1.10 2. Bertsimas 1.19 3. Bertsimas 2.4 4. Bertsimas 2.10 1 ...

  • 1 Pages

    ps6

    Lehigh, IE 406

    Excerpt: ... Problem Set 7 IE406 Introduction to Mathematical Programming Dr. Ralphs Due October 24, 2007 1. AMPL Chapter 1, problem 4 2. Bertsimas 5.1 3. Bertimas 5.5 4. Bertsimas 5.10 1 ...

  • 1 Pages

    ps1

    Lehigh, IE 406

    Excerpt: ... Problem Set 1 IE406 Introduction to Mathematical Programming Dr. Ralphs Due Sept 5, 2007 1. Bertsimas 1.6 2. Bertsimas 1.7 3. Bertsimas 1.9 4. Bertsimas 1.18 5. Bertsimas 2.1 1 ...

  • 1 Pages

    ps4

    Lehigh, IE 406

    Excerpt: ... Problem Set 4 IE406 Introduction to Mathematical Programming Dr. Ralphs Due October 3, 2007 1. 3.7 2. 3.12 3. 3.20 4. 3.27 1 ...

  • 2 Pages

    sd311outline

    W. Alabama, SD 311

    Excerpt: ... UNIVERSITY OF WATERLOO Department of Systems Design Engineering SD311 Engineering Optimization Spring 2009 INSTRUCTOR: Paul Calamai, Oce Hours: M/W/F 9-9:30, DC-2623 (ext 33182), phcalama@uwaterloo.ca This course deals with the mutual interaction between Systems Modeling and Design Optimization. Teaching Assistant (grading & tutorials) Mario Ventresca, Oce Hours: 4 to 6 on Wednesdays Required Texts Applied Mathematical Programming , S. Bradley, A. Hax and T. Magnanti, Addison-Wesley, 1977 QA402.5.B7 1977. This text is available on the web! Supplementary Material Optimization in Operations Research, R.L. Rardin, Prentice-Hall, 1998 T57.7.R37 1998 Operations Research: Deterministic Optimization Models, K.G. Murty, Prentice-Hall, 1995 T57.74.M88 1995 Introduction to Operations Research, F.S. Hillier and G.J. Lieberman, McGraw-Hill, 1990 T57.6.H54 2005 Linear Programming: Foundations and Extensions, R.J. Vanderbei, International Series in Operations Research and Management Science, ...

  • 11 Pages

    Lecture5

    Lehigh, IE 406

    Excerpt: ... Introduction to Mathematical Programming IE406 Lecture 5 Dr. Ted Ralphs IE406 Lecture 5 1 Reading for This Lecture Bertsimas 2.5-2.7 IE406 Lecture 5 2 Existence of Extreme Points Definition 1. A polyhedron P Rn contains a line if there exists a vector x P and a nonzero vector d Rn such that x + d P R. Theorem 1. Suppose that the polyhedron P = {x Rn|Ax b} is nonempty. Then the following are equivalent: The polyhedron P has at least one extreme point. The polyhedron P does not contain a line. There exist n rows of A that are linearly independent. IE406 Lecture 5 3 Optimality of Extreme Points Theorem 2. Let P Rn be a polyhedron and consider the problem minxP c x for a given c Rn. If P has at least one extreme point and there exists an optimal solution, then there exists an optimal solution that is an extreme point. Proof: IE406 Lecture 5 4 Optimality in Linear Programming For linear optimization, a finite optimal cost is equivalent to the existen ...

  • 9 Pages

    Lecture19

    Lehigh, IE 417

    Excerpt: ... Advanced Mathematical Programming IE417 Lecture 19 Dr. Ted Ralphs IE417 Lecture 19 1 Reading for this lecture Sections 9.1-9.2 IE417 Lecture 19 2 Constrained Optimization In Chapter 9, we look at methods based on applying unconstrained methods to constrained problems. Idea: Penalize violations of the constraints in the objective function. Consider min f (x) s.t. h(x) = 0 Try min{f (x) + [h(x)]2}. Will this work? IE417 Lecture 19 3 Penalty Functions A suitable penalty function is (x) = (gi(x) + (hi(x) where (y) = 0 if y 0 (y) > 0 if y > 0 (y) = 0 if y = 0 (y) > 0 if y = 0 IE417 Lecture 19 4 Performance of Penalty Methods Suppose we simply solve min{f (x) + (x) : x X} for some suitable penalty function and some > 0. Consider solving the following problem: max () s.t. 0 where () = inf {f (x) + (x) : x X}. What will the solution be? IE417 Lecture 19 5 Main result on Penalty Methods As long as the following assumptio ...

  • 8 Pages

    M3RPObRep

    Western Michigan, NSFM 3

    Excerpt: ... Findings from Observations of Mathematics Lessons in M3RP Teacher Leader Classrooms During the 2000-01 School Year Prepared by SAMPI-Western Michigan University July 2001 The Michigan Middle Schools Mathematics Reform Project (M3RP) is a four-year c ...

  • 3 Pages

    Lecture5

    University of Florida, AEB 6182

    Excerpt: ... AEB 6182: Lecture V Transformations of Risk Aversion and E-V Versus Direct Utility Maximization I. Interpretations and Transformations of Scale for the Pratt-Arrow Absolute Risk Aversion coefficient: Implications for Generalized Stochastic Dominance A. To this point, we have discussed technical manifestations of risk aversion such as where the risk aversion coefficient comes from and how the utility of income is derived. However, I want to start turning to the question: How do we apply the concept of risk aversion? B. Several procedures exist for integrating risk into the decision making process such as direct application of expected utility, mathematical programming using the expected value-variance approximation, or the use of stochastic dominance. All of these approaches, however, require some notion of the relative size of risk aversion. 1. Risk aversion directly uses a risk aversion coefficient toparameterize the negative exponential or power utility functions. 2. Mathematical programming uses the concep ...