#### 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 ...

#### 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 ...

#### mathprogramming

Caltech, CS 20
Excerpt: ... Mathematical Programming : Fundamentals CS20b 1 Outline The problem domain Definitions Fundamental theorems Introduction to Linear Programming CS 20b http:/www.cs.caltech.edu/~cs20/ Mathematical Programming 27 February 2003 2 Problem Domain: Optimization Maximize f(x) Subject to g(x) =< c Where f(x) is a function from a vector space to the reals And g(x) is a function from the vector space to a vector of length m and c is a constant vector of length m. f(x) is called the objective function g(x) = < c is called the set of constraints. CS 20b http:/www.cs.caltech.edu/~cs20/ Mathematical Programming 27 February 2003 3 Problem Domain: Optimization Maximize f(x) Subject to g(x) = < c This problem is the same as: Maximize f(x) over all x in the set {x | g(x) = < c} The set {x | g(x) = < c} is called the feasible region. CS 20b http:/www.cs.caltech.edu/~cs20/ Mathematical Programming 27 February 2003 4 Definitions: Convex Set Convex set is a set of points in a vector space such that for any two ...

#### 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) ...

#### 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 ...

#### 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 ...

CSU Chico, CSCI 151
Excerpt: ... This directory contains transparency masters (which may also be used asstudy guides) for the book "Data Structures and Program Design in C+"by Robert L. Kruse and Alexander J. Ryba, copyright (C) 1999 byPrentice-Hall, Inc., Upper Saddle River, New Je ...

#### 565

University of Illinois, Urbana Champaign, ECON 565
Excerpt: ... Course Schedule - Spring 2007 Economics 565 Math Econ I Credit: 4 hours. Studies quantitative techniques useful in economic analysis and decision making; mathematical programming ; input-output analysis; point-set theory and game theory; existence, optimality, and stability conditions for static general equilibrium; and activity analysis, including welfare economics. Prerequisite: MATH 415; ECON 502 and ECON 503, or equivalent. CRN 47392 Type lecturediscussion Section M Time 02:30 PM - 04:20 PM Days MW Location room 219 David Kinley Hall Instructor Polborn, M Page 1 - Economics, Spring 2007 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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 ...

#### 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, ...

#### 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 ...

#### 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 ...

#### 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 ...

#### mathematical_programming

Sanford-Brown Institute, CSCI 1410
Excerpt: ... CS 141 Introduction to AI Greenwald Lecture 08: Mathematical Programming TBA Contents 1 Overview 2 Sample Linear Programs 3 Sample Integer Linear Programs 3.1 3.2 Sample Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 4 1 Overview A mathematical program is an optimization problem of the form: xRn min f (x) subject to Ax b where A = (aij )1im,1jn is a matrix of real-valued constants b = (bj )1jm is a vector of real-valued constants x = (xi )1in is a vector of variables f is a function If f is linear, then the program is called a linear program (LP). If f is linear, and for all i, xi Z, then the program is called an integer linear program (ILP). If f is linear, and for some i, xi R, while for other i, xi Z, then the program is called a mixed integer linear program. If f is linear, and for all i, xi {0, 1}, then the program is called a binary integer line ...

#### Mathematical_Programming

UPenn, OPIM 910
Excerpt: ... MP in Action The Newsletter of Mathematical Programming in Industry and Commerce May 1995 How to Build a Mathematical Programming Model by Robert Simons This months article continues an occasional series on aspects of the practice of Mathematical Programming . This article is intended to provide guidance on three points: is my problem likely to be amenable to Mathematical Programming ? how should I set about formulating my model? what should I include and what should I leave out? It does not attempt to address issues of what software is most appropriate, on which guidance was offered in November 1994s and January 1995s MP in Action. Nor is it a substitute for a training course or a textbook such as H.P.Williams Model Building in Mathematical Programming (Wiley, 1993). Rather it is intended as practical guidance for those setting out to tackle a problem. restrictive in what it can represent than other techniques. Nor should it be imagined that it really does find the best solution to the real-world ...

#### 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 ...

#### lecture01

Lehigh, IE 426
Excerpt: ... Evaluation IE 426 Optimization models and applications instructor: phone: office hrs: email: web page: Pietro Belotti Mohler Lab #322 (610) 758 3865 Tue/Thu, 10am 12pm (or by appt.) belotti at lehigh dot edu http:/www.lehigh.edu/~pib208 Homework: Quiz #1: Quiz #2: Case study: Final exam: 20% 10% 15% 25% 30% (four) ( Sep 25) ( Nov 5) Case studies Material Select chapters of "Introduction to Operations Research" by F.S. Hillier and G.J. Lieberman, McGraw-Hill: New York, NY, 1990; Select chapters of "Introduction to Mathematical Programming : Applications and Algorithms", Volume 1, by W.L. Winston and M. Venkataramanan; Select chapters of "Operations Research: Applications and Algorithms" by Wayne L. Winston, PWS-Kent Pub. Co., 1991; three chapters of a book on modeling by Robert Fourer, available for download. modeling language: "AMPL: A Modeling Language for Mathematical Programming " by Robert Fourer, David M. Gay, and Brian W. Kernighan. Some handouts. Groups of three-four people study an Opt ...