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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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, ...

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 ...

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 ...

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 ...