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 ...
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 ...
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 ...
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 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 ...
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 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 ...
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 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 ...
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 ...
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 ...
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 ...
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 ...