Kentucky, EE 640

Excerpt: ... LECTURE NOTES DAY 19 ECE EE640 Optimum Decision Boundaries MAP, Neyman Pearson, Lagrange Multipliers 3-31-05 1 ...

SMU - Cox School of Business, PHYSICS 6321

Excerpt: ... Lagrange multipliers - Wikipedia, the free encyclopedia http:/en.wikipedia.org/wiki/Lagrange_multipliers Lagrange multipliers From Wikipedia, the free encyclopedia In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the maximum/minimum of a function subject to constraints. For example (see Figure 1 on the right), consider the optimization problem maximize subject to We introduce a new variable () called a Lagrange multiplier, and study the Lagrange function defined by Figure 1: Find x and y to maximize f(x,y) subject to a constraint (shown in red) g(x,y) = c. If (x,y) is a maximum for the original constrained problem, then there exists a such that (x,y,) is a stationary point for the Lagrange function (stationary points are those points where the partial derivatives of are zero). However, not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yields a necessary ...

Idaho, ME 435

Excerpt: ... Simulation of thermal energy systems, how to write the steady-state equations for the following: -system specifications (cycle), -component performance descriptions, -fluid properties, and -basic equations for fluids, heat transfer and thermodynamics. How to tell when the system of equations is closed Solution techniques (know how to set them up and how to do them) for 1-d and higher order: -successive substitution -Newton-Raphson Taking numerical derivatives: -backward difference -forward difference -central difference Parametric studies, why and how to do them Optimization: -setting it up: objective function, design variables, constraints -finding the min or max by methods of calculus or Lagrange Multipliers -sensitivity analysis for Lagrange multiplier analysis ...

Stanford, EE 363

Excerpt: ... EE363 Winter 2008-09 Lecture 2 LQR via Lagrange multipliers useful matrix identities linearly constrained optimization LQR via constrained optimization 21 Some useful matrix identities let's start with a simple one: Z(I + Z)-1 = I - (I + Z)-1 (provided I + Z is invertible) to verify this identity, we start with I = (I + Z)(I + Z)-1 = (I + Z)-1 + Z(I + Z)-1 re-arrange terms to get identity LQR via Lagrange multipliers 22 an identity that's a bit more complicated: (I + XY )-1 = I - X(I + Y X)-1Y (if either inverse exists, then the other does; in fact det(I + XY ) = det(I + Y X) to verify: I - X(I + Y X)-1Y (I + XY ) = I + XY - X(I + Y X)-1Y (I + XY ) = I + XY - X(I + Y X)-1(I + Y X)Y = I + XY - XY = I LQR via Lagrange multipliers 23 another identity: Y (I + XY )-1 = (I + Y X)-1Y to verify this one, start with Y (I + XY ) = (I + Y X)Y then multiply on left by (I + Y X)-1, on right by (I + XY )-1 note dimensions of inverses not necessarily the same mnemonic: lefthand Y moves into inve ...

SMU - Cox School of Business, PHYSICS 6321

Excerpt: ... An Introduction to Lagrange Multipliers http:/www.slimy.com/~steuard/teaching/tutorials/Lagrange.html An Introduction to Lagrange Multipliers http:/www.slimy.com/~steuard/teaching/tutorials/Lagrange.html An Introduction to Lagrange Multipliers by Steuard Jensen Lagrange multipliers are a very useful technique in multivariable calculus, but all too often they are poorly taught and poorly understood. With luck, this overview will help to make the concept and its applications a bit clearer. Be warned: this page may not be what you're looking for! If you're looking for detailed proofs, I recommend consulting your favorite textbook on multivariable calculus: my focus here is on concepts, not mechanics. (Comes of being a physicist rather than a mathematician, I guess.) If you want to know about Lagrange multipliers in the calculus of variations, as often used in Lagrangian mechanics in physics, this page only discusses them very briefly. Here's a basic outline of this discussion: When are Lagrange multipliers ...

Vanderbilt, MATH 175

Excerpt: ... Math 175, Final Exam Review Information Limits and continuity Partial and directional derivatives Tangent planes The chain rule Maxima and minima Lagrange multipliers Double integrals Triple integrals Change of variables Vector fields Math 175, Final Exam Review Math 175, Final Exam Review Information Limits and continuity Partial and directional derivatives Tangent planes The chain rule Maxima and minima Lagrange multipliers Double integrals Triple integrals Change of variables Vector fields These slides will be posted on my webpage www.math.vanderbilt.edu/mrinalr There will be another version of these for printing. Do not print these slides. It is a waste of ink, paper, and electricity. If I see you with a printout of these on-screen slides, I will deduct 50 points from your exam! Exam information Math 175, Final Exam Review 120 minutes Information Limits and continuity Partial and directional derivatives Tangent planes The chain rule Maxima and minima Lagrange multipliers Double integrals Triple i ...

Georgia Tech, AE 6531

Excerpt: ... EE363 Winter 2005-06 Lecture 2 LQR via Lagrange multipliers useful matrix identities linearly constrained optimization LQR via constrained optimization 21 Some useful matrix identities let's start with a simple one: Z(I + Z)-1 = I - (I + Z)-1 (provided I + Z is invertible) to verify this identity, we start with I = (I + Z)(I + Z)-1 = (I + Z)-1 + Z(I + Z)-1 re-arrange terms to get identity LQR via Lagrange multipliers 22 an identity that's a bit more complicated: (I + XY )-1 = I - X(I + Y X)-1Y (if either inverse exists, then the other does; in fact det(I + XY ) = det(I + Y X) to verify: I - X(I + Y X)-1Y (I + XY ) = I + XY - X(I + Y X)-1Y (I + XY ) = I + XY - X(I + Y X)-1(I + Y X)Y = I + XY - XY = I LQR via Lagrange multipliers 23 another identity: Y (I + XY )-1 = (I + Y X)-1Y to verify this one, start with Y (I + XY ) = (I + Y X)Y then multiply on left by (I + Y X)-1, on right by (I + XY )-1 note dimensions of inverses not necessarily the same mnemonic: lefthand Y moves into inve ...

Colorado, ASEN 5519

Excerpt: ... FORTUNE EIGHT FORTUNE EIGHT Aerospace Industries, Inc. International Technical Services Original Lecture: 2002 Feb 6 MEMORANDUM To: From: CMA Class Chauncey Uphoff Subject: Class Notes for Lecture #4 I was somewhat disappointed with my presentation ...

Minnesota, MATH 4428

Excerpt: ... 4428: Exam-1 Topics 1. Optimization. Constrained and unconstrained optimization, the method of Lagrange multipliers . Study material and examples: class notes and the theoretical parts of Sect. 1.1-1.3, 2.1-2.3. Some linear optimization problems (Sect. 3.3) may be included if they can be solved by hand (no computers of programmable calculators shall be used at the exam). 2. Dierential equations and dynamical systems. Stability and instability of equilibria for dierential equations and discrete-time dynamical systems by linearization (eigenvalue method). Solutions of initial value problems for linear dierential equations and discrete time dynamical systems using eigenvalues and eigenvectors. Study material and examples: Class notes and the theoretical parts of Sect. 4.1-4.3, 5.1-5.2. 1 ...

University of Illinois, Urbana Champaign, CS 598

Excerpt: ... sis j can be set to g j , which is perpendicular to the level set g j (r1,., rn ) = 0. References Lagrange Multipliers For a good overview see: "An Introduction to Physically Based Modeling: Constrained Dynamics" - Andrew Witkin http:/www-2.cs.cmu.edu/~baraff/pbm/constraints.pdf http:/www.pixar.com/companyinfo/research/pbm2001/notesa.pdf ...

UCLA, STATS 231

Excerpt: ... 1. Stat 231. A.L. Yuille. Fall 2004 Linear Separation and Margins. Non-Separable and Slack Variables. Duality and Support Vectors. Read 5.10, A.3, 5.11 Duda, Hart, Stork. Or better: 12.1, 12.2, Hastie, Tibshirani, Friedman. Lecture notes for Stat 231: Pattern Recognition and Machine Learning 2. Separation by Hyperplanes Data Hyperplane Linear Classifier: By simple geometry, the signed distance of a point x to the plane is The line through x perpendicular to the plane is: Hits the plane when which implies thatPattern Recognition andis the distance Lecture notes for Stat 231: Machine Learning 3. Margin, Support Vectors We introduce new concepts: (I) Margin, (II) Support Vectors. This will enable us to understand performance in non-seperable case. Technical methods: quadratic optimization with linear constraints, Lagrange multipliers and duality. Margins will also be important when studying generalization. Everything in this lecture can be extended beyond hyperplanes (next lecture). L ...

UMass (Amherst), MIE 697

Excerpt: ... 1 Lecture 15: Lagrangian Relaxation and the Maximum Covering Location Model 1 Introduction In this section we describe the Lagrangian relaxation technique in detail as we apply it to the Maximum Covering Location problem. 2 Lagrangian Relaxation Approach Lagrangian relaxation is an approach to solving difficult problems (such as integer programming problems). The approach is outlined below in cast of solving a maximization problem. 1. Relax one or more constraints by multiplying the constraints by a Lagrange multiplier and bringing the constraint(s) to the objective function. The constraints to relax must be such that the relaxed problem can be solved very easily for fixed values of the Lagrange multipliers . 2. Solve the resulting relaxed problem finding the optimal values of the decision variables. 3. (Optional) Use the decision variables from the solution to the relaxed problem found in step 2 to find a feasible solution to the original problem. This can often be done fairly easily and provides a lower ...

Sveriges lantbruksuniversitet, ECON 331

Excerpt: ... ECON 331 Study Questions for Chapter 12: Lagrange Multiplier Method Kevin Wainwright SHORT ANSWER. Show all your work. In each case, check 2nd order conditions 1) Use the method of Lagrange multipliers to find the critical points of f(x, y, z) = 2x + 4y - 4z subject to the constraint x2 + y2 + z2 = 9. 2) Use the method of Lagrange multipliers to determine the critical points of f(x, y, z) = x2 - 3 y2 - z2 + 6 subject to the constraint 5x - 3y + z = 21. 3) The Cobb-Douglas production function for a company is given by P(k, l) = 163k1/5l 4/5 where P is the monthly production value when k is the number of units of capital and l is the number of units of labor. Suppose that capital costs $105 per unit, labor costs $70 per unit, and the total cost of capital and labor is limited to $152,250. Use Lagrange Multiplier's to write the system of equations you would use to find the number of units of capital and labor that maximize production. 4) The Cobb-Douglas production function for a company is given by P(k, l) = ...