Linear Programming: Data Fitting Steve Gu Mar 21, 2008.

Slides:

Advertisements

Similar presentations

3.4 Linear Programming 10/31/2008. Optimization: finding the solution that is either a minimum or maximum.

Advertisements

Linear Programming, 1 Max c 1 *X 1 +…+ c n *X n = z s.t. a 11 *X 1 +…+ a 1n *X n  b 1 … a m1 *X 1 +…+ a mn *X n  b m X 1, X n  0 Standard form.

© 2003 Anita Lee-Post Linear Programming Part 2 By Anita Lee-Post.

Linear Programming Problem

Solving IPs – Cutting Plane Algorithm General Idea: Begin by solving the LP relaxation of the IP problem. If the LP relaxation results in an integer solution,

Linear Programming Graphical Solution Procedure. Two Variable Linear Programs When a linear programming model consists of only two variables, a graphical.

Nonlinear Programming

Linear Programming – Simplex Method

Use MATLAB to solve linear programs LI Xiao-lei. MATLAB format for linear programs MATLAB uses the following format for linear programs: min z = f T x.

MS&E 211 Quadratic Programming Ashish Goel. A simple quadratic program Minimize (x 1 ) 2 Subject to: -x 1 + x 2 ≥ 3 -x 1 – x 2 ≥ -2.

Linear Programming?!?! Sec Linear Programming In management science, it is often required to maximize or minimize a linear function called an objective.

COMP 116: Introduction to Scientific Programming Lecture 11: Linear Regression.

Chapter 2 Linear Programming Models: Graphical and Computer Methods © 2007 Pearson Education.

Nonlinear Programming

CSCI 3160 Design and Analysis of Algorithms Tutorial 6 Fei Chen.

Linear Programming Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions.

Support Vector Machines Formulation  Solve the quadratic program for some : min s. t.,, denotes where or membership.  Different error functions and measures.

Branch and Bound Algorithm for Solving Integer Linear Programming

Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar Slides available online

1 Linear Programming Supplements (Optional). 2 Standard Form LP (a.k.a. First Primal Form) Strictly ≤ All x j 's are non-negative.

1 1 Slides by John Loucks St. Edward’s University Modifications by A. Asef-Vaziri.

Linear Programming Operations Research – Engineering and Math Management Sciences – Business Goals for this section  Modeling situations in a linear environment.

Chapter 0 – Objectives Solve for a variable in one-variable equations ex: Solve -3(x+5) = 10 Solve for a variable in two-variable equations ex: Solve.

3.4 Linear Programming p Optimization - Finding the minimum or maximum value of some quantity. Linear programming is a form of optimization where.

1 1 Slide © 2005 Thomson/South-Western Chapter 2 Introduction to Linear Programming n Linear Programming Problem n Problem Formulation n A Maximization.

Game theory & Linear Programming Steve Gu Mar 28, 2008.

Linear Programming – Simplex Method

Notes 7.5 – Systems of Inequalities. I. Half-Planes A.) Given the inequality, the line is the boundary, and the half- plane “below” the boundary is the.

1 Max 8X 1 + 5X 2 (Weekly profit) subject to 2X 1 + 1X 2  1000 (Plastic) 3X 1 + 4X 2  2400 (Production Time) X 1 + X 2  700 (Total production) X 1.

Linear Programming Problem. Definition A linear programming problem is the problem of optimizing (maximizing or minimizing) a linear function (a function.

Linear Programming Advanced Math Topics Mrs. Mongold.

CVX_class Cvx tool setup Search for CVX tool ( ) Dezip to your assigned directory Key cvx_setup in the.

Monday WARM-UP: TrueFalseStatementCorrected Statement F 1. Constraints are conditions written as a system of equations Constraints are conditions written.

3.4 – Linear Programming. Ex. 1 Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the max & min values.

3.4: Linear Programming Objectives: Students will be able to… Use linear inequalities to optimize the value of some quantity To solve linear programming.

Warm-upWarm-up Sketch the region bounded by the system of inequalities: 1) 2) Sketch the region bounded by the system of inequalities: 1) 2)

LINEAR PROGRAMMING 3.4 Learning goals represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret.

Linear Programming (LP) Problems MAX (or MIN): c 1 X 1 + c 2 X 2 + … + c n X n Subject to:a 11 X 1 + a 12 X 2 + … + a 1n X n

Calculus III Hughes-Hallett Chapter 15 Optimization.

Submodularity Reading Group Submodular Function Minimization via Linear Programming M. Pawan Kumar

3-5: Linear Programming. Learning Target I can solve linear programing problem.

Sullivan Algebra and Trigonometry: Section 12.9 Objectives of this Section Set Up a Linear Programming Problem Solve a Linear Programming Problem.

3.4 Linear Programming p Optimization - Finding the minimum or maximum value of some quantity. Linear programming is a form of optimization where.

Chapter 2 Linear Programming Models: Graphical and Computer Methods

An Introduction to Linear Programming Pertemuan 4

Problem 1 Demand Total There are 20 full time employees, each can produce 10.

Introduction to Linear Programs

Systems of Equations and Inequalities

Linear Programming: The Graphical Method

آشنايی با اصول و پايه های يک آزمايش

3.2 Linear Programming 3 Credits AS

3-3 Optimization with Linear Programming

Linear Programming.

3.4 – Linear Programming.

8.4 Linear Programming p

ENGM 435/535 Optimization Adapting to Non-standard forms.

Graphical Solution Procedure

Max Z = x1 + x2 2 x1 + 3 x2  6 (1) x2  1.5 (2) x1 - x2  2 (3)

INFM 718A / LBSC 705 Information For Decision Making

7.5 – Constrained Optimization: The Method of Lagrange Multipliers

Warm Up Solve for x:

Linear Programming Example: Maximize x + y x and y are called

Linear Programming Problem

9.3 Linear programming and 2 x 2 games : A geometric approach

Lecture 19 Linear Program

Graphical solution A Graphical Solution Procedure (LPs with 2 decision variables can be solved/viewed this way.) 1. Plot each constraint as an equation.

1.6 Linear Programming Pg. 30.

Presentation transcript:

Linear Programming: Data Fitting Steve Gu Mar 21, 2008

Outline Data fitting (Ref: pp 250) Data fitting using LP

Data fitting X123 Y258 X Y

How to fit the model to the following data set using Chebyshev criterion? Chebyshev’s criterion: minimize the largest absolute deviation: X123 Y258

Chebyshev Criterion A min max problem!

More Formal Statement X123 Y258 subject to:

Interpret Geometrically c r Feasible region

Interpret Geometrically c r Feasible region Optimal solution Therefore, 2.5 is the optimal value of c and the residual error is 0.5

More general problem Given N points (X 1,Y 1 ),(X 2,Y 2 ),…,(X N,Y N ), How to fit the model y=cx using Chebyshev criterion?

Data fitting using Linear Programming What’s the Goal? What are the Unknowns? – r and c What are the constraints? – Compare to 3-points case:

Data fitting using Linear Programming

LP in MATLAB Use command: – linprog – Do type “help linprog” ! Example: – X=linprog(f,A,b) solves the linear programming problem: min f'*x subject to: A*x <= b

LP in MATLAB

Results: Fitting data using LP

c= r=

End: Q&A Thanks