LECTURE NOTES Duality Theory, Sensitivity Analysis, and Parametric Programming

Size: px
Start display at page:

Download "LECTURE NOTES Duality Theory, Sensitivity Analysis, and Parametric Programming"

Transcription

1 CEE 60 Davd Rosenberg p. LECTURE NOTES Dualty Theory, Senstvty Analyss, and Parametrc Programmng Learnng Objectves. Revew the prmal LP model formulaton 2. Formulate the Dual Problem of an LP problem (TUES) 3. Relate the dual program outputs to the prmal problem (TUES) 4. Identfy how LP outputs change wth changes n model nputs (TUES/THURS) o Rght-hand sde constrant values o Objectve functon coeffcents o Addng a new constrant o Systematcally changng a model nput. Revew Optmzaton and LP model formulaton General Formulaton Maxmze (or mnmze) Z = f(x) Subject to: g x b, m m LP Formulaton m LP Formulaton (vector-matrx format) c = by n row vector of objectve functon coeffcents x = n by column vector of decson varable values A = m by n matrx of constrant coeffcents b = m by column vector of constrant rght-hand-sde values

2 CEE 60 Davd Rosenberg p. 2 The PRIMAL formulaton s the formulaton we have been talkng about all along so far: How does the soluton change (the outputs Z and x) f we change varous nputs such as b, c, and A? 2. Formulate the Dual Problem of an LP Every prmal LP problem (.e., every LP problem!) has a correspondng dual formulaton. Prmal Dual n decsons (actvty levels) m constrants (resource avalabltes) Max Zx = cx s.t. Ax b x = vector of actvty levels c = unt product per actvty level ($/level) b = resource levels A = Constrant coeffcents (resource/actvty) Zx = (product/level)*(level) = product (e.g., $) Zy = b T y = Maxmze benefts from actvtes At the optmum,

3 CEE 60 Davd Rosenberg p. 3 Example. What s the dual of the followng prmal problem? Max Z x = 6 x egg + 7 x tom xegg s.t. 4 3 xtom 2000 Dual: Example 2. Solve and compare the solutons to the prmal and dual problems n Example #.

4 CEE 60 Davd Rosenberg p. 4 Example 3. What s the dual of the followng prmal problem? Mn Z x = 0 x + 4 x 2 s.t x x Dual:

5 CEE 60 Davd Rosenberg p Relate the dual program outputs to the prmal problem Interpretaton of dual varable y: Margnal mprovement to objectve functon Your wllngness to Shadow Value Complementary Slackness How are the shadow value and constrant slack varables related? At optmalty, f a prmary constrant s slack (slack varable > 0), then the correspondng shadow value for the constrant = 0. Smlarly, f a prmary constrant s bndng, then the correspondng shadow value > 0. Smlarly for the converse. Ether the slack varable or shadow value have to zero! Interpretaton: Bndng constrants reflect lmted (scare) resources. Scare resources have value. The value of the scarce resource s the shadow value. Relaxng a bndng constrant wll mprove the objectve functon value by the shadow value for the constrant. Show graphcally You can solve ether the Prmal or Dual problem to get all the outputs for both problems (Z, x, y). What s the dual of the dual formulaton?

6 CEE 60 Davd Rosenberg p Identfy how LP outputs change wth changes n model nputs General Method Determne whch decson varables are Determne whch constrants are A. Changes n Rght Hand Sde Constrant Values How? [Hnt look at the Dual Problem outputs] Example 4. How wll the objectve functon value n Example change f: () The rght-hand value n the frst constrant s decreased from 4,000,000 to 3,900,000 gallons? () The rght-hand value of the second constrant s decreased from 2,000 to,900 sq ft? Soluton: In Excel, called Shadow Prce. In GAMS, margnal assocated wth constrants. In Smplex, the objectve functon coeffcents of slack varables n the fnal tableau. Range of bass the change n rght-hand-sde coeffcent values allowed before the bass soluton wll change. In Excel, ths s called the allowable ncrease and allowable decrease. In GAMS, follows nstructons at ex. B. Changes n Objectve Functon Coeffcents Graphcally the rotaton of the objectve functon hyperplane (slope) that wll move the optmal soluton to a new bass. Rotaton so slope s less than (greater than) slope of one of the bndng constrants Mathematcally the Reduced Cost From smplex, the reduced cost s the coeffcent n the fnal row assocated wth the objectve functon

7 CEE 60 Davd Rosenberg p. 7 Can also calculate drectly as ~ c c constrant n the dual formulaton! m j Allowable ncrease before change the bass: a j y j,.e., the slack assocated wth the th c new ~ c c Reduced Cost n Excel; In GAMS, Margnal assocated wth decson varables Interpretaton: C. Addng a New Constrant Two optons:. Optmal soluton stll feasble (satsfes the new constrant) => same soluton 2. Optmal soluton now nfeasble New optmal soluton Objectve functon value wll decrease (for a maxmzaton problem) => smaller feasble regon D. Parametrc Programmng Systematcally change one nput parameter value and see how results (optmal objectve functon and decson varable values) change. Most easly done by solvng the optmzaton model multple tmes (.e., embed n a programmng loop). Brute force approach to senstvty analyss. Cases to use: Change multple model parameters smultaneously Changes for one parameter value are far beyond ranges of values allowed by margnal (shadow value and reduced cost) analyss. Do not use when: Changes n parameter values are very small and wthn the ranges covered by margnal (shadow value and reduced cost) analyss. Example 5. How wll the objectve functon and decson varable values n Example change f the water requrement for tomatoes decreases from 2,000 gal/plant to,500, 000, and 500 gal/plant? At what water requrement level does the soluton bass change? Soluton: See GAMS fle Ex2--parametrc.gms.

SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING USING A CALCULATOR

SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING USING A CALCULATOR SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING USING A CALCULATOR Judth Aronow Rchard Jarvnen Independent Consultant Dept of Math/Stat 559 Frost Wnona State Unversty Beaumont, TX 7776 Wnona, MN 55987 aronowju@hal.lamar.edu

More information

5 The Primal-Dual Method

5 The Primal-Dual Method 5 The Prmal-Dual Method Orgnally desgned as a method for solvng lnear programs, where t reduces weghted optmzaton problems to smpler combnatoral ones, the prmal-dual method (PDM) has receved much attenton

More information

Support Vector Machines

Support Vector Machines /9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.

More information

GSLM Operations Research II Fall 13/14

GSLM Operations Research II Fall 13/14 GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are

More information

Solving two-person zero-sum game by Matlab

Solving two-person zero-sum game by Matlab Appled Mechancs and Materals Onlne: 2011-02-02 ISSN: 1662-7482, Vols. 50-51, pp 262-265 do:10.4028/www.scentfc.net/amm.50-51.262 2011 Trans Tech Publcatons, Swtzerland Solvng two-person zero-sum game by

More information

Multicriteria Decision Making

Multicriteria Decision Making Multcrtera Decson Makng Andrés Ramos (Andres.Ramos@comllas.edu) Pedro Sánchez (Pedro.Sanchez@comllas.edu) Sonja Wogrn (Sonja.Wogrn@comllas.edu) Contents 1. Basc concepts 2. Contnuous methods 3. Dscrete

More information

Optimization Methods: Integer Programming Integer Linear Programming 1. Module 7 Lecture Notes 1. Integer Linear Programming

Optimization Methods: Integer Programming Integer Linear Programming 1. Module 7 Lecture Notes 1. Integer Linear Programming Optzaton Methods: Integer Prograng Integer Lnear Prograng Module Lecture Notes Integer Lnear Prograng Introducton In all the prevous lectures n lnear prograng dscussed so far, the desgn varables consdered

More information

Kent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming

Kent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming CS 4/560 Desgn and Analyss of Algorthms Kent State Unversty Dept. of Math & Computer Scence LECT-6 Dynamc Programmng 2 Dynamc Programmng Dynamc Programmng, lke the dvde-and-conquer method, solves problems

More information

Classification / Regression Support Vector Machines

Classification / Regression Support Vector Machines Classfcaton / Regresson Support Vector Machnes Jeff Howbert Introducton to Machne Learnng Wnter 04 Topcs SVM classfers for lnearly separable classes SVM classfers for non-lnearly separable classes SVM

More information

NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS

NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS ARPN Journal of Engneerng and Appled Scences 006-017 Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana

More information

A mathematical programming approach to the analysis, design and scheduling of offshore oilfields

A mathematical programming approach to the analysis, design and scheduling of offshore oilfields 17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 A mathematcal programmng approach to the analyss, desgn and

More information

Proposed Simplex Method For Fuzzy Linear Programming With Fuzziness at the Right Hand Side

Proposed Simplex Method For Fuzzy Linear Programming With Fuzziness at the Right Hand Side IOSR Journal of Mathematcs (IOSR-JM) e-issn: 8-8, p-issn: 9-X. Volume, Issue Ver. II (May - Jun. ), PP 8- www.osrournals.org Proposed Smple Method For Fuzzy Lnear Programmng Wth Fuzzness at the Rght Hand

More information

Intra-Parametric Analysis of a Fuzzy MOLP

Intra-Parametric Analysis of a Fuzzy MOLP Intra-Parametrc Analyss of a Fuzzy MOLP a MIAO-LING WANG a Department of Industral Engneerng and Management a Mnghsn Insttute of Technology and Hsnchu Tawan, ROC b HSIAO-FAN WANG b Insttute of Industral

More information

Analysis of Continuous Beams in General

Analysis of Continuous Beams in General Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support,

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 5 Luca Trevisan September 7, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 5 Luca Trevisan September 7, 2017 U.C. Bereley CS294: Beyond Worst-Case Analyss Handout 5 Luca Trevsan September 7, 207 Scrbed by Haars Khan Last modfed 0/3/207 Lecture 5 In whch we study the SDP relaxaton of Max Cut n random graphs. Quc

More information

Sum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints

Sum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints Australan Journal of Basc and Appled Scences, 2(4): 1204-1208, 2008 ISSN 1991-8178 Sum of Lnear and Fractonal Multobjectve Programmng Problem under Fuzzy Rules Constrants 1 2 Sanjay Jan and Kalash Lachhwan

More information

An Application of Network Simplex Method for Minimum Cost Flow Problems

An Application of Network Simplex Method for Minimum Cost Flow Problems BALKANJM 0 (0) -0 Contents lsts avalable at BALKANJM BALKAN JOURNAL OF MATHEMATICS journal homepage: www.balkanjm.com An Applcaton of Network Smplex Method for Mnmum Cost Flow Problems Ergun EROGLU *a

More information

An Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation

An Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation 17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed

More information

11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS Copng wth NP-completeness 11. APPROXIMATION ALGORITHMS load balancng center selecton prcng method: vertex cover LP roundng: vertex cover generalzed load balancng knapsack problem Q. Suppose I need to solve

More information

Greedy Technique - Definition

Greedy Technique - Definition Greedy Technque Greedy Technque - Defnton The greedy method s a general algorthm desgn paradgm, bult on the follong elements: confguratons: dfferent choces, collectons, or values to fnd objectve functon:

More information

Scheduling with Integer Time Budgeting for Low-Power Optimization

Scheduling with Integer Time Budgeting for Low-Power Optimization Schedlng wth Integer Tme Bdgetng for Low-Power Optmzaton We Jang, Zhr Zhang, Modrag Potkonjak and Jason Cong Compter Scence Department Unversty of Calforna, Los Angeles Spported by NSF, SRC. Otlne Introdcton

More information

Ecient Computation of the Most Probable Motion from Fuzzy. Moshe Ben-Ezra Shmuel Peleg Michael Werman. The Hebrew University of Jerusalem

Ecient Computation of the Most Probable Motion from Fuzzy. Moshe Ben-Ezra Shmuel Peleg Michael Werman. The Hebrew University of Jerusalem Ecent Computaton of the Most Probable Moton from Fuzzy Correspondences Moshe Ben-Ezra Shmuel Peleg Mchael Werman Insttute of Computer Scence The Hebrew Unversty of Jerusalem 91904 Jerusalem, Israel Emal:

More information

Support Vector Machines. CS534 - Machine Learning

Support Vector Machines. CS534 - Machine Learning Support Vector Machnes CS534 - Machne Learnng Perceptron Revsted: Lnear Separators Bnar classfcaton can be veed as the task of separatng classes n feature space: b > 0 b 0 b < 0 f() sgn( b) Lnear Separators

More information

The Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique

The Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique //00 :0 AM Outlne and Readng The Greedy Method The Greedy Method Technque (secton.) Fractonal Knapsack Problem (secton..) Task Schedulng (secton..) Mnmum Spannng Trees (secton.) Change Money Problem Greedy

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Decson surface s a hyperplane (lne n 2D) n feature space (smlar to the Perceptron) Arguably, the most mportant recent dscovery n machne learnng In a nutshell: map the data to a predetermned

More information

Smoothing Spline ANOVA for variable screening

Smoothing Spline ANOVA for variable screening Smoothng Splne ANOVA for varable screenng a useful tool for metamodels tranng and mult-objectve optmzaton L. Rcco, E. Rgon, A. Turco Outlne RSM Introducton Possble couplng Test case MOO MOO wth Game Theory

More information

An Application of LFP Method for Sintering Ore Ratio

An Application of LFP Method for Sintering Ore Ratio An Applcaton of LFP Method for Snterng Ore Rato X Cheng, Kalng Pan, and Yunfeng Ma School of Management, Wuhan Unversty of Scence and Technology, P.R.Chna, 408 suxn49@63.com Abstract. The proper rato decson

More information

Course Introduction. Algorithm 8/31/2017. COSC 320 Advanced Data Structures and Algorithms. COSC 320 Advanced Data Structures and Algorithms

Course Introduction. Algorithm 8/31/2017. COSC 320 Advanced Data Structures and Algorithms. COSC 320 Advanced Data Structures and Algorithms Course Introducton Course Topcs Exams, abs, Proects A quc loo at a few algorthms 1 Advanced Data Structures and Algorthms Descrpton: We are gong to dscuss algorthm complexty analyss, algorthm desgn technques

More information

Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations

Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations Fxng Max-Product: Convergent Message Passng Algorthms for MAP LP-Relaxatons Amr Globerson Tomm Jaakkola Computer Scence and Artfcal Intellgence Laboratory Massachusetts Insttute of Technology Cambrdge,

More information

OPL: a modelling language

OPL: a modelling language OPL: a modellng language Carlo Mannno (from OPL reference manual) Unversty of Oslo, INF-MAT60 - Autumn 00 (Mathematcal optmzaton) ILOG Optmzaton Programmng Language OPL s an Optmzaton Programmng Language

More information

Loop Transformations for Parallelism & Locality. Review. Scalar Expansion. Scalar Expansion: Motivation

Loop Transformations for Parallelism & Locality. Review. Scalar Expansion. Scalar Expansion: Motivation Loop Transformatons for Parallelsm & Localty Last week Data dependences and loops Loop transformatons Parallelzaton Loop nterchange Today Scalar expanson for removng false dependences Loop nterchange Loop

More information

A Facet Generation Procedure. for solving 0/1 integer programs

A Facet Generation Procedure. for solving 0/1 integer programs A Facet Generaton Procedure for solvng 0/ nteger programs by Gyana R. Parja IBM Corporaton, Poughkeepse, NY 260 Radu Gaddov Emery Worldwde Arlnes, Vandala, Oho 45377 and Wlbert E. Wlhelm Teas A&M Unversty,

More information

Efficient Load-Balanced IP Routing Scheme Based on Shortest Paths in Hose Model. Eiji Oki May 28, 2009 The University of Electro-Communications

Efficient Load-Balanced IP Routing Scheme Based on Shortest Paths in Hose Model. Eiji Oki May 28, 2009 The University of Electro-Communications Effcent Loa-Balance IP Routng Scheme Base on Shortest Paths n Hose Moel E Ok May 28, 2009 The Unversty of Electro-Communcatons Ok Lab. Semnar, May 28, 2009 1 Outlne Backgroun on IP routng IP routng strategy

More information

An Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices

An Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices Internatonal Mathematcal Forum, Vol 7, 2012, no 52, 2549-2554 An Applcaton of the Dulmage-Mendelsohn Decomposton to Sparse Null Space Bases of Full Row Rank Matrces Mostafa Khorramzadeh Department of Mathematcal

More information

Modeling and Solving Nontraditional Optimization Problems Session 2a: Conic Constraints

Modeling and Solving Nontraditional Optimization Problems Session 2a: Conic Constraints Modelng and Solvng Nontradtonal Optmzaton Problems Sesson 2a: Conc Constrants Robert Fourer Industral Engneerng & Management Scences Northwestern Unversty AMPL Optmzaton LLC 4er@northwestern.edu 4er@ampl.com

More information

Biostatistics 615/815

Biostatistics 615/815 The E-M Algorthm Bostatstcs 615/815 Lecture 17 Last Lecture: The Smplex Method General method for optmzaton Makes few assumptons about functon Crawls towards mnmum Some recommendatons Multple startng ponts

More information

Loop Permutation. Loop Transformations for Parallelism & Locality. Legality of Loop Interchange. Loop Interchange (cont)

Loop Permutation. Loop Transformations for Parallelism & Locality. Legality of Loop Interchange. Loop Interchange (cont) Loop Transformatons for Parallelsm & Localty Prevously Data dependences and loops Loop transformatons Parallelzaton Loop nterchange Today Loop nterchange Loop transformatons and transformaton frameworks

More information

CHAPTER 2 PROPOSED IMPROVED PARTICLE SWARM OPTIMIZATION

CHAPTER 2 PROPOSED IMPROVED PARTICLE SWARM OPTIMIZATION 24 CHAPTER 2 PROPOSED IMPROVED PARTICLE SWARM OPTIMIZATION The present chapter proposes an IPSO approach for multprocessor task schedulng problem wth two classfcatons, namely, statc ndependent tasks and

More information

OPTIMIZATION OF PROCESS PARAMETERS USING AHP AND TOPSIS WHEN TURNING AISI 1040 STEEL WITH COATED TOOLS

OPTIMIZATION OF PROCESS PARAMETERS USING AHP AND TOPSIS WHEN TURNING AISI 1040 STEEL WITH COATED TOOLS Internatonal Journal of Mechancal Engneerng and Technology (IJMET) Volume 7, Issue 6, November December 2016, pp.483 492, Artcle ID: IJMET_07_06_047 Avalable onlne at http://www.aeme.com/jmet/ssues.asp?jtype=ijmet&vtype=7&itype=6

More information

CS 534: Computer Vision Model Fitting

CS 534: Computer Vision Model Fitting CS 534: Computer Vson Model Fttng Sprng 004 Ahmed Elgammal Dept of Computer Scence CS 534 Model Fttng - 1 Outlnes Model fttng s mportant Least-squares fttng Maxmum lkelhood estmaton MAP estmaton Robust

More information

Introduction to linear programming

Introduction to linear programming Introducton to lnear programmng Overvew Lnear programmng Mathematcal and lnear programs Lnear programmng and resoluton methods; Modelng n (nteger) lnear programmng; Schedulng problems; Optmzaton problems

More information

Programming Assignment Six. Semester Calendar. 1D Excel Worksheet Arrays. Review VBA Arrays from Excel. Programming Assignment Six May 2, 2017

Programming Assignment Six. Semester Calendar. 1D Excel Worksheet Arrays. Review VBA Arrays from Excel. Programming Assignment Six May 2, 2017 Programmng Assgnment Sx, 07 Programmng Assgnment Sx Larry Caretto Mechancal Engneerng 09 Computer Programmng for Mechancal Engneers Outlne Practce quz for actual quz on Thursday Revew approach dscussed

More information

Improving Low Density Parity Check Codes Over the Erasure Channel. The Nelder Mead Downhill Simplex Method. Scott Stransky

Improving Low Density Parity Check Codes Over the Erasure Channel. The Nelder Mead Downhill Simplex Method. Scott Stransky Improvng Low Densty Party Check Codes Over the Erasure Channel The Nelder Mead Downhll Smplex Method Scott Stransky Programmng n conjuncton wth: Bors Cukalovc 18.413 Fnal Project Sprng 2004 Page 1 Abstract

More information

Radial Basis Functions

Radial Basis Functions Radal Bass Functons Mesh Reconstructon Input: pont cloud Output: water-tght manfold mesh Explct Connectvty estmaton Implct Sgned dstance functon estmaton Image from: Reconstructon and Representaton of

More information

Problem Definitions and Evaluation Criteria for Computational Expensive Optimization

Problem Definitions and Evaluation Criteria for Computational Expensive Optimization Problem efntons and Evaluaton Crtera for Computatonal Expensve Optmzaton B. Lu 1, Q. Chen and Q. Zhang 3, J. J. Lang 4, P. N. Suganthan, B. Y. Qu 6 1 epartment of Computng, Glyndwr Unversty, UK Faclty

More information

Feedback Min-Max Model Predictive Control Based on a Quadratic Cost Function

Feedback Min-Max Model Predictive Control Based on a Quadratic Cost Function Proceedngs of the 26 Amercan Control Conference Mnneapols, Mnnesota, USA, June 14-16, 26 WeC5.5 Feedback Mn-Max Model Predctve Control Based on a Quadratc Cost Functon D. Muñoz de la Peña,T.Alamo, A. Bemporad

More information

UNIT 2 : INEQUALITIES AND CONVEX SETS

UNIT 2 : INEQUALITIES AND CONVEX SETS UNT 2 : NEQUALTES AND CONVEX SETS ' Structure 2. ntroducton Objectves, nequaltes and ther Graphs Convex Sets and ther Geometry Noton of Convex Sets Extreme Ponts of Convex Set Hyper Planes and Half Spaces

More information

An interactive fuzzy multi-objective optimization method for engineering design

An interactive fuzzy multi-objective optimization method for engineering design ARTICLE IN PRESS Engneerng Applcatons of Artfcal Intellgence 19 (2006) 451 460 www.elsever.com/locate/engappa An nteractve fuzzy mult-objectve optmzaton method for engneerng desgn Hong-Zhong Huang a,,

More information

MODULE - 9 LECTURE NOTES 1 FUZZY OPTIMIZATION

MODULE - 9 LECTURE NOTES 1 FUZZY OPTIMIZATION Water Resources Systems Plannng an Management: vance Tocs Fuzzy Otmzaton MODULE - 9 LECTURE NOTES FUZZY OPTIMIZTION INTRODUCTION The moels scusse so far are crs an recse n nature. The term crs means chotonomous.e.,

More information

Monte Carlo Rendering

Monte Carlo Rendering Monte Carlo Renderng Last Tme? Modern Graphcs Hardware Cg Programmng Language Gouraud Shadng vs. Phong Normal Interpolaton Bump, Dsplacement, & Envronment Mappng Cg Examples G P R T F P D Today Does Ray

More information

Face Recognition University at Buffalo CSE666 Lecture Slides Resources:

Face Recognition University at Buffalo CSE666 Lecture Slides Resources: Face Recognton Unversty at Buffalo CSE666 Lecture Sldes Resources: http://www.face-rec.org/algorthms/ Overvew of face recognton algorthms Correlaton - Pxel based correspondence between two face mages Structural

More information

Machine Learning. Support Vector Machines. (contains material adapted from talks by Constantin F. Aliferis & Ioannis Tsamardinos, and Martin Law)

Machine Learning. Support Vector Machines. (contains material adapted from talks by Constantin F. Aliferis & Ioannis Tsamardinos, and Martin Law) Machne Learnng Support Vector Machnes (contans materal adapted from talks by Constantn F. Alfers & Ioanns Tsamardnos, and Martn Law) Bryan Pardo, Machne Learnng: EECS 349 Fall 2014 Support Vector Machnes

More information

11. HARMS How To: CSV Import

11. HARMS How To: CSV Import and Rsk System 11. How To: CSV Import Preparng the spreadsheet for CSV Import Refer to the spreadsheet template to ad algnng spreadsheet columns wth Data Felds. The spreadsheet s shown n the Appendx, an

More information

Solving Mixed Integer Formulation of the KS Maximization Problem Dual Based Methods and Results from Large Practical Problems

Solving Mixed Integer Formulation of the KS Maximization Problem Dual Based Methods and Results from Large Practical Problems Solvng Mxed Integer Formulaton of the KS Maxmzaton Problem Dual ased Methods and Results from Large Practcal Problems Debashsh Sarkar Management Scences roup CIT, New Jersey, USA (August 24, 2005) 1 Abstract

More information

MILP. LP: max cx ' MILP: some integer. ILP: x integer BLP: x 0,1. x 1. x 2 2 2, c ,

MILP. LP: max cx ' MILP: some integer. ILP: x integer BLP: x 0,1. x 1. x 2 2 2, c , MILP LP: max cx ' s.t. Ax b x 0 MILP: some nteger x max 6x 8x s.t. x x x 7 x, x 0 c A 6 8, 0 b 7 ILP: x nteger BLP: x 0, x 4 x, cx * * 0 4 5 6 x 06 Branch and Bound x 4 0 max 6x 8x s.t. xx x 7 x, x 0 x,

More information

TPL-Aware Displacement-driven Detailed Placement Refinement with Coloring Constraints

TPL-Aware Displacement-driven Detailed Placement Refinement with Coloring Constraints TPL-ware Dsplacement-drven Detaled Placement Refnement wth Colorng Constrants Tao Ln Iowa State Unversty tln@astate.edu Chrs Chu Iowa State Unversty cnchu@astate.edu BSTRCT To mnmze the effect of process

More information

INTEGER PROGRAMMING MODELING FOR THE CHINESE POSTMAN PROBLEMS

INTEGER PROGRAMMING MODELING FOR THE CHINESE POSTMAN PROBLEMS INTEGER PROGRAMMING MODELING FOR THE CHINESE POSTMAN PROBLEMS ABSTRACT Feng Junwen School of Economcs and Management, Nanng Unversty of Scence and Technology, Nanng, 2009, Chna As far as the tradtonal

More information

Message-Passing Algorithms for Quadratic Programming Formulations of MAP Estimation

Message-Passing Algorithms for Quadratic Programming Formulations of MAP Estimation Message-Passng Algorthms for Quadratc Programmng Formulatons of MAP Estmaton Akshat Kumar Department of Computer Scence Unversty of Massachusetts Amherst akshat@cs.umass.edu Shlomo Zlbersten Department

More information

Design of Structure Optimization with APDL

Design of Structure Optimization with APDL Desgn of Structure Optmzaton wth APDL Yanyun School of Cvl Engneerng and Archtecture, East Chna Jaotong Unversty Nanchang 330013 Chna Abstract In ths paper, the desgn process of structure optmzaton wth

More information

Parallel Numerics. 1 Preconditioning & Iterative Solvers (From 2016)

Parallel Numerics. 1 Preconditioning & Iterative Solvers (From 2016) Technsche Unverstät München WSe 6/7 Insttut für Informatk Prof. Dr. Thomas Huckle Dpl.-Math. Benjamn Uekermann Parallel Numercs Exercse : Prevous Exam Questons Precondtonng & Iteratve Solvers (From 6)

More information

Lecture 15: Memory Hierarchy Optimizations. I. Caches: A Quick Review II. Iteration Space & Loop Transformations III.

Lecture 15: Memory Hierarchy Optimizations. I. Caches: A Quick Review II. Iteration Space & Loop Transformations III. Lecture 15: Memory Herarchy Optmzatons I. Caches: A Quck Revew II. Iteraton Space & Loop Transformatons III. Types of Reuse ALSU 7.4.2-7.4.3, 11.2-11.5.1 15-745: Memory Herarchy Optmzatons Phllp B. Gbbons

More information

AIR FORCE INSTITUTE OF TECHNOLOGY

AIR FORCE INSTITUTE OF TECHNOLOGY `` PLANNING COVERAGE OF POINTS OF INTEREST VIA MULTIPLE IMAGING SURVEILLANCE ASSETS: A MULTI-MODAL APPROACH THESIS Sarah E. Jackson, Captan, USAF AFIT/GOR/ENS/03-11 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY

More information

Range images. Range image registration. Examples of sampling patterns. Range images and range surfaces

Range images. Range image registration. Examples of sampling patterns. Range images and range surfaces Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples

More information

LU Decomposition Method Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America

LU Decomposition Method Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America nbm_sle_sm_ludecomp.nb 1 LU Decomposton Method Jame Trahan, Autar Kaw, Kevn Martn Unverst of South Florda Unted States of Amerca aw@eng.usf.edu nbm_sle_sm_ludecomp.nb 2 Introducton When solvng multple

More information

Life Tables (Times) Summary. Sample StatFolio: lifetable times.sgp

Life Tables (Times) Summary. Sample StatFolio: lifetable times.sgp Lfe Tables (Tmes) Summary... 1 Data Input... 2 Analyss Summary... 3 Survval Functon... 5 Log Survval Functon... 6 Cumulatve Hazard Functon... 7 Percentles... 7 Group Comparsons... 8 Summary The Lfe Tables

More information

How Accurately Can We Model Timing In A Placement Engine?

How Accurately Can We Model Timing In A Placement Engine? How Accurately Can We Model Tmng In A Placement Engne? Amt Chowdhary, Karth Raagopal, Satsh Venatesan, Tung Cao, Vladmr Tourn, Yegna Parasuram, Bll Halpn Intel Corporaton Serra Desgn Automaton Synplcty,

More information

Cooperative UAV Trajectory Planning with Multiple Dynamic Targets

Cooperative UAV Trajectory Planning with Multiple Dynamic Targets AIAA Gudance, avgaton, and Control Conference 2-5 August 200, Toronto, Ontaro Canada AIAA 200-8437 Cooperatve UAV Trajectory Plannng wth Multple Dynamc Targets Zhenshen Qu and Xangmng X 2 Harbn Insttute

More information

陳申岳 S-Y. Chen, 2007, Gradient-Based Structural and CFD Global Shape Optimization with SmartDO and the Response Smoothing Technology, Proceeding of

陳申岳 S-Y. Chen, 2007, Gradient-Based Structural and CFD Global Shape Optimization with SmartDO and the Response Smoothing Technology, Proceeding of 陳申岳 S-Y. Chen, 2007, Gradent-Based Structural and CFD Global Shape Optmzaton wth SmartDO and the Response Smoothng Technology, Proceedng of the 7 th World Congress of Structural and Multdscplnary Optmzaton

More information

Programming in Fortran 90 : 2017/2018

Programming in Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Exercse 1 : Evaluaton of functon dependng on nput Wrte a program who evaluate the functon f (x,y) for any two user specfed values

More information

Taxonomy of Large Margin Principle Algorithms for Ordinal Regression Problems

Taxonomy of Large Margin Principle Algorithms for Ordinal Regression Problems Taxonomy of Large Margn Prncple Algorthms for Ordnal Regresson Problems Amnon Shashua Computer Scence Department Stanford Unversty Stanford, CA 94305 emal: shashua@cs.stanford.edu Anat Levn School of Computer

More information

Solutions to Programming Assignment Five Interpolation and Numerical Differentiation

Solutions to Programming Assignment Five Interpolation and Numerical Differentiation College of Engneerng and Coputer Scence Mechancal Engneerng Departent Mechancal Engneerng 309 Nuercal Analyss of Engneerng Systes Sprng 04 Nuber: 537 Instructor: Larry Caretto Solutons to Prograng Assgnent

More information

A New Approach For the Ranking of Fuzzy Sets With Different Heights

A New Approach For the Ranking of Fuzzy Sets With Different Heights New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts Pushpnder Sngh School of Mathematcs Computer pplcatons Thapar Unversty, Patala-7 00 Inda pushpndersnl@gmalcom STCT ankng of fuzzy sets plays

More information

Multiobjective fuzzy optimization method

Multiobjective fuzzy optimization method Buletnul Ştnţfc al nverstăţ "Poltehnca" dn Tmşoara Sera ELECTRONICĂ ş TELECOMNICAŢII TRANSACTIONS on ELECTRONICS and COMMNICATIONS Tom 49(63, Fasccola, 24 Multobjectve fuzzy optmzaton method Gabrel Oltean

More information

Polyhedral Compilation Foundations

Polyhedral Compilation Foundations Polyhedral Complaton Foundatons Lous-Noël Pouchet pouchet@cse.oho-state.edu Dept. of Computer Scence and Engneerng, the Oho State Unversty Feb 8, 200 888., Class # Introducton: Polyhedral Complaton Foundatons

More information

Design for Reliability: Case Studies in Manufacturing Process Synthesis

Design for Reliability: Case Studies in Manufacturing Process Synthesis Desgn for Relablty: Case Studes n Manufacturng Process Synthess Y. Lawrence Yao*, and Chao Lu Department of Mechancal Engneerng, Columba Unversty, Mudd Bldg., MC 473, New York, NY 7, USA * Correspondng

More information

CS246: Mining Massive Datasets Jure Leskovec, Stanford University

CS246: Mining Massive Datasets Jure Leskovec, Stanford University CS46: Mnng Massve Datasets Jure Leskovec, Stanford Unversty http://cs46.stanford.edu /19/013 Jure Leskovec, Stanford CS46: Mnng Massve Datasets, http://cs46.stanford.edu Perceptron: y = sgn( x Ho to fnd

More information

Alternating Direction Method of Multipliers Implementation Using Apache Spark

Alternating Direction Method of Multipliers Implementation Using Apache Spark Alternatng Drecton Method of Multplers Implementaton Usng Apache Spark Deterch Lawson June 4, 2014 1 Introducton Many applcaton areas n optmzaton have benefted from recent trends towards massve datasets.

More information

Evaluation of Two Lagrangian Dual Optimization Algorithms for Large-Scale Unit Commitment Problems

Evaluation of Two Lagrangian Dual Optimization Algorithms for Large-Scale Unit Commitment Problems Journal of Electrcal Engneerng & Technology Vol. 7, No., pp. 7~22, 202 7 http://dx.do.org/0.5370/jeet.202.7..7 Evaluaton of Two Lagrangan Dual Optmzaton Algorthms for Large-Scale Unt Commtment Problems

More information

Meta-heuristics for Multidimensional Knapsack Problems

Meta-heuristics for Multidimensional Knapsack Problems 2012 4th Internatonal Conference on Computer Research and Development IPCSIT vol.39 (2012) (2012) IACSIT Press, Sngapore Meta-heurstcs for Multdmensonal Knapsack Problems Zhbao Man + Computer Scence Department,

More information

The Codesign Challenge

The Codesign Challenge ECE 4530 Codesgn Challenge Fall 2007 Hardware/Software Codesgn The Codesgn Challenge Objectves In the codesgn challenge, your task s to accelerate a gven software reference mplementaton as fast as possble.

More information

3. CR parameters and Multi-Objective Fitness Function

3. CR parameters and Multi-Objective Fitness Function 3 CR parameters and Mult-objectve Ftness Functon 41 3. CR parameters and Mult-Objectve Ftness Functon 3.1. Introducton Cogntve rados dynamcally confgure the wreless communcaton system, whch takes beneft

More information

Learning to Project in Multi-Objective Binary Linear Programming

Learning to Project in Multi-Objective Binary Linear Programming Learnng to Project n Mult-Objectve Bnary Lnear Programmng Alvaro Serra-Altamranda Department of Industral and Management System Engneerng, Unversty of South Florda, Tampa, FL, 33620 USA, amserra@mal.usf.edu,

More information

9. BASIC programming: Control and Repetition

9. BASIC programming: Control and Repetition Am: In ths lesson, you wll learn: H. 9. BASIC programmng: Control and Repetton Scenaro: Moz s showng how some nterestng patterns can be generated usng math. Jyot [after seeng the nterestng graphcs]: Usng

More information

LARRY SNYDER DEPT. OF INDUSTRIAL AND SYSTEMS ENGINEERING CENTER FOR VALUE CHAIN RESEARCH LEHIGH UNIVERSITY

LARRY SNYDER DEPT. OF INDUSTRIAL AND SYSTEMS ENGINEERING CENTER FOR VALUE CHAIN RESEARCH LEHIGH UNIVERSITY Faclty Locaton Models: An Overvew 1 LARRY SNYDER DEPT. OF INDUSTRIAL AND SYSTEMS ENGINEERING CENTER FOR VALUE CHAIN RESEARCH LEHIGH UNIVERSITY EWO SEMINAR SERIES APRIL 21, 2010 Outlne Introducton Taxonomy

More information

5.0 Quality Assurance

5.0 Quality Assurance 5.0 Dr. Fred Omega Garces Analytcal Chemstry 25 Natural Scence, Mramar College Bascs of s what we do to get the rght answer for our purpose QA s planned and refers to planned and systematc producton processes

More information

A Saturation Binary Neural Network for Crossbar Switching Problem

A Saturation Binary Neural Network for Crossbar Switching Problem A Saturaton Bnary Neural Network for Crossbar Swtchng Problem Cu Zhang 1, L-Qng Zhao 2, and Rong-Long Wang 2 1 Department of Autocontrol, Laonng Insttute of Scence and Technology, Benx, Chna bxlkyzhangcu@163.com

More information

Data Mining For Multi-Criteria Energy Predictions

Data Mining For Multi-Criteria Energy Predictions Data Mnng For Mult-Crtera Energy Predctons Kashf Gll and Denns Moon Abstract We present a data mnng technque for mult-crtera predctons of wnd energy. A mult-crtera (MC) evolutonary computng method has

More information

Review of approximation techniques

Review of approximation techniques CHAPTER 2 Revew of appromaton technques 2. Introducton Optmzaton problems n engneerng desgn are characterzed by the followng assocated features: the objectve functon and constrants are mplct functons evaluated

More information

Shape Optimization of Shear-type Hysteretic Steel Damper for Building Frames using FEM-Analysis and Heuristic Approach

Shape Optimization of Shear-type Hysteretic Steel Damper for Building Frames using FEM-Analysis and Heuristic Approach The Seventh Chna-Japan-Korea Jont Symposum on Optmzaton of Structural and Mechancal Systems Huangshan, June, 18-21, 2012, Chna Shape Optmzaton of Shear-type Hysteretc Steel Damper for Buldng Frames usng

More information

SLAM Summer School 2006 Practical 2: SLAM using Monocular Vision

SLAM Summer School 2006 Practical 2: SLAM using Monocular Vision SLAM Summer School 2006 Practcal 2: SLAM usng Monocular Vson Javer Cvera, Unversty of Zaragoza Andrew J. Davson, Imperal College London J.M.M Montel, Unversty of Zaragoza. josemar@unzar.es, jcvera@unzar.es,

More information

LESSON 15: BODE PLOTS OF TRANSFER FUNCTIONS

LESSON 15: BODE PLOTS OF TRANSFER FUNCTIONS 10/8/015 1 LESSON 15: BODE PLOTS OF TRANSFER FUNCTIONS ET 438a Automatc Control Systems Technology Learnng Objectves After ths presentaton you wll be able to: Compute the magntude of a transfer functon

More information

Measuring the efficiency of Portuguese hospitals with DEA: an approach using the General Algebraic Modeling System

Measuring the efficiency of Portuguese hospitals with DEA: an approach using the General Algebraic Modeling System Measurng the effcency of Portuguese hosptals wth DEA: an approach usng the General Algebrac Modelng System ANTÓNIO XAVIER Faculdade de Cêncas e Tecnologas and CEFAGE-UE (Center For Advanced Studes n Management

More information

Abstract Ths paper ponts out an mportant source of necency n Smola and Scholkopf's Sequental Mnmal Optmzaton (SMO) algorthm for SVM regresson that s c

Abstract Ths paper ponts out an mportant source of necency n Smola and Scholkopf's Sequental Mnmal Optmzaton (SMO) algorthm for SVM regresson that s c Improvements to SMO Algorthm for SVM Regresson 1 S.K. Shevade S.S. Keerth C. Bhattacharyya & K.R.K. Murthy shrsh@csa.sc.ernet.n mpessk@guppy.mpe.nus.edu.sg cbchru@csa.sc.ernet.n murthy@csa.sc.ernet.n 1

More information

Dijkstra s Single Source Algorithm. All-Pairs Shortest Paths. Dynamic Programming Solution. Performance. Decision Sequence.

Dijkstra s Single Source Algorithm. All-Pairs Shortest Paths. Dynamic Programming Solution. Performance. Decision Sequence. All-Pars Shortest Paths Gven an n-vertex drected weghted graph, fnd a shortest path from vertex to vertex for each of the n vertex pars (,). Dstra s Sngle Source Algorthm Use Dstra s algorthm n tmes, once

More information

6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour

6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour 6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the

More information

INTRODUCTION INTRODUCTION. Moisès Graells Semi-continuous processes

INTRODUCTION INTRODUCTION. Moisès Graells Semi-continuous processes INTRODUCTION Mosès Graells (moses.graells@upc.edu) Barcelona / Catalona / Span Unverstat Poltècnca de Catalunya CEPIMA, PSE research group Emertus Prof. Lus Puganer IECR Specal Issue INTRODUCTION Sem-contnuous

More information

Lobachevsky State University of Nizhni Novgorod. Polyhedron. Quick Start Guide

Lobachevsky State University of Nizhni Novgorod. Polyhedron. Quick Start Guide Lobachevsky State Unversty of Nzhn Novgorod Polyhedron Quck Start Gude Nzhn Novgorod 2016 Contents Specfcaton of Polyhedron software... 3 Theoretcal background... 4 1. Interface of Polyhedron... 6 1.1.

More information

Optimal Caching Placement for D2D Assisted Wireless Caching Networks

Optimal Caching Placement for D2D Assisted Wireless Caching Networks Optmal Cachng Placement for DD Asssted Wreless Cachng Networks Jun Rao, ao Feng, Chenchen Yang, Zhyong Chen, and Bn Xa Department of Electronc Engneerng, Shangha Jao Tong Unversty, Shangha, P. R. Chna

More information

Harvard University CS 101 Fall 2005, Shimon Schocken. Assembler. Elements of Computing Systems 1 Assembler (Ch. 6)

Harvard University CS 101 Fall 2005, Shimon Schocken. Assembler. Elements of Computing Systems 1 Assembler (Ch. 6) Harvard Unversty CS 101 Fall 2005, Shmon Schocken Assembler Elements of Computng Systems 1 Assembler (Ch. 6) Why care about assemblers? Because Assemblers employ some nfty trcks Assemblers are the frst

More information

Quality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation

Quality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation Intellgent Informaton Management, 013, 5, 191-195 Publshed Onlne November 013 (http://www.scrp.org/journal/m) http://dx.do.org/10.36/m.013.5601 Qualty Improvement Algorthm for Tetrahedral Mesh Based on

More information