Introduction to Lexicographic Reverse Search: lrs
|
|
|
- Bertha Bennett
- 5 years ago
- Views:
Transcription
1 Introduction to Lexicographic Reverse Search: lrs June 29, 2012 Jayant Apte ASPITRG
2 Outline Introduction Lexicographic Simplex Algorithm Lex-positive and Lex min bases The pitfalls in reverse search Lexicographic Reverse search June 29,2012 Jayant Apte. ASPITRG 2
3 Introduction Representations of a Polyhedron Reverse Search: Some Trivia Reverse Search: The high level idea June 29,2012 Jayant Apte. ASPITRG 3
4 H-Representation of a Polyhedron June 29,2012 Jayant Apte. ASPITRG 4
5 V-Representation of a Polyhedron June 29,2012 Jayant Apte. ASPITRG 5
6 Switching between the two representations H-representation V-representation: The Vertex Enumeration problem Methods: Reverse Search, Lexicographic Reverse Search Double-description method Reverse search is the most robust method June 29,2012 Jayant Apte. ASPITRG 6
7 Brief history of Reverse Search Presented by David Avis and Komei Fukuda in a 1992 article A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra" A C implementation: rs Modified by Avis in 1998 to use lexicographic pivoting and implemented in rational arithmetic A C implementation: lrs June 29,2012 Jayant Apte. ASPITRG 7
8 Applications Finding the vertices and extreme rays given the H- representation of polyhedron Facet enumeration: V-representation to H- representation Enumerating all veritces of Voronoi diagram Computing volume of convex hull of set of points Estimation of number of vertices of polyhedron Nash equilibrium in non-cooperative bi-matrix games June 29,2012 Jayant Apte. ASPITRG 8
9 Relationship between Simplex Algorithm and Reverse Search Simplex Algorithm is used for solving linear programs Simplex Algorithm starts at any vertex of polyhedron defined by the constraint set Travels along the edges of polyhedron to find the optimum Reverse search runs simplex in reverse Traces all possible paths taken by Simplex algorithm in reverse June 29,2012 Jayant Apte. ASPITRG 9
10 Relationship between Simplex Algorithm and Reverse Search Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm June 29,2012 Jayant Apte. ASPITRG 10
11 Background: Lexicographic Order June 29,2012 Jayant Apte. ASPITRG 11
12 Vectors Comparing two vectors in lexicographic sense e.g. [5, 4, 1, 7, 4] [5, 4, 0, 9, 8 ] Lex-positive vector First non-zero element is positive e.g. [0, 4, -1, -7, -4], [5, -4, 1, -7, 4] are lexicographically positive vectors e.g. [ ], [ ] are lexicographically negative vectors Lex-min vector Out of [ ],[ ], [ ]; [ ] is lex-min vector June 29,2012 Jayant Apte. ASPITRG 12
13 Lexicographic Simplex Method June 29,2012 Jayant Apte. ASPITRG 13
14 The Linear Programming Problem Objective Function Constraint Set d-decision variables m-constraints June 29,2012 Jayant Apte. ASPITRG 14
15 Example June 29,2012 Jayant Apte. ASPITRG 15
16 Nature of Constrained Region (0.5,0.5,1.5) (1,1,1) (0,1,1) (0,0,1) (1,1,0) (0,1,0) (1,0,0) (0,0,0) June 29,2012 Jayant Apte. ASPITRG 16
17 Relationship between inequalities and Vertex: Geometric intuition of Degeneracy Highlighted inequalities are tight at (0,0,1) (0,0,1) June 29,2012 Jayant Apte. ASPITRG 17
18 Relationship between inequalities and Vertex: Geometric intuition of Degeneracy (0,0,1) June 29,2012 Jayant Apte. ASPITRG 18
19 Relationship between Co-Basis and Vertex: Geometric intuition of Degeneracy (0,0,1) is a degenerate vertex There are ways of specifying the vertex (0,0,1) In general this number are ways of representing a degenerate vertex, where n=total number of inequalities that are tight at that vertex d=number of dimensions This number can get very large very quickly. One of the places where simplex and by implication reverse search runs into problems. June 29,2012 Jayant Apte. ASPITRG 19
20 How to solve this LP? Rearrange inequalities to decide a starting vertex Introduce slack variables Convert to matrix form Convert to dictionary form June 29,2012 Jayant Apte. ASPITRG 20
21 Starting Simplex at a vertex: Rearrangement of the inequalities Rearrange the inequalities so that last 'd' inequalities contain initial vertex (0.5,0.5,1.5) June 29,2012 Jayant Apte. ASPITRG 21
22 Define Slack Variables Convert inequalities to equalities by introducing slack variables Original variables are called decision variables June 29,2012 Jayant Apte. ASPITRG 22
23 Our Example June 29,2012 Jayant Apte. ASPITRG 23
24 Our Example June 29,2012 Jayant Apte. ASPITRG 24
25 Terminology d decision variables m constraints Hence m slack variables Basis: B: Ordered m-tuple indexing set of m affinely independent columns of A Co-basis: N: Ordered d-tuple indexing remaining d columns : Sub-matrix of A indexed by B June 29,2012 Jayant Apte. ASPITRG 25
26 Our example We define initial basis as Hence initial co-basis is June 29,2012 Jayant Apte. ASPITRG 26
27 Our example is invertible Is a Basic Feasible Solution(BFS) June 29,2012 Jayant Apte. ASPITRG 27
28 Add a cost row Add a cost row at the top June 29,2012 Jayant Apte. ASPITRG 28
29 Our example June 29,2012 Jayant Apte. ASPITRG 29
30 Converting Linear program to the dictionary format June 29,2012 Jayant Apte. ASPITRG 30
31 Derive the dictionary from matrix form June 29,2012 Jayant Apte. ASPITRG 31
32 Our example Cost function value At current vertex The vertex June 29,2012 Jayant Apte. ASPITRG 32
33 Dictionary Format Change in Cost Corresponding to each Co-basic Variable Cost Corresponding To Dictionary Basic Indices Co-Basic Indices Solution Corresponding To Dictionary June 29,2012 Jayant Apte. ASPITRG 33
34 Redefine a bit Basis: B: Ordered m+1-tuple indexing set of m+1 affinely independent columns of dictionary & Dictionary is to serve as starting point Call this the solution of the initial dictionary or the initial Call this the initial implying that initial basis is identity matrix June 29,2012 Jayant Apte. ASPITRG 34
35 Our example Basis Cost The vertex June 29,2012 Jayant Apte. ASPITRG 35
36 Simplex Algorithm: The Ingredients It travels from one vertex to another, or more specifically one BFS to another BFS For that it uses pivoting operation to obtain the dictionary corresponding to new BFS. A co-basic variable assumes a nonzero value and is said to 'enter' the basis A basic variable 'leaves' the basis We get a new dictionary which is 'equivalent' to current dictionary in algebraic terms, i.e. It carries the same solution This new dictionary is obtained by means of 'pivot operation'. June 29,2012 Jayant Apte. ASPITRG 36
37 Choosing the entering variable Choose the entering variable that brings about an increase in cost One of these is supposed to assume a nonzero value Only a negative coefficient at will do as it will mean that is positive and results in an increase in cost If there are many such co-basic variables, choose the one with least subscript If there are no such co-basic variables, current dictionary is optimal and current BFS is the optimum June 29,2012 Jayant Apte. ASPITRG 37
38 Choosing entering variable Basis June 29,2012 Jayant Apte. ASPITRG 38
39 Choosing the leaving variable Recall lexicographic order Construct following matrix: June 29,2012 Jayant Apte. ASPITRG 39
40 Choosing the leaving variable Let and consider the column. If this column defines an extreme ray. Otherwise, let be the index such that is lexicographically minimum vector of: Such a minimum is unique as has full row rank This notation can be abbreviated as, In case there is no such we set June 29,2012 Jayant Apte. ASPITRG 40
41 Choosing the Leaving Variable Basis June 29,2012 Jayant Apte. ASPITRG 41
42 Lexicographic choice of leaving variable June 29,2012 Jayant Apte. ASPITRG 42
43 Where to get after each successive iteration? Let be current basis inverse and be the next basis inverse and differ only in 1 column We can take advantage of this fact and write Pre-multiplication by elementary matrix corresponds to row operations Hence, we get new basis inverse by some row operations June 29,2012 Jayant Apte. ASPITRG 43
44 The pivot Operation June 29,2012 Jayant Apte. ASPITRG 44
45 This is how to get after each is given as: successive iteration June 29,2012 Jayant Apte. ASPITRG 45
46 This is how to get after each successive iteration Row multiplication Operation Row corresponding to pivot element is divided by pivot element Hence pivot operation automatically yields new basis inverse June 29,2012 Jayant Apte. ASPITRG 46
47 Result of pivot (5,11) Basis: Co-Basis: Basis Identity column June 29,2012 Jayant Apte. ASPITRG 47
48 Solving Linear Programs: Simplex Algorithm June 29,2012 Jayant Apte. ASPITRG 48
49 Iteration 1 You Are Here. (0.5,0.5,1.5) (1,1,1) (0,1,1) (0,0,1) (1,1,0) (0,1,0) (1,0,0) (0,0,0) June 29,2012 Jayant Apte. ASPITRG 49
50 Iteration Basis Cost The vertex June 29,2012 Jayant Apte. ASPITRG 50
51 Pivot(5,11) June 29,2012 Jayant Apte. ASPITRG 51
52 Iteration 2 You Are Here. (0.5,0.5,1.5) (1,1,1) (0,1,1) (0,0,1) (1,1,0) (0,1,0) (1,0,0) (0,0,0) June 29,2012 Jayant Apte. ASPITRG 52
53 Iteration 2 Basis: Co-Basis: Basis June 29,2012 Jayant Apte. ASPITRG 53
54 pivot(4,10) June 29,2012 Jayant Apte. ASPITRG 54
55 Iteration 3 (0.5,0.5,1.5) (1,1,1) You Are Still Here. Blame degeneracy! (0,1,1) (0,0,1) (1,1,0) (0,1,0) (1,0,0) (0,0,0) June 29,2012 Jayant Apte. ASPITRG 55
56 Iteration 3 Basis: Co-Basis: Basis June 29,2012 Jayant Apte. ASPITRG 56
57 pivot(6,12) June 29,2012 Jayant Apte. ASPITRG 57
58 Iteration 4 (0.5,0.5,1.5) (1,1,1) (0,1,1) (0,0,1) (1,1,0) You Are Here. (0,1,0) (1,0,0) (0,0,0) June 29,2012 Jayant Apte. ASPITRG 58
59 Iteration 4 Basis: Co-Basis:4 5 6 Basis June 29,2012 Jayant Apte. ASPITRG 59
60 Observations Lexicographic simplex method doesn't travel through all the bases corresponding to a vertex It only visits bases that are lex-positive Recall A basis is lex-positive if the rows indexed d+1,...,m of D are lex-positive Lexicographic pivot rule is reversible! is reversible June 29,2012 Jayant Apte. ASPITRG 60
61 Reverse Search: The Recipe Start with dictionary corresponding to the optimal vertex Ask yourself 'What pivot would have landed me at this dictionary if i was running simplex?' Go to that dictionary by applying reverse pivot to current dictionary Ask the same question again Generate the so-called 'reverse search tree' June 29,2012 Jayant Apte. ASPITRG 61
62 What pivot would have landed me at this dictionary if i was running simplex? June 29,2012 Jayant Apte. ASPITRG 62
63 Pitfalls Initial basis not being the unique optimum basis Degeneracy of other vertices: One vertex corresponding to many (actually too many) bases Can result in program outputting a vertex more than once Earlier version rs suffered from all these June 29,2012 Jayant Apte. ASPITRG 63
64 Initial Basis not the unique optimum basis Reverse search must begin at the optimal vertex corresponding to any cost. In rs, one would have to find all possible bases corresponding to initial vertex and perform reverse search on them. In lrs, we define initial cost in such a way that initial vertex has a unique Basis We define cost as: This cost makes the top row of the dictionary June 29,2012 Jayant Apte. ASPITRG 64
65 Degeneracy of vertices Define lex-min bases Clearly if (i)-(iii) was to be true, we could do and get that is lexicographically smaller than summarizes the operation June 29,2012 Jayant Apte. ASPITRG 65
66 Evolution of lrs All possible Bases Very few vertices rs BFS >> Vertices A reduced set of bases Called lex-positive Bases lex-positive BFS> Vertices A reduced set of bases Called lex-min Bases Lex-min BFS = Vertices June 29,2012 Jayant Apte. ASPITRG 66
67 Recall these functions June 29,2012 Jayant Apte. ASPITRG 67
68 Pseudo-code June 29,2012 Jayant Apte. ASPITRG 68
69 Pseudo-code Examine each cobasic column of dictionary by reverse to see if there exists a lex positive pivot using this column Reverse will be executed d times for each dictionary X Reverse(a)=p Cobasis={a,b,c} Cobasis={p,b,c} 1 June 29,2012 Jayant Apte. ASPITRG 69
70 Pseudo-code X Reverse(a)=0 Cobasis={a,b,c} Reverse(b)=? June 29,2012 Jayant Apte. ASPITRG 70
71 Pseudo-code When whie loop terminates, return to the parent of current node: backtrack Correct j is restored by selectpivot Subsequent j=j+1 means next cobasic element of the parent comes under consideration Backtrack X Cobasis={a,b,c} Reverse(c)=0 Reverse(a)=0 Reverse(b)=0 June 29,2012 Jayant Apte. ASPITRG 71
72 Pseudo-code Execution continues until all the columns of the starting dictionary are examined June 29,2012 Jayant Apte. ASPITRG 72
73 Lexicographic Reverse Search on our example (0.5,0.5,1.5) (1,1,1) (0,1,1) (0,0,1) (1,1,0) You Are Here. (0,1,0) (1,0,0) (0,0,0) June 29,2012 Jayant Apte. ASPITRG 73
74 V=(0 0 0) N=( ) reverse(4)=0 V=(1 0 1) N=(4 11 8) reverse(11)=5 pivot(5,11) V=(1 0 0) N=( ) reverse(12)=8 pivot(8,12) reverse(11)=0 V=(1 1 0) N=(4 5 12) reverse(10)=4 pivot(4,10) reverse(10)=4 pivot(4,10) V=(0 1 1) N=(10 5 6) reverse(11)=5 pivot(5,11) V=0 1 0) N=( ) reverse(12)=9 pivot(9,12) reverse(12)=6reverse(5)=0 pivot(6,12) reverse(9)=0 V=(0 0 1) N=( ) reverse(10)=7 pivot(7,10) reverse(11)=6 pivot(6,11) V=(0 0 1) N=(7 11 9) reverse(8)=0 V=(0 1 1) N=(10 6 9) reverse(9)=0 reverse(12)=0 V=( ) N=(7 8 9) reverse(11)=8 pivot(8,11) reverse(9)=8 pivot(8,9) reverse(7)=0 V=(1 1 1) N=(5 6 9) reverse(8)=0 reverse(7)=6 pivot(6,7) reverse(9)=0 V=(1 0 1) N=(8 12 9) reverse(5)=0 reverse(6)=0 reverse(9)=0 V=( ) N=(6 8 9) reverse(6)=0 reverse(9)=5 pivot(5,9) reverse(8)=0 reverse(8)=0 reverse(12)=0 reverse(9)=0 V=(1 0 0) N=(6 8 5) reverse(6)=0 reverse(8)=0 reverse(5)=0 June 29,2012 Jayant Apte. ASPITRG 74
75 Conclusion Lexicographic reverse search is an improved reverse search for vertex enumeration It is robust- not affected by degeneracy Yet, it travels more bases than there are vertices, hence requiring lot of computational power June 29,2012 Jayant Apte. ASPITRG 75
76 Starting dictionary June 29,2012 Jayant Apte. ASPITRG 76
77 Result of pivot(4,10) (real notation) June 29,2012 Jayant Apte. ASPITRG 77
78 Result of pivot(8,12) June 29,2012 Jayant Apte. ASPITRG 78
79 Result of pivot(5,11) June 29,2012 Jayant Apte. ASPITRG 79
80 Now backtrack all the way to root June 29,2012 Jayant Apte. ASPITRG 80
81 Result of pivot(5,11) June 29,2012 Jayant Apte. ASPITRG 81
82 Result of pivot(4,10) June 29,2012 Jayant Apte. ASPITRG 82
83 Backtrack 1 step: pivot(10,4) June 29,2012 Jayant Apte. ASPITRG 83
84 Result of pivot(6,12) June 29,2012 Jayant Apte. ASPITRG 84
85 Backtrack all the way to root June 29,2012 Jayant Apte. ASPITRG 85
86 Result of pivot(9,12) June 29,2012 Jayant Apte. ASPITRG 86
87 Result of pivot (7,10) June 29,2012 Jayant Apte. ASPITRG 87
88 Result of pivot(8,11) June 29,2012 Jayant Apte. ASPITRG 88
89 Result of pivot(6,7) June 29,2012 Jayant Apte. ASPITRG 89
90 Result of pivot(6,10) June 29,2012 Jayant Apte. ASPITRG 90
91 Backtrack: pivot(9,5),pivot(7,6),pivot(11,8) June 29,2012 Jayant Apte. ASPITRG 91
92 Result of pivot(8,9) June 29,2012 Jayant Apte. ASPITRG 92
93 Backtrack pivot(9,8),pivot(10,7) June 29,2012 Jayant Apte. ASPITRG 93
94 Result of pivot(6,11) June 29,2012 Jayant Apte. ASPITRG 94
95 Backtrack to root June 29,2012 Jayant Apte. ASPITRG 95
96 Proof of uniqueness of initial basis Suppose there exists another lex-positive optimum basis B Recall D matrix. June 29,2012 Jayant Apte. ASPITRG 96
97 Proof of uniqueness of initial basis B B-B* B*-B B*={0,...m} N*={m+1,...,m+d} Consider pivoting from B* to B. Indices B-B*= must be brought into basis Every pivot will involve subtracting a certain row vector from cost row Since B is also optimum It follows that is lexicographicallly negative June 29,2012 Jayant Apte. ASPITRG 97
of a bimatrix game David Avis McGill University Gabriel Rosenberg Yale University Rahul Savani University of Warwick
Finding all Nash equilibria of a bimatrix game David Avis McGill University Gabriel Rosenberg Yale University Rahul Savani University of Warwick Bernhard von Stengel London School of Economics Nash equilibria
LP. Lecture 3. Chapter 3: degeneracy. degeneracy example cycling the lexicographic method other pivot rules the fundamental theorem of LP
LP. Lecture 3. Chapter 3: degeneracy. degeneracy example cycling the lexicographic method other pivot rules the fundamental theorem of LP 1 / 23 Repetition the simplex algorithm: sequence of pivots starting
Part 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.2. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Basic Feasible Solutions Key to the Algebra of the The Simplex Algorithm
1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
OPERATIONS RESEARCH. Linear Programming Problem
OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com MODULE - 2: Simplex Method for
Lecture slides by Kevin Wayne
LINEAR PROGRAMMING I a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm Lecture slides by Kevin Wayne Last updated on 7/25/17 11:09 AM Linear programming
Linear Programming. Linear Programming I. Lecture 1. Linear Programming. Linear Programming
Linear Programming Linear Programming Lecture Linear programming. Optimize a linear function subject to linear inequalities. (P) max " c j x j n j= n s. t. " a ij x j = b i # i # m j= x j 0 # j # n (P)
Week 2. The Simplex method was developed by Dantzig in the late 40-ties.
1 The Simplex method Week 2 The Simplex method was developed by Dantzig in the late 40-ties. 1.1 The standard form The simplex method is a general description algorithm that solves any LPproblem instance.
Supplementary lecture notes on linear programming. We will present an algorithm to solve linear programs of the form. maximize.
Cornell University, Fall 2016 Supplementary lecture notes on linear programming CS 6820: Algorithms 26 Sep 28 Sep 1 The Simplex Method We will present an algorithm to solve linear programs of the form
LINEAR PROGRAMMING I. a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm
Linear programming Linear programming. Optimize a linear function subject to linear inequalities. (P) max c j x j n j= n s. t. a ij x j = b i i m j= x j 0 j n (P) max c T x s. t. Ax = b Lecture slides
CSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming
CSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming January 26, 2018 1 / 38 Liability/asset cash-flow matching problem Recall the formulation of the problem: max w c 1 + p 1 e 1 = 150
Linear Programming and the Simplex method
Linear Programming and the Simplex method Harald Enzinger, Michael Rath Signal Processing and Speech Communication Laboratory Jan 9, 2012 Harald Enzinger, Michael Rath Jan 9, 2012 page 1/37 Outline Introduction
Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta
Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau
TIM 206 Lecture 3: The Simplex Method
TIM 206 Lecture 3: The Simplex Method Kevin Ross. Scribe: Shane Brennan (2006) September 29, 2011 1 Basic Feasible Solutions Have equation Ax = b contain more columns (variables) than rows (constraints),
Simplex method(s) for solving LPs in standard form
Simplex method: outline I The Simplex Method is a family of algorithms for solving LPs in standard form (and their duals) I Goal: identify an optimal basis, as in Definition 3.3 I Versions we will consider:
3. THE SIMPLEX ALGORITHM
Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).
IE 400: Principles of Engineering Management. Simplex Method Continued
IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for
Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method
The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 The perturbation method Recap max c T x (P ) s.t. Ax = b x 0 Assumption: B is a feasible
Simplex Algorithm Using Canonical Tableaus
41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau
Termination, Cycling, and Degeneracy
Chapter 4 Termination, Cycling, and Degeneracy We now deal first with the question, whether the simplex method terminates. The quick answer is no, if it is implemented in a careless way. Notice that we
2.098/6.255/ Optimization Methods Practice True/False Questions
2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence
The Simplex Algorithm and Goal Programming
The Simplex Algorithm and Goal Programming In Chapter 3, we saw how to solve two-variable linear programming problems graphically. Unfortunately, most real-life LPs have many variables, so a method is
ORF 522. Linear Programming and Convex Analysis
ORF 5 Linear Programming and Convex Analysis Initial solution and particular cases Marco Cuturi Princeton ORF-5 Reminder: Tableaux At each iteration, a tableau for an LP in standard form keeps track of....................
The Simplex Method for Solving a Linear Program Prof. Stephen Graves
The Simplex Method for Solving a Linear Program Prof. Stephen Graves Observations from Geometry feasible region is a convex polyhedron an optimum occurs at a corner point possible algorithm - search over
In Chapters 3 and 4 we introduced linear programming
SUPPLEMENT The Simplex Method CD3 In Chapters 3 and 4 we introduced linear programming and showed how models with two variables can be solved graphically. We relied on computer programs (WINQSB, Excel,
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 22th June 2005 Chapter 8: Finite Termination Recap On Monday, we established In the absence of degeneracy, the simplex method will
IE 400 Principles of Engineering Management. The Simplex Algorithm-I: Set 3
IE 4 Principles of Engineering Management The Simple Algorithm-I: Set 3 So far, we have studied how to solve two-variable LP problems graphically. However, most real life problems have more than two variables!
Math 16 - Practice Final
Math 16 - Practice Final May 28th 2007 Name: Instructions 1. In Part A, attempt every question. In Part B, attempt two of the five questions. If you attempt more you will only receive credit for your best
The Avis-Kalunzy Algorithm is designed to find a basic feasible solution (BFS) of a given set of constraints. Its input: A R m n and b R m such that
Lecture 4 Avis-Kaluzny and the Simplex Method Last time, we discussed some applications of Linear Programming, such as Max-Flow, Matching, and Vertex-Cover. The broad range of applications to Linear Programming
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, You wish to solve the IP below with a cutting plane technique.
IP Cut Homework from J and B Chapter 9: 14, 15, 16, 23, 24, 31 14. You wish to solve the IP below with a cutting plane technique. Maximize 4x 1 + 2x 2 + x 3 subject to 14x 1 + 10x 2 + 11x 3 32 10x 1 +
Introduction to Linear and Combinatorial Optimization (ADM I)
Introduction to Linear and Combinatorial Optimization (ADM I) Rolf Möhring based on the 20011/12 course by Martin Skutella TU Berlin WS 2013/14 1 General Remarks new flavor of ADM I introduce linear and
A Review of Linear Programming
A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex
Optimization WS 13/14:, by Y. Goldstein/K. Reinert, 9. Dezember 2013, 16: Linear programming. Optimization Problems
Optimization WS 13/14:, by Y. Goldstein/K. Reinert, 9. Dezember 2013, 16:38 2001 Linear programming Optimization Problems General optimization problem max{z(x) f j (x) 0,x D} or min{z(x) f j (x) 0,x D}
Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:
Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx
min 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,
(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. Problem 1 Consider
Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Module - 03 Simplex Algorithm Lecture 15 Infeasibility In this class, we
ECE 307 Techniques for Engineering Decisions
ECE 7 Techniques for Engineering Decisions Introduction to the Simple Algorithm George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ECE 7 5 9 George
IE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 3: Linear Programming, Continued Prof. John Gunnar Carlsson September 15, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 15, 2010
IE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 5: The Simplex method, continued Prof. John Gunnar Carlsson September 22, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 22, 2010
ORF 307: Lecture 2. Linear Programming: Chapter 2 Simplex Methods
ORF 307: Lecture 2 Linear Programming: Chapter 2 Simplex Methods Robert Vanderbei February 8, 2018 Slides last edited on February 8, 2018 http://www.princeton.edu/ rvdb Simplex Method for LP An Example.
CS Algorithms and Complexity
CS 50 - Algorithms and Complexity Linear Programming, the Simplex Method, and Hard Problems Sean Anderson 2/15/18 Portland State University Table of contents 1. The Simplex Method 2. The Graph Problem
+ 5x 2. = x x. + x 2. Transform the original system into a system x 2 = x x 1. = x 1
University of California, Davis Department of Agricultural and Resource Economics ARE 5 Optimization with Economic Applications Lecture Notes Quirino Paris The Pivot Method for Solving Systems of Equations...................................
Distributed Real-Time Control Systems. Lecture Distributed Control Linear Programming
Distributed Real-Time Control Systems Lecture 13-14 Distributed Control Linear Programming 1 Linear Programs Optimize a linear function subject to a set of linear (affine) constraints. Many problems can
Optimization (168) Lecture 7-8-9
Optimization (168) Lecture 7-8-9 Jesús De Loera UC Davis, Mathematics Wednesday, April 2, 2012 1 DEGENERACY IN THE SIMPLEX METHOD 2 DEGENERACY z =2x 1 x 2 + 8x 3 x 4 =1 2x 3 x 5 =3 2x 1 + 4x 2 6x 3 x 6
Lecture 5. x 1,x 2,x 3 0 (1)
Computational Intractability Revised 2011/6/6 Lecture 5 Professor: David Avis Scribe:Ma Jiangbo, Atsuki Nagao 1 Duality The purpose of this lecture is to introduce duality, which is an important concept
Week 3: Simplex Method I
Week 3: Simplex Method I 1 1. Introduction The simplex method computations are particularly tedious and repetitive. It attempts to move from one corner point of the solution space to a better corner point
Theory and Internet Protocols
Game Lecture 2: Linear Programming and Zero Sum Nash Equilibrium Xiaotie Deng AIMS Lab Department of Computer Science Shanghai Jiaotong University September 26, 2016 1 2 3 4 Standard Form (P) Outline
Linear Programming Redux
Linear Programming Redux Jim Bremer May 12, 2008 The purpose of these notes is to review the basics of linear programming and the simplex method in a clear, concise, and comprehensive way. The book contains
Review Solutions, Exam 2, Operations Research
Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To
Slack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0
Simplex Method Slack Variable Max Z= 3x 1 + 4x 2 + 5X 3 Subject to: X 1 + X 2 + X 3 20 3x 1 + 4x 2 + X 3 15 2X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Standard Form Max Z= 3x 1 +4x 2 +5X 3 + 0S 1 + 0S 2
Prelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions.
Prelude to the Simplex Algorithm The Algebraic Approach The search for extreme point solutions. 1 Linear Programming-1 x 2 12 8 (4,8) Max z = 6x 1 + 4x 2 Subj. to: x 1 + x 2
Sensitivity Analysis
Dr. Maddah ENMG 500 /9/07 Sensitivity Analysis Changes in the RHS (b) Consider an optimal LP solution. Suppose that the original RHS (b) is changed from b 0 to b new. In the following, we study the affect
Summary of the simplex method
MVE165/MMG631,Linear and integer optimization with applications The simplex method: degeneracy; unbounded solutions; starting solutions; infeasibility; alternative optimal solutions Ann-Brith Strömberg
9.1 Linear Programs in canonical form
9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems
Lecture 11 Linear programming : The Revised Simplex Method
Lecture 11 Linear programming : The Revised Simplex Method 11.1 The Revised Simplex Method While solving linear programming problem on a digital computer by regular simplex method, it requires storing
Math 354 Summer 2004 Solutions to review problems for Midterm #1
Solutions to review problems for Midterm #1 First: Midterm #1 covers Chapter 1 and 2. In particular, this means that it does not explicitly cover linear algebra. Also, I promise there will not be any proofs.
Linear Algebra Review: Linear Independence. IE418 Integer Programming. Linear Algebra Review: Subspaces. Linear Algebra Review: Affine Independence
Linear Algebra Review: Linear Independence IE418: Integer Programming Department of Industrial and Systems Engineering Lehigh University 21st March 2005 A finite collection of vectors x 1,..., x k R n
Introduction to Mathematical Programming IE406. Lecture 21. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2 IE406 Lecture 21 2 Branch and Bound Branch
Math Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations
The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear
15.081J/6.251J Introduction to Mathematical Programming. Lecture 18: The Ellipsoid method
15.081J/6.251J Introduction to Mathematical Programming Lecture 18: The Ellipsoid method 1 Outline Efficient algorithms and computational complexity Slide 1 The key geometric result behind the ellipsoid
February 17, Simplex Method Continued
15.053 February 17, 2005 Simplex Method Continued 1 Today s Lecture Review of the simplex algorithm. Formalizing the approach Alternative Optimal Solutions Obtaining an initial bfs Is the simplex algorithm
Math 273a: Optimization The Simplex method
Math 273a: Optimization The Simplex method Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 material taken from the textbook Chong-Zak, 4th Ed. Overview: idea and approach If a standard-form
Lectures 6, 7 and part of 8
Lectures 6, 7 and part of 8 Uriel Feige April 26, May 3, May 10, 2015 1 Linear programming duality 1.1 The diet problem revisited Recall the diet problem from Lecture 1. There are n foods, m nutrients,
CO 250 Final Exam Guide
Spring 2017 CO 250 Final Exam Guide TABLE OF CONTENTS richardwu.ca CO 250 Final Exam Guide Introduction to Optimization Kanstantsin Pashkovich Spring 2017 University of Waterloo Last Revision: March 4,
AM 121: Intro to Optimization
AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript
LP Duality: outline. Duality theory for Linear Programming. alternatives. optimization I Idea: polyhedra
LP Duality: outline I Motivation and definition of a dual LP I Weak duality I Separating hyperplane theorem and theorems of the alternatives I Strong duality and complementary slackness I Using duality
THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS. Operations Research I
LN/MATH2901/CKC/MS/2008-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS Operations Research I Definition (Linear Programming) A linear programming (LP) problem is characterized by linear functions
Lesson 27 Linear Programming; The Simplex Method
Lesson Linear Programming; The Simplex Method Math 0 April 9, 006 Setup A standard linear programming problem is to maximize the quantity c x + c x +... c n x n = c T x subject to constraints a x + a x
ORF 522. Linear Programming and Convex Analysis
ORF 522 Linear Programming and Convex Analysis The Simplex Method Marco Cuturi Princeton ORF-522 1 Reminder: Basic Feasible Solutions, Extreme points, Optima Some important theorems last time for standard
Chapter 1. Preliminaries
Introduction This dissertation is a reading of chapter 4 in part I of the book : Integer and Combinatorial Optimization by George L. Nemhauser & Laurence A. Wolsey. The chapter elaborates links between
Operations Research Lecture 2: Linear Programming Simplex Method
Operations Research Lecture 2: Linear Programming Simplex Method Notes taken by Kaiquan Xu@Business School, Nanjing University Mar 10th 2016 1 Geometry of LP 1.1 Graphical Representation and Solution Example
III. Linear Programming
III. Linear Programming Thomas Sauerwald Easter 2017 Outline Introduction Standard and Slack Forms Formulating Problems as Linear Programs Simplex Algorithm Finding an Initial Solution III. Linear Programming
LOWER BOUNDS FOR THE MAXIMUM NUMBER OF SOLUTIONS GENERATED BY THE SIMPLEX METHOD
Journal of the Operations Research Society of Japan Vol 54, No 4, December 2011, pp 191 200 c The Operations Research Society of Japan LOWER BOUNDS FOR THE MAXIMUM NUMBER OF SOLUTIONS GENERATED BY THE
The simplex algorithm
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,
Example. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive
3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically
Simplex Method for LP (II)
Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:
Lecture 2: The Simplex method. 1. Repetition of the geometrical simplex method. 2. Linear programming problems on standard form.
Lecture 2: The Simplex method. Repetition of the geometrical simplex method. 2. Linear programming problems on standard form. 3. The Simplex algorithm. 4. How to find an initial basic solution. Lecture
The augmented form of this LP is the following linear system of equations:
1 Consider the following LP given in standard form: max z = 5 x_1 + 2 x_2 Subject to 3 x_1 + 2 x_2 2400 x_2 800 2 x_1 1200 x_1, x_2 >= 0 The augmented form of this LP is the following linear system of
Metode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize
Metode Kuantitatif Bisnis Week 4 Linear Programming Simplex Method - Minimize Outlines Solve Linear Programming Model Using Graphic Solution Solve Linear Programming Model Using Simplex Method (Maximize)
From the Zonotope Construction to the Minkowski Addition of Convex Polytopes
From the Zonotope Construction to the Minkowski Addition of Convex Polytopes Komei Fukuda School of Computer Science, McGill University, Montreal, Canada Abstract A zonotope is the Minkowski addition of
Integer Programming ISE 418. Lecture 8. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 8 Dr. Ted Ralphs ISE 418 Lecture 8 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Duality for Mixed-Integer
Optimization Methods in Management Science
Optimization Methods in Management Science MIT 15.05 Recitation 8 TAs: Giacomo Nannicini, Ebrahim Nasrabadi At the end of this recitation, students should be able to: 1. Derive Gomory cut from fractional
Contents. 4.5 The(Primal)SimplexMethod NumericalExamplesoftheSimplexMethod
Contents 4 The Simplex Method for Solving LPs 149 4.1 Transformations to be Carried Out On an LP Model Before Applying the Simplex Method On It... 151 4.2 Definitions of Various Types of Basic Vectors
Decision Procedures An Algorithmic Point of View
An Algorithmic Point of View ILP References: Integer Programming / Laurence Wolsey Deciding ILPs with Branch & Bound Intro. To mathematical programming / Hillier, Lieberman Daniel Kroening and Ofer Strichman
Developing an Algorithm for LP Preamble to Section 3 (Simplex Method)
Moving from BFS to BFS Developing an Algorithm for LP Preamble to Section (Simplex Method) We consider LP given in standard form and let x 0 be a BFS. Let B ; B ; :::; B m be the columns of A corresponding
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1 Prof. Yiling Chen Fall 2018 Here are some practice questions to help to prepare for the midterm. The midterm will
AM 121: Intro to Optimization Models and Methods
AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 2: Intro to LP, Linear algebra review. Yiling Chen SEAS Lecture 2: Lesson Plan What is an LP? Graphical and algebraic correspondence Problems
OPRE 6201 : 3. Special Cases
OPRE 6201 : 3. Special Cases 1 Initialization: The Big-M Formulation Consider the linear program: Minimize 4x 1 +x 2 3x 1 +x 2 = 3 (1) 4x 1 +3x 2 6 (2) x 1 +2x 2 3 (3) x 1, x 2 0. Notice that there are
Linear programming: algebra
: algebra CE 377K March 26, 2015 ANNOUNCEMENTS Groups and project topics due soon Announcements Groups and project topics due soon Did everyone get my test email? Announcements REVIEW geometry Review geometry
Linear Programming. Jie Wang. University of Massachusetts Lowell Department of Computer Science. J. Wang (UMass Lowell) Linear Programming 1 / 47
Linear Programming Jie Wang University of Massachusetts Lowell Department of Computer Science J. Wang (UMass Lowell) Linear Programming 1 / 47 Linear function: f (x 1, x 2,..., x n ) = n a i x i, i=1 where
OPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM
OPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM Abstract These notes give a summary of the essential ideas and results It is not a complete account; see Winston Chapters 4, 5 and 6 The conventions and notation
Ω R n is called the constraint set or feasible set. x 1
1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We
3 The Simplex Method. 3.1 Basic Solutions
3 The Simplex Method 3.1 Basic Solutions In the LP of Example 2.3, the optimal solution happened to lie at an extreme point of the feasible set. This was not a coincidence. Consider an LP in general form,
Lecture Simplex Issues: Number of Pivots. ORIE 6300 Mathematical Programming I October 9, 2014
ORIE 6300 Mathematical Programming I October 9, 2014 Lecturer: David P. Williamson Lecture 14 Scribe: Calvin Wylie 1 Simplex Issues: Number of Pivots Question: How many pivots does the simplex algorithm
Lecture: Algorithms for LP, SOCP and SDP
1/53 Lecture: Algorithms for LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html wenzw@pku.edu.cn Acknowledgement:
The Graphical Method & Algebraic Technique for Solving LP s. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 1
The Graphical Method & Algebraic Technique for Solving LP s Métodos Cuantitativos M. En C. Eduardo Bustos Farías The Graphical Method for Solving LP s If LP models have only two variables, they can be
1 Simplex and Matrices
1 Simplex and Matrices We will begin with a review of matrix multiplication. A matrix is simply an array of numbers. If a given array has m rows and n columns, then it is called an m n (or m-by-n) matrix.