Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound
|
|
- Jodie Griffin
- 5 years ago
- Views:
Transcription
1 Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Samir Elhedhli Department of Management Sciences, University of Waterloo, Canada Page of 4 McMaster University, March 22, 24
2 . Motivation Interior-point methods are quite useful for solving largescale LPs, but they are not widely employed in branchand-bound algorithms because it is cumbersome to reoptimize an LP with an interior-point code after rows or columns have been added. E.L. Johnson, G. Nemhauser and M.W.P. Savelsbergh, Progress in linear programmingbased algorithms for integer programming: An exposition, Informs JOC,2:, 2. the simplex method has a significant advantage over interiorpoint methods within a branch and bound algorithm. In the branch-and-bound algorithm we can warm start the solution of each new subproblem with the optimal solution from the parent subproblem. Using the simplex method, a new optimal solution can easily be found after a few simplex iterations. With interior-point methods, warm starts are not as effective. J. E. Mitchell and B. Borchers Using an interior point method in a branch-and-bound Page 2 of 4 algorithm for integer programming, 99, revised 992, Rensselaer Polytechnic Institute, Troy, N.Y.
3 2. Branch-and-bound: LP bound Simplex Optimal Primal Optima Dual Basic NB branching constraints (Primal) Page 3 of 4 First Primal First Dual Basic N B Dual simplex
4 Replacing the simplex by an IPM is not going to work. (you may sacrifice warm starting). crossover is a prtial solution (Cplex) How should we apply IPMs? Save an approximate solution before termination (Gondzio et al Math. Prog.) Cutting plane/column generation context (Elhedhli and Goffin. 23. Math. Prog.) Page 4 of 4
5 3. A cutting plane context: Lagrangean Relaxation [MIP ] : min c T x s.t. Ax b (complicating) λ Bx d (easy) x x j integer for j J Lagrangean subproblem: [MIP λ ] : b T λ + min (c Ax) T λ s.t. Bx d x x j integer for j J Dual Lagrangean problem The dual master problem { max b T λ + v(ip λ ) } λ Page 5 of 4 max b T λ + λ s.t. (c Ax i ) T λ λ i I
6 max b T λ + λ s.t. λ + (Ax i ) T λ c T x i i I Where x i are the integer feasible points to the bounded set: {x : Bx d, x j integer for j J, x } The Dantzig-Wolfe master problem min s.t. i I (c T x i )α i α i (Ax i ) b i I α i = i I i I α i, i I. Page 6 of 4
7 4. Facts: The LR (DW) bound is at least as good as the LP bound. The subproblems are usually decomposable: solved much faster & allow the addition of multiple cuts (columns). Columns can be used as a basis for generating good feasible solutions (better incumbents). Exponential number of cuts (columns): need a cutting plane (column generation) approach May suffer from tailing effects. selected with care. Cuts (columns) should be Page 7 of 4
8 5. Using the LR bound within branch and bound: branchand-price Columns satisfying left branching Optimal Bound : zl Analytic center : a a a ( x, y, s ) Columns satisfying right branching Page 8 of 4 zl is valid Analytic center : a a ( y, s ) is dual feasible Dual IPM
9 6. Issues Careful branching: you may make the subproblerms more difficult to solve (considerably distort their structure) Early termination of the column gneeration/cutting plane scheme Warm starting Primal and dual heuristics Page 9 of 4
10 7. Example: The (binary) cutting stock problem Given: Rolls of length V and demand d l for items of lengths D l l =,..., L Find: the minimum number of rolls necessary to meet the demand D Page of 4 V D l If d l is either zero or one, we get the bin packing problem
11 8. MIP model for the bin packing problem { zk = when bin k is used Binary variables : y kl = when item l is assigned to bin k [BP P ] min s.t. K z k k= K y kl = l =,.., L Single sourcing constraints k= L D l y kl V z k k =,..., K Capacity constraints z k {, } k =,..., K; y kl {, } k =,..., K; l =,..., L; Page of 4 Question: Which decomposition to use? Answer: Depends on the quality of the bound.
12 9. The LP bound: LP The dual: [LP ] min s.t. K z k k= Dual variable K y kl = l ν l k= L D l y kl V z k k µ k z k k ρ k y kl k, l max s.t. L ν l K ρ k k= ν l D l µ k k, l V µ k ρ k = k µ k, ρ k k Solution: ρ k = ; ν l = D l V ; µ k = V ; L LP bound: D l V. Page 2 of 4
13 . Lagrangean relaxation of capacity constraints: LR2 min s.t. K z k k= K y kl = k= l L D l y kl V z k k :µ k z k {, } k y kl {, } k l min s.t. K k= L µ k D l y kl + K ( µ k V )z k K y kl = k= k= z k {, } k y kl {, } k l Optimal: µ k = V K L (Geoffrion): LR bound: V D ly kl = L k= V D l l K y kl = L k= D l V = LP Page 3 of 4
14 . Lagrangean relaxation of the single sourcing constraints: LR min s.t. K z k k= K y kl = l :λ l k= L D l y kl V z k k z k {, } k y kl {, } k l = The dual Lagrangean problem is : max λ Master problem: max Kθ + L s.t. λ l [KP λ ] min y,z ( λ l )y l + z l s.t. D l y l V z l y l, z =, l K identical - knapsacks λ l + Kv(KP λ ) L ( λ l )yl h + z h θ h =,..., H H is the index set of the integer solution to the sets l Page 4 of 4
15 { (y l, z) : L } D l y l V z; y l, z =, ; l =,.., L Dual: the Dantzig-Wolfe master problem Bound: min s.t. H h= ( z h ) α h H α h = K h= H yl hα h = l =,.., L h= α h h =,..., H. v(lp ) = v(lr 2 ) v(lr ) Page 5 of 4
16 2. Lagrangean decomposition: LD min z k k s.t. x kl = k l Two sets of subproblems [SP k y ] min s.t. L x kl = y kl k l :π kl D l y kl V k k l y kl, z k =, k, l π kl y kl + z k L D l y kl V z k y kl, z k =, l The dual Lagrangean problem: K max v(spy k ) + λ k= \qquad L v(spx) l [SP l x] min s.t. K k= π kl x kl K x kl = k= x kl =, k Page 6 of 4
17 The full master problem max K k= s.t. θ k + l θ k + L θ l (π kl )y h kl zh k h =,..., H k y. θ l k π kl x h kl h =,..., Hl x. where H k y and H l x are as defined previously. Easy structure of [SPx] l = ( K L ) max π kl ykl h + zk h + π = max π k= K k= min h H k y min h H k y ( L ) π kl ykl h + zk h + Which yields a smaller master problem max K k= s.t. θ k + l θ k + L θ l L L ( ) min π kl x h h Hx l kl ( ) min π kl k=,..,k (π kl )y h kl zh k h =,..., H k y ; k =,..., K Page 7 of 4 θ l π kl k =,.., K; l =,.., L
18 3. Lagrangean decomposition: LD2 Instead of copying variable y, let us copy D l y kl, leading to [BP P ] min z k k s.t. y kl = k l x kl = D l y kl k; l :ω kl x kl V z k k l y kl, z k =, k; l x kl D l k; l Two sets of subproblems Page 8 of 4 [SPy k ] min s.t. K D l y kl k= y kl = k y kl =, l \qquad [SP l x] min s.t. L ω kl x kl + z k x kl V z k l z k =, k x kl D l k
19 The dual Lagrangean problem is max λ K v(spy k ) + k= The full master problem max K k= s.t. θ k + l θ l k θ k + L Ranking of the bounds θ l L v(spx) l (ω kl )y h kl zh k h =,..., H k y. ω kl x h kl h =,..., Hl x. v(lr2) = v(lp ) v(ld2) v(lr) = V (LD) Page 9 of 4
20 4. Using v(lr) in an interior-point branch-and-price framework Master problem max Kθ + L s.t. λ l L ( λ l )yl h + z h θ h =,..., H H { is the index set of the integer points: } L (y l, z) : D l y l V z; y l, z =, ; l =,.., L Dual: the Dantzig-Wolfe master problem min s.t. H h= ( z h ) α h H α h K h= H yl hα h = l =,.., L h= α h h =,..., H. Page 2 of 4
21 As H α h h= = K is redundant, ( we assume that we have a sufficient number of available bins), We get min s.t. H h= ( z h ) α h H yl hα h = l =,.., L h= α h h =,..., H. The Gilmore-Gomory formulation for the cutting stock problem Page 2 of 4
22 5. Cutting plane methods Relaxed master problem: max Kθ + L s.t. λ l L ( λ l )yl h + z h θ h H H Solution: Cutting Plane Method Get a query point λ from the relaxed master problem Check for optimality at the full master problem. If not optimal, add a new set of constraints Solve the K subproblems [SP λ ] m Get lower bound (z l ) : λ l + Kv(RMP ) Form a new relaxed master problem Page 22 of 4
23 6. ACCPM The choice of the query point gives the different variants of the cutting plane algorithms. Kelley s cutting plane method (Kelly, 96): the query point is the optimal solution of [RMP ]. Analytic Centre Cutting Plane Method (ACCPM): (Goffin, Haurie & Vial, 992) the query point is the analytic centre of the localization set: (θ, λ) : Kθ + m λ l z l ; F D (z l ) = θ + m yl hλ l z h ; h H, Page 23 of 4
24 7. ACCPM Kelley s (usually Simplex methods) ACCPM (interior point methods) Cut Localization Set ac New ac Lower bound New LB Page 24 of 4 Extreme point
25 8. Computing the Analytic Centre The analytic centre is the point that maximizes the distance from the boundaries of the localization set F q D (z l). is the unique point maximizing The weighted dual potential. The weighted primal potential The weighted primal dual potential satisfies the first order conditions Page 25 of 4 S x = Ne ν () Ã x =, x > (2) Ã T y + s = c, s > (3)
26 9. Waram starting Computing the analytic centre after adding cuts Primal IPM. Primal-dual IPM. Computing the analytic cemnte after branching Dual IPM. Page 26 of 4
27 2. The IP-B&P algorithm Columns satisfying left branching Optimal Bound : zl Analytic center : a a a ( x, y, s ) Columns satisfying right branching Page 27 of 4 zl is valid Analytic center : a a ( y, s ) is dual feasible Dual IPM
28 Initilaization: Intial upper bound: UB =. Initial set of nodes to explore: S = {}. Initial lower bound for node : LB =. Initial matrix for node : A is empty. Iteration: While there are nodes to explore (S is not empty) End while. Pick a node, say node n 2. Get a lower bound LB n from the full master problem. Use the matrix A n as a starting matrix. Use the lower bound LBn as an intial lower bound Get a lower bound LB n by applying a cutting plane/column generation method (ACCPM, Kelley s method or Bundle methods ). 3. If possible, generate a feasible solution. This gives an upper bound UB n 4. Update upper bound: UB = min (UB n, UB) 5. If LB n UB 6. Else Fathom node n: S = S\{n}. Branch: Create two new nodes n and n 2 : S = S {n, n 2 } 7. Save warm starting information: Use the branching rule to split the columns of A n into two matrices A n and A n 2. Use them as initial matrices for the child nodes n and n 2. The lower bound LB n is valid for the child nodes. Use it to initialize the lower bounds at child nodes n and n 2. LBn = LB n and LBn 2 = LB n. Page 28 of 4 Figure : The main steps of the branch-and-price algorithm
29 2. Branching rule Pick two items l and l 2 that are once put in the same bin k and once put in diffeent bins (search for an other bin k 2 that has one but not the other.) This corresponds to identifying the pattern x k l = x k l 2 = x k2 l = ; x k2 l 2 = Branching constraints x kl = x kl2 x kl + x kl2 Page 29 of 4
30 Page 3 of 4 l h
31 22. Generating feasible solutions Dual heuristics: Rounding α kh Rounding the original variables y kl = h y h kl α kh min δ s.t. n k= h H k Primal heuristics ( m ) c kl ykl h δ kh δ kh = h H k n ykl h δ kh = k= h H k δ kh Use the set of generated columns. k =,..., n l =,..., m h H k, k =,..., n Page 3 of 4
32 23. Numerical Testing Matlab 6. Cplex 7.5 Problems Triplets Random On a Sun Ultra-/44 workstation. Page 32 of 4
33 24. ACCPM: x 5 The progress of the bounds Bound upper bound lower bound best lower bound Newton iters to compute the Analytic centre Newton iters Newton iters to find a primal feasible point Newton iters 5 Page 33 of Phase I Phase II Figure 2: The progress of the lower and upper bounds in ACCPM
34 25. Comparsion between AC-BP, K-BP and Cplex-MIP 7.5 LP Lag. AC- BP Cplex-MIP 7.5 K-BP bound bound Nodes CPU b Nodes Gap CPU b Nodes CPU b BinG BinG BinG %() a BinG BinG %() BinG BinG BinG %() BinG %() BinG BinG BinG %(7) BinG %(6) BinG BinG Min Max Average (.) a : Difference between lower and upper bound. (.) b : All CPU s in minutes. : Failed to find a feasible solution within the allowed time. # : Did not get past node. (.) d : found lower bound as a percentage of best lower bound. : Gap =. Page 34 of 4
35 26. Comparsion between AC-BP, K-BP and Cplex-MIP 7.5 LP Lag. AC- BP Cplex-MIP 7.5 K-BP bound bound Nodes CPU b Nodes Gap CPU b Nodes CPU b BinT BinT BinT %() a > 2.89 BinT %() > BinT %() > BinT %() > BinT %() > BinT %() > BinT %() > BinT %() > 2 #(99.72%) d > 2 BinT %() > 2 #(99.9%) > 2 BinT %() > 2 #(99.5%) > 2 BinT > 2 #(99.52%) > 2 BinT > 2 #(82.75%) > 2 Min.2 c 2.2 c.6 c Max c 276 c c Average c c c (.) a : Difference between lower and upper bound. (.) b : All CPU s in minutes. (.) c : Taken over the first 9 instances (those solved successfully by AC-BP and K-BP). : Failed to find a feasible solution within the allowed time. # : Did not get past node. (.) d : found lower bound as a percentage of best lower bound. : Gap =. Page 35 of 4
36 27. Comparsion between AC-BP and K-BP: Warm starting Problem A-BP K-BP SP SPrest CPU CPUrest SP SPrest CPU CPUrest BinG BinG BinG BinG BinG BinG BinG BinG BinG BinG BinG BinG BinG BinG BinG Min Max Average # : Did not get past node. a : SPrest as a percent of SP b : CPUrest as a percent of CPU : No branching was done. Page 36 of 4
37 28. Comparsion between AC-BP and K-BP: Warm starting Problem A-BP K-BP SP SPrest CPU CPUrest SP SPrest CPU CPUrest BinT BinT BinT BinT BinT BinT BinT BinT BinT BinT # BinT # BinT # BinT # BinT # Min Max Average # : Did not get past node. a : SPrest as a percent of SP b : CPUrest as a percent of CPU : No branching was done. Page 37 of 4
38 29. IP B&P vs Classical B&P: Nodes solved and CPU 35 CPU (mins) Classical B&B (Kelley) IP B&B (ACCPM) Problem number Number of nodes Classical B&B (Kelley) IP B&B (ACCPM) Page 38 of Problem number
39 3. IP B&P vs Classical B&P: Subproblems called. Subporoblems called at node IP B&P (ACCPM) Classical B&P (Kelley) Problem number Subporoblems called at rest of nodes IP B&P (ACCPM) Classical B&P (Kelley) Problem number Page 39 of 4
40 3. Conclusion Simpelx: LP bounding scheme IPMs: Lagrangean bounding scheme: Dantzig-Wolfe formulation Decomp. approaches Branch & Price ACCPM IP B&P IP B&P Branch & Bound IP B&B Page 4 of 4 Interior point methods
41 Warm starting can be done efficciently with IPMs. IPM-based Branch-and-Bound is promising to solve large MIPs Still to do: An efficient implementation (framework??) Branching Dual heuristics Preprocessing, valid cuts, etc Testing... Page 4 of 4
A CONIC DANTZIG-WOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING
A CONIC DANTZIG-WOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING Kartik Krishnan Advanced Optimization Laboratory McMaster University Joint work with Gema Plaza Martinez and Tamás
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective 1 / 24 Cutting stock problem 2 / 24 Problem description
More informationIntroduction to Bin Packing Problems
Introduction to Bin Packing Problems Fabio Furini March 13, 2015 Outline Origins and applications Applications: Definition: Bin Packing Problem (BPP) Solution techniques for the BPP Heuristic Algorithms
More informationColumn Generation. MTech Seminar Report. Soumitra Pal Roll No: under the guidance of
Column Generation MTech Seminar Report by Soumitra Pal Roll No: 05305015 under the guidance of Prof. A. G. Ranade Computer Science and Engineering IIT-Bombay a Department of Computer Science and Engineering
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective Vehicle routing problem 1 / 33 Cutting stock
More information1 Column Generation and the Cutting Stock Problem
1 Column Generation and the Cutting Stock Problem In the linear programming approach to the traveling salesman problem we used the cutting plane approach. The cutting plane approach is appropriate when
More informationLagrangian Relaxation in MIP
Lagrangian Relaxation in MIP Bernard Gendron May 28, 2016 Master Class on Decomposition, CPAIOR2016, Banff, Canada CIRRELT and Département d informatique et de recherche opérationnelle, Université de Montréal,
More informationLarge-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view
Large-scale optimization and decomposition methods: outline I Solution approaches for large-scaled problems: I Delayed column generation I Cutting plane methods (delayed constraint generation) 7 I Problems
More informationLecture 9: Dantzig-Wolfe Decomposition
Lecture 9: Dantzig-Wolfe Decomposition (3 units) Outline Dantzig-Wolfe decomposition Column generation algorithm Relation to Lagrangian dual Branch-and-price method Generated assignment problem and multi-commodity
More informationmaxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2
ex-5.-5. Foundations of Operations Research Prof. E. Amaldi 5. Branch-and-Bound Given the integer linear program maxz = x +x x +x 6 x +x 9 x,x integer solve it via the Branch-and-Bound method (solving
More informationMath Models of OR: Branch-and-Bound
Math Models of OR: Branch-and-Bound John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA November 2018 Mitchell Branch-and-Bound 1 / 15 Branch-and-Bound Outline 1 Branch-and-Bound
More informationPedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2
Pedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2 Outline Vehicle routing problem; How interior point methods can help; Interior point branch-price-and-cut:
More informationColumn Generation for Extended Formulations
1 / 28 Column Generation for Extended Formulations Ruslan Sadykov 1 François Vanderbeck 2,1 1 INRIA Bordeaux Sud-Ouest, France 2 University Bordeaux I, France ISMP 2012 Berlin, August 23 2 / 28 Contents
More informationOutline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lagrangian Relaxation Lecture 12 Single Machine Models, Column Generation 2. Dantzig-Wolfe Decomposition Dantzig-Wolfe Decomposition Delayed Column
More informationSection Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018
Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections
More informationImproving Branch-And-Price Algorithms For Solving One Dimensional Cutting Stock Problem
Improving Branch-And-Price Algorithms For Solving One Dimensional Cutting Stock Problem M. Tech. Dissertation Submitted in partial fulfillment of the requirements for the degree of Master of Technology
More informationDecomposition Methods for Integer Programming
Decomposition Methods for Integer Programming J.M. Valério de Carvalho vc@dps.uminho.pt Departamento de Produção e Sistemas Escola de Engenharia, Universidade do Minho Portugal PhD Course Programa Doutoral
More informationOperations Research Lecture 6: Integer Programming
Operations Research Lecture 6: Integer Programming Notes taken by Kaiquan Xu@Business School, Nanjing University May 12th 2016 1 Integer programming (IP) formulations The integer programming (IP) is the
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making (discrete optimization) problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock,
More informationmin3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0.
ex-.-. Foundations of Operations Research Prof. E. Amaldi. Dual simplex algorithm Given the linear program minx + x + x x + x + x 6 x + x + x x, x, x. solve it via the dual simplex algorithm. Describe
More informationInterior Point Cutting Plane Methods in Integer Programming
Interior Point Cutting Plane Methods in Integer Programming by Joe Naoum-Sawaya A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy
More informationInteger Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform
Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform François Vanderbeck B. Detienne, F. Clautiaux, R. Griset, T. Leite, G. Marques, V. Nesello, A. Pessoa,
More informationColumn Generation. ORLAB - Operations Research Laboratory. Stefano Gualandi. June 14, Politecnico di Milano, Italy
ORLAB - Operations Research Laboratory Politecnico di Milano, Italy June 14, 2011 Cutting Stock Problem (from wikipedia) Imagine that you work in a paper mill and you have a number of rolls of paper of
More information18 hours nodes, first feasible 3.7% gap Time: 92 days!! LP relaxation at root node: Branch and bound
The MIP Landscape 1 Example 1: LP still can be HARD SGM: Schedule Generation Model Example 157323 1: LP rows, still can 182812 be HARD columns, 6348437 nzs LP relaxation at root node: 18 hours Branch and
More informationNotes on Dantzig-Wolfe decomposition and column generation
Notes on Dantzig-Wolfe decomposition and column generation Mette Gamst November 11, 2010 1 Introduction This note introduces an exact solution method for mathematical programming problems. The method is
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock, crew scheduling, vehicle
More informationIntroduction to optimization and operations research
Introduction to optimization and operations research David Pisinger, Fall 2002 1 Smoked ham (Chvatal 1.6, adapted from Greene et al. (1957)) A meat packing plant produces 480 hams, 400 pork bellies, and
More informationThe Fixed Charge Transportation Problem: A Strong Formulation Based On Lagrangian Decomposition and Column Generation
The Fixed Charge Transportation Problem: A Strong Formulation Based On Lagrangian Decomposition and Column Generation Yixin Zhao, Torbjörn Larsson and Department of Mathematics, Linköping University, Sweden
More informationAcceleration and Stabilization Techniques for Column Generation
Acceleration and Stabilization Techniques for Column Generation Zhouchun Huang Qipeng Phil Zheng Department of Industrial Engineering & Management Systems University of Central Florida Sep 26, 2014 Outline
More informationInteger program reformulation for robust branch-and-cut-and-price
Integer program reformulation for robust branch-and-cut-and-price Marcus Poggi de Aragão Informática PUC-Rio Eduardo Uchoa Engenharia de Produção Universidade Federal Fluminense Outline of the talk Robust
More informationA Hub Location Problem with Fully Interconnected Backbone and Access Networks
A Hub Location Problem with Fully Interconnected Backbone and Access Networks Tommy Thomadsen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark tt@imm.dtu.dk
More informationBenders Decomposition Methods for Structured Optimization, including Stochastic Optimization
Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund May 2, 2001 Block Ladder Structure Basic Model minimize x;y c T x + f T y s:t: Ax = b Bx +
More informationIP Duality. Menal Guzelsoy. Seminar Series, /21-07/28-08/04-08/11. Department of Industrial and Systems Engineering Lehigh University
IP Duality Department of Industrial and Systems Engineering Lehigh University COR@L Seminar Series, 2005 07/21-07/28-08/04-08/11 Outline Duality Theorem 1 Duality Theorem Introduction Optimality Conditions
More informationNetwork Flows. 6. Lagrangian Relaxation. Programming. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 6. Lagrangian Relaxation 6.3 Lagrangian Relaxation and Integer Programming Fall 2010 Instructor: Dr. Masoud Yaghini Integer Programming Outline Branch-and-Bound Technique
More informationInteger 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
More informationA BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY
APPLICATIONES MATHEMATICAE 23,2 (1995), pp. 151 167 G. SCHEITHAUER and J. TERNO (Dresden) A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY Abstract. Many numerical computations
More informationAM 121: Intro to Optimization! Models and Methods! Fall 2018!
AM 121: Intro to Optimization Models and Methods Fall 2018 Lecture 13: Branch and Bound (I) Yiling Chen SEAS Example: max 5x 1 + 8x 2 s.t. x 1 + x 2 6 5x 1 + 9x 2 45 x 1, x 2 0, integer 1 x 2 6 5 x 1 +x
More informationCutting Plane Separators in SCIP
Cutting Plane Separators in SCIP Kati Wolter Zuse Institute Berlin DFG Research Center MATHEON Mathematics for key technologies 1 / 36 General Cutting Plane Method MIP min{c T x : x X MIP }, X MIP := {x
More informationApplications. Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang
Introduction to Large-Scale Linear Programming and Applications Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang Daniel J. Epstein Department of Industrial and Systems Engineering, University of
More informationFeasibility Pump Heuristics for Column Generation Approaches
1 / 29 Feasibility Pump Heuristics for Column Generation Approaches Ruslan Sadykov 2 Pierre Pesneau 1,2 Francois Vanderbeck 1,2 1 University Bordeaux I 2 INRIA Bordeaux Sud-Ouest SEA 2012 Bordeaux, France,
More informationScenario grouping and decomposition algorithms for chance-constrained programs
Scenario grouping and decomposition algorithms for chance-constrained programs Yan Deng Shabbir Ahmed Jon Lee Siqian Shen Abstract A lower bound for a finite-scenario chance-constrained problem is given
More informationDisconnecting Networks via Node Deletions
1 / 27 Disconnecting Networks via Node Deletions Exact Interdiction Models and Algorithms Siqian Shen 1 J. Cole Smith 2 R. Goli 2 1 IOE, University of Michigan 2 ISE, University of Florida 2012 INFORMS
More informationA Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem. Ryutsu Keizai University Naoto KATAYAMA
A Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem Ryutsu Keizai University Naoto KATAYAMA Problems 2006 1 Multi-Commodity Network Design Problem The basic model for
More informationThe two-dimensional bin-packing problem is the problem of orthogonally packing a given set of rectangles
INFORMS Journal on Computing Vol. 19, No. 1, Winter 2007, pp. 36 51 issn 1091-9856 eissn 1526-5528 07 1901 0036 informs doi 10.1287/ijoc.1060.0181 2007 INFORMS Using Decomposition Techniques and Constraint
More informationA generic view of Dantzig Wolfe decomposition in mixed integer programming
Operations Research Letters 34 (2006) 296 306 Operations Research Letters www.elsevier.com/locate/orl A generic view of Dantzig Wolfe decomposition in mixed integer programming François Vanderbeck a,,
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms
MVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2017 04 07 Lecture 8 Linear and integer optimization with applications
More informationDecomposition Branching for Mixed Integer Programming
Decomposition Branching for Mixed Integer Programming Baris Yildiz 1, Natashia Boland 2, and Martin Savelsbergh 2 1 Department of Industrial Engineering, Koc University, Istanbul, Turkey 2 H. Milton Stewart
More informationAdvanced linear programming
Advanced linear programming http://www.staff.science.uu.nl/~akker103/alp/ Chapter 10: Integer linear programming models Marjan van den Akker 1 Intro. Marjan van den Akker Master Mathematics TU/e PhD Mathematics
More informationAn Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory
An Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory by Troels Martin Range Discussion Papers on Business and Economics No. 10/2006 FURTHER INFORMATION Department of Business
More informationCutting Planes in SCIP
Cutting Planes in SCIP Kati Wolter Zuse-Institute Berlin Department Optimization Berlin, 6th June 2007 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2
More informationLessons from MIP Search. John Hooker Carnegie Mellon University November 2009
Lessons from MIP Search John Hooker Carnegie Mellon University November 2009 Outline MIP search The main ideas Duality and nogoods From MIP to AI (and back) Binary decision diagrams From MIP to constraint
More informationLecture 23 Branch-and-Bound Algorithm. November 3, 2009
Branch-and-Bound Algorithm November 3, 2009 Outline Lecture 23 Modeling aspect: Either-Or requirement Special ILPs: Totally unimodular matrices Branch-and-Bound Algorithm Underlying idea Terminology Formal
More informationScenario Grouping and Decomposition Algorithms for Chance-constrained Programs
Scenario Grouping and Decomposition Algorithms for Chance-constrained Programs Siqian Shen Dept. of Industrial and Operations Engineering University of Michigan Joint work with Yan Deng (UMich, Google)
More informationIntroduction to Integer Linear Programming
Lecture 7/12/2006 p. 1/30 Introduction to Integer Linear Programming Leo Liberti, Ruslan Sadykov LIX, École Polytechnique liberti@lix.polytechnique.fr sadykov@lix.polytechnique.fr Lecture 7/12/2006 p.
More informationParallel PIPS-SBB Multi-level parallelism for 2-stage SMIPS. Lluís-Miquel Munguia, Geoffrey M. Oxberry, Deepak Rajan, Yuji Shinano
Parallel PIPS-SBB Multi-level parallelism for 2-stage SMIPS Lluís-Miquel Munguia, Geoffrey M. Oxberry, Deepak Rajan, Yuji Shinano ... Our contribution PIPS-PSBB*: Multi-level parallelism for Stochastic
More informationto work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting
Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the
More informationInteger Programming ISE 418. Lecture 16. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 16 Dr. Ted Ralphs ISE 418 Lecture 16 1 Reading for This Lecture Wolsey, Chapters 10 and 11 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.3.7, II.5.4 CCZ Chapter 8
More informationwhere X is the feasible region, i.e., the set of the feasible solutions.
3.5 Branch and Bound Consider a generic Discrete Optimization problem (P) z = max{c(x) : x X }, where X is the feasible region, i.e., the set of the feasible solutions. Branch and Bound is a general semi-enumerative
More informationKNAPSACK PROBLEMS WITH SETUPS
7 e Conférence Francophone de MOdélisation et SIMulation - MOSIM 08 - du 31 mars au 2 avril 2008 - Paris - France Modélisation, Optimisation et Simulation des Systèmes : Communication, Coopération et Coordination
More informationSection Notes 9. IP: Cutting Planes. Applied Math 121. Week of April 12, 2010
Section Notes 9 IP: Cutting Planes Applied Math 121 Week of April 12, 2010 Goals for the week understand what a strong formulations is. be familiar with the cutting planes algorithm and the types of cuts
More informationStabilized Branch-and-cut-and-price for the Generalized Assignment Problem
Stabilized Branch-and-cut-and-price for the Generalized Assignment Problem Alexandre Pigatti, Marcus Poggi de Aragão Departamento de Informática, PUC do Rio de Janeiro {apigatti, poggi}@inf.puc-rio.br
More informationThe CPLEX Library: Mixed Integer Programming
The CPLEX Library: Mixed Programming Ed Rothberg, ILOG, Inc. 1 The Diet Problem Revisited Nutritional values Bob considered the following foods: Food Serving Size Energy (kcal) Protein (g) Calcium (mg)
More informationThe Modified Integer Round-Up Property of the One-Dimensional Cutting Stock Problem
EJOR 84 (1995) 562 571 The Modified Integer Round-Up Property of the One-Dimensional Cutting Stock Problem Guntram Scheithauer and Johannes Terno Institute of Numerical Mathematics, Dresden University
More informationExact and heuristic solution approaches for the mixed integer setup knapsack problem
DePaul University From the SelectedWorks of Nezih Altay 2008 Exact and heuristic solution approaches for the mixed integer setup knapsack problem Nezih Altay, University of Richmond Powell E Robinson,
More informationSoftware for Integer and Nonlinear Optimization
Software for Integer and Nonlinear Optimization Sven Leyffer, leyffer@mcs.anl.gov Mathematics & Computer Science Division Argonne National Laboratory Roger Fletcher & Jeff Linderoth Advanced Methods and
More informationA Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price
A Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price Gleb Belov University of Dresden Adam N. Letchford Lancaster University Eduardo Uchoa Universidade Federal Fluminense August 4, 2011
More informationStrengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse
Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse Merve Bodur 1, Sanjeeb Dash 2, Otay Günlü 2, and James Luedte 3 1 Department of Mechanical and Industrial Engineering,
More informationA Branch-and-Cut-and-Price Algorithm for One-Dimensional Stock Cutting and Two-Dimensional Two-Stage Cutting
A Branch-and-Cut-and-Price Algorithm for One-Dimensional Stock Cutting and Two-Dimensional Two-Stage Cutting G. Belov,1 G. Scheithauer University of Dresden, Institute of Numerical Mathematics, Mommsenstr.
More information21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.
Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial
More informationSolving LP and MIP Models with Piecewise Linear Objective Functions
Solving LP and MIP Models with Piecewise Linear Obective Functions Zonghao Gu Gurobi Optimization Inc. Columbus, July 23, 2014 Overview } Introduction } Piecewise linear (PWL) function Convex and convex
More informationDM545 Linear and Integer Programming. Lecture 13 Branch and Bound. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 13 Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 2 Exam Tilladt Håndscanner/digital pen og ordbogsprogrammet
More informationInteger Linear Programming
Integer Linear Programming Solution : cutting planes and Branch and Bound Hugues Talbot Laboratoire CVN April 13, 2018 IP Resolution Gomory s cutting planes Solution branch-and-bound General method Resolution
More informationLOWER BOUNDS FOR INTEGER PROGRAMMING PROBLEMS
LOWER BOUNDS FOR INTEGER PROGRAMMING PROBLEMS A Thesis Presented to The Academic Faculty by Yaxian Li In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Industrial
More informationWeighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths
Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths H. Murat AFSAR Olivier BRIANT Murat.Afsar@g-scop.inpg.fr Olivier.Briant@g-scop.inpg.fr G-SCOP Laboratory Grenoble Institute of Technology
More informationBenders Decomposition Methods for Structured Optimization, including Stochastic Optimization
Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund April 29, 2004 c 2004 Massachusetts Institute of echnology. 1 1 Block Ladder Structure We consider
More informationInteger Programming. Wolfram Wiesemann. December 6, 2007
Integer Programming Wolfram Wiesemann December 6, 2007 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration
More informationAn Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling Problems
Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 An Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling
More informationLinear integer programming and its application
Linear integer programming and its application Presented by Dr. Sasthi C. Ghosh Associate Professor Advanced Computing & Microelectronics Unit Indian Statistical Institute Kolkata, India Outline Introduction
More informationSolving quadratic multicommodity problems through an interior-point algorithm
Solving quadratic multicommodity problems through an interior-point algorithm Jordi Castro Department of Statistics and Operations Research Universitat Politècnica de Catalunya Pau Gargallo 5, 08028 Barcelona
More informationReformulation and Decomposition of Integer Programs
Reformulation and Decomposition of Integer Programs François Vanderbeck 1 and Laurence A. Wolsey 2 (Reference: CORE DP 2009/16) (1) Université Bordeaux 1 & INRIA-Bordeaux (2) Université de Louvain, CORE.
More informationBenders Decomposition
Benders Decomposition Yuping Huang, Dr. Qipeng Phil Zheng Department of Industrial and Management Systems Engineering West Virginia University IENG 593G Nonlinear Programg, Spring 2012 Yuping Huang (IMSE@WVU)
More informationA Horizon Decomposition approach for the Capacitated Lot-Sizing Problem with Setup Times
Submitted to INFORMS Journal on Computing manuscript (Please, provide the mansucript number!) A Horizon Decomposition approach for the Capacitated Lot-Sizing Problem with Setup Times Fragkos Ioannis Rotterdam
More informationFixed-charge transportation problems on trees
Fixed-charge transportation problems on trees Gustavo Angulo * Mathieu Van Vyve * gustavo.angulo@uclouvain.be mathieu.vanvyve@uclouvain.be November 23, 2015 Abstract We consider a class of fixed-charge
More informationBranch-and-Price-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows
Branch-and-Price-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows Guy Desaulniers École Polytechnique de Montréal and GERAD Column Generation 2008 Aussois, France Outline Introduction
More informationIntroduction to Integer Programming
Lecture 3/3/2006 p. /27 Introduction to Integer Programming Leo Liberti LIX, École Polytechnique liberti@lix.polytechnique.fr Lecture 3/3/2006 p. 2/27 Contents IP formulations and examples Total unimodularity
More informationOverview of course. Introduction to Optimization, DIKU Monday 12 November David Pisinger
Introduction to Optimization, DIKU 007-08 Monday November David Pisinger Lecture What is OR, linear models, standard form, slack form, simplex repetition, graphical interpretation, extreme points, basic
More informationLagrangean relaxation
Lagrangean relaxation Giovanni Righini Corso di Complementi di Ricerca Operativa Joseph Louis de la Grange (Torino 1736 - Paris 1813) Relaxations Given a problem P, such as: minimize z P (x) s.t. x X P
More informationNovel update techniques for the revised simplex method (and their application)
Novel update techniques for the revised simplex method (and their application) Qi Huangfu 1 Julian Hall 2 Others 1 FICO 2 School of Mathematics, University of Edinburgh ERGO 30 November 2016 Overview Background
More informationDecomposition and Reformulation in Integer Programming
and Reformulation in Integer Programming Laurence A. WOLSEY 7/1/2008 / Aussois and Reformulation in Integer Programming Outline 1 Resource 2 and Reformulation in Integer Programming Outline Resource 1
More informationLP based heuristics for the multiple knapsack problem. with assignment restrictions
LP based heuristics for the multiple knapsack problem with assignment restrictions Geir Dahl and Njål Foldnes Centre of Mathematics for Applications and Department of Informatics, University of Oslo, P.O.Box
More informationOutline. Outline. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Scheduling CPM/PERT Resource Constrained Project Scheduling Model
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING Lecture 3 and Mixed Integer Programg Marco Chiarandini 1. Resource Constrained Project Model 2. Mathematical Programg 2 Outline Outline 1. Resource Constrained
More informationResource Constrained Project Scheduling Linear and Integer Programming (1)
DM204, 2010 SCHEDULING, TIMETABLING AND ROUTING Lecture 3 Resource Constrained Project Linear and Integer Programming (1) Marco Chiarandini Department of Mathematics & Computer Science University of Southern
More informationExact Solution to Bandwidth Packing Problem with Queuing Delays
Exact Solution to Bandwidth Packing Problem with Queuing Delays Navneet Vidyarthi a,b,, Sachin Jayaswal c, Vikranth Babu Tirumala Chetty c a Department of Supply Chain and Business Technology Management
More informationInteger Linear Programming Models for 2-staged Two-Dimensional Knapsack Problems. Andrea Lodi, Michele Monaci
Integer Linear Programming Models for 2-staged Two-Dimensional Knapsack Problems Andrea Lodi, Michele Monaci Dipartimento di Elettronica, Informatica e Sistemistica, University of Bologna Viale Risorgimento,
More informationAN INTEGRATED COLUMN GENERATION AND LAGRANGIAN RELAXATION FOR SOLVING FLOWSHOP PROBLEMS TO MINIMIZE THE TOTAL WEIGHTED TARDINESS
International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN 1349-4198 Volume 7, Number 11, November 2011 pp. 6453 6471 AN INTEGRATED COLUMN GENERATION AND LAGRANGIAN
More informationDecision 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
More information3.10 Lagrangian relaxation
3.10 Lagrangian relaxation Consider a generic ILP problem min {c t x : Ax b, Dx d, x Z n } with integer coefficients. Suppose Dx d are the complicating constraints. Often the linear relaxation and the
More information23. Cutting planes and branch & bound
CS/ECE/ISyE 524 Introduction to Optimization Spring 207 8 23. Cutting planes and branch & bound ˆ Algorithms for solving MIPs ˆ Cutting plane methods ˆ Branch and bound methods Laurent Lessard (www.laurentlessard.com)
More informationBenders Decomposition for the Uncapacitated Multicommodity Network Design Problem
Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem 1 Carlos Armando Zetina, 1 Ivan Contreras, 2 Jean-François Cordeau 1 Concordia University and CIRRELT, Montréal, Canada
More informationBranch and Price for Hub Location Problems with Single Assignment
Branch and Price for Hub Location Problems with Single Assignment Ivan Contreras 1, Elena Fernández 2 1 Concordia University and CIRRELT, Montreal, Canada 2 Technical University of Catalonia, Barcelona,
More information