Exploiting Degeneracy in MIP
|
|
- Briana Spencer
- 5 years ago
- Views:
Transcription
1 Exploiting Degeneracy in MIP Tobias Achterberg 9 January 2018 Aussois
2 Performance Impact in Gurobi % 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 67 discarded due to inconsistent answers - 9 discarded that none of the versions can solve - speed-up measured on >100s bracket: 1015 models 2
3 Definition of Degeneracy An LP solution is called degenerate, if there are basic variables equal to their bounds Dual solution degenerate primal problem has multiple optimal solutions Non-basic variable with reduced costs zero Non-basic slack with dual solution value zero Primal solution degenerate dual problem has multiple optimal solutions Basic variable with primal solution value equal to its bound Basic slack with primal solution value zero Solutions for LPs of practical problems: typically dual and primal degenerate c c dual degeneracy (1-dimensional primal optimal face) primal degeneracy (2-dimensional dual optimal face) 3
4 6.1% 3.8% 1.2% 0.1% fraction of models 13.5% 30.7% 22.6% 41.1% 52.9% 80.0% 69.3% 62.0% 54.1% 46.2% 39.3% 34.4% 29.0% 25.4% 21.7% 17.8% 14.1% 9.1% 5.4% fraction of models 64.2% 74.8% 91.6% 96.5% 94.2% 89.2% 84.6% Degeneracy in Gurobi MIP Test Set 3485 models (excluding those solved in presolve or with infeasible LP relaxation) 3193 (91.6%) with dual degeneracy, 3362 (96.5%) with primal degeneracy 187 (5.4%) with 100% dual degeneracy, 3 (0.1%) with 100% primal degeneracy avg. dual degeneracy (#non-basic-zero/#cols): 37.3% avg. primal degeneracy (#basic-zero/#rows): 43.6% dual degeneracy primal degeneracy
5 Dual Degeneracy Multiple Primal Optima If there are non-basic variables with zero reduced costs, there may be multiple primal optimal solutions. Which one should the MIP solver select? Good starting point for primal heuristics Few integer variables with fractional LP solution? Interesting point for cutting plane separation Many integer variables with fractional LP solution? Small angles between outgoing rays? Many tight formulation constraints and few tight cuts? Interesting point for branching and strong branching Set of different optimal LP solutions may yield some insight for branching decision See Berthold and Salvagnin: "Could Branching" (CPAIOR 2013 proceedings) Anything else? Obviously: if there is an integer solution on the optimal face, this is an optimal choice Aiming for few integer variables with fractional LP solution may find this Approximation: try to move integer variables out of the basis 5
6 Integrality in Simplex Tune the simplex algorithm in order to find less fractional optimal basis Dual simplex: Pricing: pick primal infeasible variable to leave the basis defines dual ray to follow Ratio test: "first" non-basic variable that is hit by ray needs to enter the basis Both steps are subject to heuristic choices Pricing: trade-off between sparsity and numerical properties Ratio test: trade-off between numerical properties and temporary bound violations (Harris ratio test) Idea: add integrality of variables as additional criterion to the trade-off Pricing: prefer integer variables to leave the basis Ratio test: prefer continuous variables to enter the basis See talk of Gu: "Gurobi Technology" (INFORMS 2009, Gurobi Workshop, San Diego) Gurobi 2.0 6
7 Performance Impact in Gurobi % 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 67 discarded due to inconsistent answers - 9 discarded that none of the versions can solve - speed-up measured on >100s bracket: 1015 models 7
8 Primal PumpReduce Given an optimal LP solution, fix variables and slacks with non-zero reduced costs or duals Fixes solution to stay on optimal face What does "non-zero" mean in floating point arithmetics? Gurobi uses as threshold Tiny reduced cost values for variables with large domains can still lead to substantial objective changes Modify the objective function to move to different point on optimal face using primal simplex Search for integer solution similar to Feasibility Pump Try multiple objective function vectors Record least fractional basis found in the process Translate final basis of fixed LP to original LP Need to flip non-basis statuses (at lower/upper bound) to match values of variables with non-zero reduced costs Original LP solve cleans up errors we got from not fixing tiny reduced costs See talk of Achterberg: "LP Basis Selection and Cutting Planes" (MIP 2010, Atlanta) Introduced by Gu for CPLEX 11.0 Extended in CPLEX
9 Primal PumpReduce Quick Version Perform ratio tests to check if basic integer variables can be replaced by continuous variables Non-basic slacks of rows with zero dual solution value Non-basic variables with zero reduced costs Very simple pivots for slacks Can just update basis and solution directly No factorizations or linear system solves needed See talk of Christophel: "The Black Art of Pivoting" (MIP 2012, UC Davis) "Pivot Reduce": first push slacks into basis, then continuous structural variables Implemented in SCIP 4.0 with Soplex 3.0 Applied only at root node (but: not active in default settings, but now used in SCIP 5.0) 10% performance improvement for MIPLIB 2010 models, but solves 2 fewer See Maher et. al.: "The SCIP Optimization Suite 4.0" (March 2017) Gurobi applies quick version after every node LP relaxation solve 9
10 Performance Impact in Gurobi % 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 67 discarded due to inconsistent answers - 9 discarded that none of the versions can solve - speed-up measured on >100s bracket: 1015 models 10
11 Reduced Cost Strengthening Reduced costs of non-basic variables provide a lower bound on the objective change c incumbent solution c LP r j l j l j +1 u j ' u j x j Given the objective value of an incumbent solution, this can tighten bounds of variables Nothing else than propagating c T x = y LPT Ax + (c y LPT A) T x = y LPT b + r LPT x = c LP + r LPT x c* c multiple LP solutions lead to piece-wise linear lower bound x j l j u j 11
12 Lurking Bounds Gurobi stores reduced cost strengthening opportunities as "lurking bounds" table "If we find an incumbent with a certain objective value, then we can tighten some bounds" c* p j x j q j (or x j q j ) Need to store only one piece of PWL function for binary variables Also store only a linear lower bound function for general integer and continuous variables Pick the one that yields largest objective value at l j + 1 (or u j 1) Table is populated during the root cut loop Every root LP solution yields a pair (c LP, r LP ) Given the optimal LP objective value c LP, can we find a "good" r LP on the dual optimal face? Every r LP yields a set of lurking bounds Lurking bounds for individual variables may come from different r LP vectors Any dual feasible solution can be used can even leave dual optimal face (i.e., c' < c LP ) 12
13 Selecting "Good" Dual Solutions Bajgiran, Cire, and Rousseau: "A First Look at Picking Dual Variables for Maximizing Reduced Cost Fixing" (CPAIOR 2017 proceedings; talk at MIP 2017, Montréal) Pick dual solution vector that "maximizes reduced-cost-based filtering" Solve MIP to maximize the number of reduced-cost fixings, subject to dual feasibility and optimality Needs binary variables and big-m constraints for counting Often quick to solve, but sometimes pretty time-consuming Solve LP to minimize total slack needed to satisfy reduced-cost fixing conditions Usually very fast Some nice performance improvements (on a small set of models) (pictures from MIP 2017 talk) 13
14 Dual PumpReduce Primal PumpReduce Given an optimal LP solution, fix variables and slacks with non-zero reduced costs or duals Fixes primal solution to stay on primal optimal face (r j > 0 x j = 0) Modify the objective function to move to different point on primal optimal face using primal simplex Search for integer solution similar to Feasibility Pump Record least fractional basis found in the process Translate final basis of fixed LP to original LP Dual PumpReduce Given an optimal LP solution, remove bounds and rows that are not tight Fixes dual solution to stay on dual optimal face (x j > 0 r j = 0) Modify bounds to move to different point on dual optimal face using dual simplex For remaining finite bounds, randomly set either l j [1,100] or u j [-1,-100] Any better ideas? For each dual feasible basis, calculate reduced costs in original model and update lurking bounds table For each dual ray, fix variables/slacks with non-zero entry in dual ray 14
15 Degenerate Reduced Cost Tightening See talks of Christophel: "The Black Art of Pivoting" (MIP 2012, UC Davis) Polik: "More Ways to Use Dual Information in MILP" (ISMP 2015, Pittsburgh) For each primal degenerate basic variable Compute pivot row Perform a ratio test to get reduced costs after pivot If non-zero, apply reduced cost strengthening Equivalent to calculating Driebeek penalties SAS/OR also has version that leaves dual optimal face 2.5% performance improvement in SAS/OR (personal communication with Imre Polik) (picture from MIP 2012 talk) 15
16 Dual PumpReduce Quick Version For each row, perform ratio test to check if the dual variable can be increased or decreased If possible, move dual variable and update reduced costs May leave dual optimal face Immediately update global lurking bounds table Updated reduced costs may lead to additional opportunities to move dual variables Track maximum reduced costs for each variable Use for local reduced cost strengthening Need to be careful with reduced costs coming from dual solutions that are not on optimal face 16
17 Performance Impact in Gurobi % 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 67 discarded due to inconsistent answers - 9 discarded that none of the versions can solve - speed-up measured on >100s bracket: 1015 models 17
18 Degeneracy and Cutting Planes Cutting plane separation depends on current LP solution Gomory cuts are defined by current basis Other cuts like MIR or knapsack covers heuristically try to separate current LP solution from convex hull of integer solutions What is a good solution/basis to separate cuts for? After adding cuts and resolving the LP, should we stop at the first optimal basis that we encounter? Typically, for this basis many of the newly added cuts will be tight Additional cuts will be based on the previous cuts Leads to high rank cuts Numerical issues in cut separation accumulate 18
19 Degeneracy and Cutting Planes Zanette, Fischetti and Balas: "Lexicography and degeneracy: Can a pure cutting plane algorithm work?" (Math Programming, 2011) Pure Gomory fractional cut approach suffers from "cuts on top of cuts" Determinants of simplex bases grow rapidly: numerical difficulties Objective improvement stalls after few iterations Solution: select different optimal basis in each iteration Lexicographic simplex (Pictures for stein15, see Zanette et. al.) 19
20 Degeneracy and Cutting Planes Tramontani et. al.: "Concurrent Root Cut Loops to Exploit Random Performance Variability" (INFORMS 2013) 2 parallel root cut loops, each started from different optimal LP basis Share cuts and solutions between loops 5% speed-up in CPLEX Also in Gurobi since version 6.0 (2014) 2 parallel root cut loops Start from same optimal LP basis, but use different settings (including random seed) Data passed only from secondary cut loop to primary one (but not in other direction) New incumbents New global bounds New cuts 20
21 Degeneracy and Branch-and-Cut Fischetti and Monaci: "Exploiting Erraticism in Search" (Operations Research, 2014) Bet-and-run approach: make a number of short (i.e., few branch-and-bound nodes) sample runs, then pick the "most promising" and bring it to completion Carvahal et. al.: "Using diversification, communication and parallelism to solve mixed-integer linear programs" (OR Letters, 2014) Concurrent optimization with communication Also available in Gurobi and CPLEX, but without communication Communication is complicated without affecting determinism Also as internal feature of XPress Shinano et. al.: "FiberSCIP - A shared memory parallelization of SCIP" (ZIB Report, 2013; IJOC, 2017) Racing ramp-up phase 21
22 Performance Impact in Gurobi % 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 67 discarded due to inconsistent answers - 9 discarded that none of the versions can solve - speed-up measured on >100s bracket: 1015 models 22
23 Degeneracy and Interior Point Methods Interior point methods find analytic center of the optimal face min - ln x j s.t. Ax = b (with arithmetic appropriately defined on ln 0 = - ) x 0 Usually, we then apply crossover to obtain a vertex solution Analytic center for zero objective (c = 0) may find fixings Detects linearly implied equations If x j = 0 in analytic center, we can permanently fix x j := 0 Fix variable to a bound Turn inequality into equation See Berthold, Perregaard, Meszaros: "Four good reasons to use an Interior Point solver within a MIP solver" (ZIB Report, 2017) 2% speed-up due to presolving fixings from analytic center 23
24 Fixings from Analytic Center of Optimal Face We can also find fixings if we solved the original problem (with objective) with interior point If x j = 0 and r j = 0 in a relative interior solution, then x j = 0 for all feasible solutions Proof (by Gu): Assume there is a feasible solution with x j * > 0 The function obj(d) for an optimal solution with x j = d x j * is piecewise linear Say that 0 < x j ' x j * is the first break point of this PWL function, i.e., [0, x j '] is the linear piece with a fixed r j ' If r j ' > 0, then 0 < r j < r j ', since primal and dual solutions are at analytic center, contradicting our assumption If r j ' = 0, then the analytic center solution value for x j should be inside (0, x j '), which contradicts x j = 0 Fixings are linearly implied: not very powerful But may trigger other presolve reductions Exploit this directly for MIP? Barrier is numerically difficult Barrier presolve reductions may lead to solution that is not the analytic center Solution may yield hints which variable fixings should be tested explicitly (e.g., in an OBBT fashion) Use analytic center in LP crossover Do not use variables at bounds in analytic center for crossover basis 1% speed-up on barrier test set 24
25 Implied Equations using Simplex Solves Guess some fixings x j = 0 with j F Solve max {x j : j F} s.t. Ax = b x 0 x j for all j F If all x j = 0 for j F, then permanently fix all variables in F and stop Otherwise, remove non-zero variables from F and iterate 25
26 Performance Impact in Gurobi % 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 67 discarded due to inconsistent answers - 9 discarded that none of the versions can solve - speed-up measured on >100s bracket: 1015 models 26
27 Total Impact in Gurobi 7.5+ More Details 50% 47.6% % % 35% 32.8% % 25% 20% 15% 10% 5% 12.4% 17.6% 23.9% % >0 sec >1 sec >10 sec >100 sec >1000 sec Unsolved models (out of 5 x 3214): 52 vs 253 (+108 for both) Unsolved models without feasible solution: 16 vs 109 (+31 for both) #wins #losses per model for 5 random seeds for disabling all degeneracy exploits Time limit: sec. Intel Xeon CPU E GHz 4 cores, 8 hyper-threads 32 GB RAM Test set has 3257 models: - 43 discarded due to inconsistent answers - 14 discarded that none of the versions can solve 27
Cloud Branching MIP workshop, Ohio State University, 23/Jul/2014
Cloud Branching MIP workshop, Ohio State University, 23/Jul/2014 Timo Berthold Xpress Optimization Team Gerald Gamrath Zuse Institute Berlin Domenico Salvagnin Universita degli Studi di Padova This presentation
More informationWelcome to the Webinar. What s New in Gurobi 7.5
Welcome to the Webinar What s New in Gurobi 7.5 Speaker Introduction Dr. Tobias Achterberg Director of R&D at Gurobi Optimization Formerly a developer at ILOG, where he worked on CPLEX 11.0 to 12.6 Obtained
More informationThe MIP-Solving-Framework SCIP
The MIP-Solving-Framework SCIP Timo Berthold Zuse Institut Berlin DFG Research Center MATHEON Mathematics for key technologies Berlin, 23.05.2007 What Is A MIP? Definition MIP The optimization problem
More informationThe Ascendance of the Dual Simplex Method: A Geometric View
The Ascendance of the Dual Simplex Method: A Geometric View Robert Fourer 4er@ampl.com AMPL Optimization Inc. www.ampl.com +1 773-336-AMPL U.S.-Mexico Workshop on Optimization and Its Applications Huatulco
More informationParallel and Distributed Optimization with Gurobi Optimizer
Parallel and Distributed Optimization with Gurobi Optimizer Our Presenter Dr. Tobias Achterberg Developer, Gurobi Optimization 2 Parallel & Distributed Optimization 3 Terminology for this presentation
More informationGurobi Guidelines for Numerical Issues February 2017
Gurobi Guidelines for Numerical Issues February 2017 Background Models with numerical issues can lead to undesirable results: slow performance, wrong answers or inconsistent behavior. When solving a model
More informationHeuristics in Commercial MIP Solvers Part I (Heuristics in IBM CPLEX)
Andrea Tramontani CPLEX Optimization, IBM CWI, Amsterdam, June 12, 2018 Heuristics in Commercial MIP Solvers Part I (Heuristics in IBM CPLEX) Agenda CPLEX Branch-and-Bound (B&B) Primal heuristics in CPLEX
More informationApplied Mixed Integer Programming: Beyond 'The Optimum'
Applied Mixed Integer Programming: Beyond 'The Optimum' 14 Nov 2016, Simons Institute, Berkeley Pawel Lichocki Operations Research Team, Google https://developers.google.com/optimization/ Applied Mixed
More informationPrimal Heuristics in SCIP
Primal Heuristics in SCIP Timo Berthold Zuse Institute Berlin DFG Research Center MATHEON Mathematics for key technologies Berlin, 10/11/2007 Outline 1 Introduction Basics Integration Into SCIP 2 Available
More informationPivot and Gomory Cut. A MIP Feasibility Heuristic NSERC
Pivot and Gomory Cut A MIP Feasibility Heuristic Shubhashis Ghosh Ryan Hayward shubhashis@randomknowledge.net hayward@cs.ualberta.ca NSERC CGGT 2007 Kyoto Jun 11-15 page 1 problem given a MIP, find a feasible
More information3 INTEGER LINEAR PROGRAMMING
3 INTEGER LINEAR PROGRAMMING PROBLEM DEFINITION Integer linear programming problem (ILP) of the decision variables x 1,..,x n : (ILP) subject to minimize c x j j n j= 1 a ij x j x j 0 x j integer n j=
More informationGeorge Reloaded. M. Monaci (University of Padova, Italy) joint work with M. Fischetti. MIP Workshop, July 2010
George Reloaded M. Monaci (University of Padova, Italy) joint work with M. Fischetti MIP Workshop, July 2010 Why George? Because of Karzan, Nemhauser, Savelsbergh Information-based branching schemes for
More informationUsing Multiple Machines to Solve Models Faster with Gurobi 6.0
Using Multiple Machines to Solve Models Faster with Gurobi 6.0 Distributed Algorithms in Gurobi 6.0 Gurobi 6.0 includes 3 distributed algorithms Distributed concurrent LP (new in 6.0) MIP Distributed MIP
More informationMotivation for Heuristics
MIP Heuristics 1 Motivation for Heuristics Why not wait for branching? Produce feasible solutions as quickly as possible Often satisfies user demands Avoid exploring unproductive sub trees Better reduced
More informationLinear Programming. Course review MS-E2140. v. 1.1
Linear Programming MS-E2140 Course review v. 1.1 Course structure Modeling techniques Linear programming theory and the Simplex method Duality theory Dual Simplex algorithm and sensitivity analysis Integer
More informationState-of-the-Optimization using Xpress-MP v2006
State-of-the-Optimization using Xpress-MP v2006 INFORMS Annual Meeting Pittsburgh, USA November 5 8, 2006 by Alkis Vazacopoulos Outline LP benchmarks Xpress performance on MIPLIB 2003 Conclusions 3 Barrier
More informationMVE165/MMG630, Applied Optimization Lecture 8 Integer linear programming algorithms. Ann-Brith Strömberg
MVE165/MMG630, Integer linear programming algorithms Ann-Brith Strömberg 2009 04 15 Methods for ILP: Overview (Ch. 14.1) Enumeration Implicit enumeration: Branch and bound Relaxations Decomposition methods:
More informationThe Heuristic (Dark) Side of MIP Solvers. Asja Derviskadic, EPFL Vit Prochazka, NHH Christoph Schaefer, EPFL
The Heuristic (Dark) Side of MIP Solvers Asja Derviskadic, EPFL Vit Prochazka, NHH Christoph Schaefer, EPFL 1 Table of content [Lodi], The Heuristic (Dark) Side of MIP Solvers, Hybrid Metaheuristics, 273-284,
More informationAdvanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture 18 All-Integer Dual Algorithm We continue the discussion on the all integer
More informationHeuristics in MILP. Group 1 D. Assouline, N. Molyneaux, B. Morén. Supervisors: Michel Bierlaire, Andrea Lodi. Zinal 2017 Winter School
Heuristics in MILP Group 1 D. Assouline, N. Molyneaux, B. Morén Supervisors: Michel Bierlaire, Andrea Lodi Zinal 2017 Winter School 0 / 23 Primal heuristics Original paper: Fischetti, M. and Lodi, A. (2011).
More informationAdvanced Use of GAMS Solver Links
Advanced Use of GAMS Solver Links Michael Bussieck, Steven Dirkse, Stefan Vigerske GAMS Development 8th January 2013, ICS Conference, Santa Fe Standard GAMS solve Solve william minimizing cost using mip;
More informationWhat's New in Gurobi 7.0
What's New in Gurobi 7.0 What's New? New employees New features in 7.0 Major and minor Performance improvements New Gurobi Instant Cloud 2 The newest members of the Gurobi team Daniel Espinoza Senior Developer
More informationFundamentals of Integer Programming
Fundamentals of Integer Programming Di Yuan Department of Information Technology, Uppsala University January 2018 Outline Definition of integer programming Formulating some classical problems with integer
More informationThe Gurobi Optimizer. Bob Bixby
The Gurobi Optimizer Bob Bixby Outline Gurobi Introduction Company Products Benchmarks Gurobi Technology Rethinking MIP MIP as a bag of tricks 8-Jul-11 2010 Gurobi Optimization 2 Gurobi Optimization Incorporated
More informationLinear & Integer Programming: A Decade of Computation
Linear & Integer Programming: A Decade of Computation Robert E. Bixby, Mary Fenelon, Zongao Gu, Irv Lustig, Ed Rothberg, Roland Wunderling 1 Outline Progress in computing machines Linear programming (LP)
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms
MVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2018 04 24 Lecture 9 Linear and integer optimization with applications
More informationFrom final point cuts to!-polyhedral cuts
AUSSOIS 2017 From final point cuts to!-polyhedral cuts Egon Balas, Aleksandr M. Kazachkov, François Margot Tepper School of Business, Carnegie Mellon University Overview Background Generalized intersection
More informationAddressing degeneracy in the dual simplex algorithm using a decompositon approach
Addressing degeneracy in the dual simplex algorithm using a decompositon approach Ambros Gleixner, Stephen J Maher, Matthias Miltenberger Zuse Institute Berlin Berlin, Germany 16th July 2015 @sj_maher
More information5.3 Cutting plane methods and Gomory fractional cuts
5.3 Cutting plane methods and Gomory fractional cuts (ILP) min c T x s.t. Ax b x 0integer feasible region X Assumption: a ij, c j and b i integer. Observation: The feasible region of an ILP can be described
More informationThe Gurobi Solver V1.0
The Gurobi Solver V1.0 Robert E. Bixby Gurobi Optimization & Rice University Ed Rothberg, Zonghao Gu Gurobi Optimization 1 1 Oct 09 Overview Background Rethinking the MIP solver Introduction Tree of Trees
More informationSection Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017
Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures
More informationA hard integer program made easy by lexicography
Noname manuscript No. (will be inserted by the editor) A hard integer program made easy by lexicography Egon Balas Matteo Fischetti Arrigo Zanette February 16, 2011 Abstract A small but notoriously hard
More informationComputational Integer Programming. Lecture 12: Branch and Cut. Dr. Ted Ralphs
Computational Integer Programming Lecture 12: Branch and Cut Dr. Ted Ralphs Computational MILP Lecture 12 1 Reading for This Lecture Wolsey Section 9.6 Nemhauser and Wolsey Section II.6 Martin Computational
More informationExact Algorithms for Mixed-Integer Bilevel Linear Programming
Exact Algorithms for Mixed-Integer Bilevel Linear Programming Matteo Fischetti, University of Padova (based on joint work with I. Ljubic, M. Monaci, and M. Sinnl) Lunteren Conference on the Mathematics
More informationAdvanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture 16 Cutting Plane Algorithm We shall continue the discussion on integer programming,
More information56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998
56:272 Integer Programming & Network Flows Final Examination -- December 14, 1998 Part A: Answer any four of the five problems. (15 points each) 1. Transportation problem 2. Integer LP Model Formulation
More informationPure Cutting Plane Methods for ILP: a computational perspective
Pure Cutting Plane Methods for ILP: a computational perspective Matteo Fischetti, DEI, University of Padova Rorschach test for OR disorders: can you see the tree? 1 Outline 1. Pure cutting plane methods
More informationInteger Programming Theory
Integer Programming Theory Laura Galli October 24, 2016 In the following we assume all functions are linear, hence we often drop the term linear. In discrete optimization, we seek to find a solution x
More informationImplementing a B&C algorithm for Mixed-Integer Bilevel Linear Programming
Implementing a B&C algorithm for Mixed-Integer Bilevel Linear Programming Matteo Fischetti, University of Padova 8th Cargese-Porquerolles Workshop on Combinatorial Optimization, August 2017 1 Bilevel Optimization
More informationAlgorithms II MIP Details
Algorithms II MIP Details What s Inside Gurobi Optimizer Algorithms for continuous optimization Algorithms for discrete optimization Automatic presolve for both LP and MIP Algorithms to analyze infeasible
More informationExact solutions to mixed-integer linear programming problems
Exact solutions to mixed-integer linear programming problems Dan Steffy Zuse Institute Berlin and Oakland University Joint work with Bill Cook, Thorsten Koch and Kati Wolter November 18, 2011 Mixed-Integer
More informationConstraint Branching and Disjunctive Cuts for Mixed Integer Programs
Constraint Branching and Disunctive Cuts for Mixed Integer Programs Constraint Branching and Disunctive Cuts for Mixed Integer Programs Michael Perregaard Dash Optimization Constraint Branching and Disunctive
More informationUnit.9 Integer Programming
Unit.9 Integer Programming Xiaoxi Li EMS & IAS, Wuhan University Dec. 22-29, 2016 (revised) Operations Research (Li, X.) Unit.9 Integer Programming Dec. 22-29, 2016 (revised) 1 / 58 Organization of this
More informationRENS. The optimal rounding. Timo Berthold
Math. Prog. Comp. (2014) 6:33 54 DOI 10.1007/s12532-013-0060-9 FULL LENGTH PAPER RENS The optimal rounding Timo Berthold Received: 25 April 2012 / Accepted: 2 October 2013 / Published online: 1 November
More informationSelected Topics in Column Generation
Selected Topics in Column Generation February 1, 2007 Choosing a solver for the Master Solve in the dual space(kelly s method) by applying a cutting plane algorithm In the bundle method(lemarechal), a
More informationConflict Analysis in Mixed Integer Programming
Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany TOBIAS ACHTERBERG Conflict Analysis in Mixed Integer Programming URL: http://www.zib.de/projects/integer-optimization/mip
More informationLinear and Integer Programming :Algorithms in the Real World. Related Optimization Problems. How important is optimization?
Linear and Integer Programming 15-853:Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Linear or Integer programming maximize z = c T x
More informationLECTURE 6: INTERIOR POINT METHOD. 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm
LECTURE 6: INTERIOR POINT METHOD 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm Motivation Simplex method works well in general, but suffers from exponential-time
More informationCS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm Instructor: Shaddin Dughmi Algorithms for Convex Optimization We will look at 2 algorithms in detail: Simplex and Ellipsoid.
More informationHow to use your favorite MIP Solver: modeling, solving, cannibalizing. Andrea Lodi University of Bologna, Italy
How to use your favorite MIP Solver: modeling, solving, cannibalizing Andrea Lodi University of Bologna, Italy andrea.lodi@unibo.it January-February, 2012 @ Universität Wien A. Lodi, How to use your favorite
More informationIntroduction to Mathematical Programming IE496. Final Review. Dr. Ted Ralphs
Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling
More information15.083J Integer Programming and Combinatorial Optimization Fall Enumerative Methods
5.8J Integer Programming and Combinatorial Optimization Fall 9 A knapsack problem Enumerative Methods Let s focus on maximization integer linear programs with only binary variables For example: a knapsack
More informationOn Mixed-Integer (Linear) Programming and its connection with Data Science
On Mixed-Integer (Linear) Programming and its connection with Data Science Andrea Lodi Canada Excellence Research Chair École Polytechnique de Montréal, Québec, Canada andrea.lodi@polymtl.ca January 16-20,
More informationInvestigating Mixed-Integer Hulls using a MIP-Solver
Investigating Mixed-Integer Hulls using a MIP-Solver Matthias Walter Otto-von-Guericke Universität Magdeburg Joint work with Volker Kaibel (OvGU) Aussois Combinatorial Optimization Workshop 2015 Outline
More informationAlgorithms for Decision Support. Integer linear programming models
Algorithms for Decision Support Integer linear programming models 1 People with reduced mobility (PRM) require assistance when travelling through the airport http://www.schiphol.nl/travellers/atschiphol/informationforpassengerswithreducedmobility.htm
More informationAssessing Performance of Parallel MILP Solvers
Assessing Performance of Parallel MILP Solvers How Are We Doing, Really? Ted Ralphs 1 Stephen J. Maher 2, Yuji Shinano 3 1 COR@L Lab, Lehigh University, Bethlehem, PA USA 2 Lancaster University, Lancaster,
More informationBenders in a nutshell Matteo Fischetti, University of Padova
Benders in a nutshell Matteo Fischetti, University of Padova ODS 2017, Sorrento, September 2017 1 Benders decomposition The original Benders decomposition from the 1960s uses two distinct ingredients for
More informationOutline. CS38 Introduction to Algorithms. Linear programming 5/21/2014. Linear programming. Lecture 15 May 20, 2014
5/2/24 Outline CS38 Introduction to Algorithms Lecture 5 May 2, 24 Linear programming simplex algorithm LP duality ellipsoid algorithm * slides from Kevin Wayne May 2, 24 CS38 Lecture 5 May 2, 24 CS38
More information/ Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang 9.1 Linear Programming Suppose we are trying to approximate a minimization
More informationTopics. Introduction. Specific tuning/troubleshooting topics "It crashed" Gurobi parameters The tuning tool. LP tuning. MIP tuning
Tuning II Topics Introduction Gurobi parameters The tuning tool Specific tuning/troubleshooting topics "It crashed" The importance of being specific LP tuning The importance of scaling MIP tuning Performance
More informationPenalty Alternating Direction Methods for Mixed- Integer Optimization: A New View on Feasibility Pumps
Penalty Alternating Direction Methods for Mixed- Integer Optimization: A New View on Feasibility Pumps Björn Geißler, Antonio Morsi, Lars Schewe, Martin Schmidt FAU Erlangen-Nürnberg, Discrete Optimization
More informationLecture 3. Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets. Tepper School of Business Carnegie Mellon University, Pittsburgh
Lecture 3 Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets Gérard Cornuéjols Tepper School of Business Carnegie Mellon University, Pittsburgh January 2016 Mixed Integer Linear Programming
More informationRestrict-and-relax search for 0-1 mixed-integer programs
EURO J Comput Optim (23) :2 28 DOI.7/s3675-3-7-y ORIGINAL PAPER Restrict-and-relax search for - mixed-integer programs Menal Guzelsoy George Nemhauser Martin Savelsbergh Received: 2 September 22 / Accepted:
More informationA Nonlinear Presolve Algorithm in AIMMS
A Nonlinear Presolve Algorithm in AIMMS By Marcel Hunting marcel.hunting@aimms.com November 2011 This paper describes the AIMMS presolve algorithm for nonlinear problems. This presolve algorithm uses standard
More informationBayesian network model selection using integer programming
Bayesian network model selection using integer programming James Cussens Leeds, 2013-10-04 James Cussens IP for BNs Leeds, 2013-10-04 1 / 23 Linear programming The Belgian diet problem Fat Sugar Salt Cost
More informationAgenda. Understanding advanced modeling techniques takes some time and experience No exercises today Ask questions!
Modeling 2 Agenda Understanding advanced modeling techniques takes some time and experience No exercises today Ask questions! Part 1: Overview of selected modeling techniques Background Range constraints
More informationFebruary 19, Integer programming. Outline. Problem formulation. Branch-andbound
Olga Galinina olga.galinina@tut.fi ELT-53656 Network Analysis and Dimensioning II Department of Electronics and Communications Engineering Tampere University of Technology, Tampere, Finland February 19,
More information16.410/413 Principles of Autonomy and Decision Making
16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT)
More informationSolving lexicographic multiobjective MIPs with Branch-Cut-Price
Solving lexicographic multiobjective MIPs with Branch-Cut-Price Marta Eso (The Hotchkiss School) Laszlo Ladanyi (IBM T.J. Watson Research Center) David Jensen (IBM T.J. Watson Research Center) McMaster
More informationModern Benders (in a nutshell)
Modern Benders (in a nutshell) Matteo Fischetti, University of Padova (based on joint work with Ivana Ljubic and Markus Sinnl) Lunteren Conference on the Mathematics of Operations Research, January 17,
More information5.4 Pure Minimal Cost Flow
Pure Minimal Cost Flow Problem. Pure Minimal Cost Flow Networks are especially convenient for modeling because of their simple nonmathematical structure that can be easily portrayed with a graph. This
More informationComparisons of Commercial MIP Solvers and an Adaptive Memory (Tabu Search) Procedure for a Class of 0-1 Integer Programming Problems
Comparisons of Commercial MIP Solvers and an Adaptive Memory (Tabu Search) Procedure for a Class of 0-1 Integer Programming Problems Lars M. Hvattum The Norwegian University of Science and Technology Trondheim,
More informationInteger Programming Chapter 9
1 Integer Programming Chapter 9 University of Chicago Booth School of Business Kipp Martin October 30, 2017 2 Outline Branch and Bound Theory Branch and Bound Linear Programming Node Selection Strategies
More informationOptimization Methods in Management Science
Problem Set Rules: Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 6, Due: Thursday April 11th, 2013 1. Each student should hand in an individual problem set. 2. Discussing
More informationColumn Generation Based Primal Heuristics
Column Generation Based Primal Heuristics C. Joncour, S. Michel, R. Sadykov, D. Sverdlov, F. Vanderbeck University Bordeaux 1 & INRIA team RealOpt Outline 1 Context Generic Primal Heuristics The Branch-and-Price
More informationAlgorithms for Integer Programming
Algorithms for Integer Programming Laura Galli November 9, 2016 Unlike linear programming problems, integer programming problems are very difficult to solve. In fact, no efficient general algorithm is
More informationSolving a Challenging Quadratic 3D Assignment Problem
Solving a Challenging Quadratic 3D Assignment Problem Hans Mittelmann Arizona State University Domenico Salvagnin DEI - University of Padova Quadratic 3D Assignment Problem Quadratic 3D Assignment Problem
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationSome Advanced Topics in Linear Programming
Some Advanced Topics in Linear Programming Matthew J. Saltzman July 2, 995 Connections with Algebra and Geometry In this section, we will explore how some of the ideas in linear programming, duality theory,
More informationOutline. Modeling. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Models Lecture 5 Mixed Integer Programming Models and Exercises
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lecture 5 Mixed Integer Programming and Exercises Marco Chiarandini 2. 3. 2 Outline Modeling 1. Min cost flow Shortest path 2. Max flow Assignment
More informationLinear Programming. Linear Programming. Linear Programming. Example: Profit Maximization (1/4) Iris Hui-Ru Jiang Fall Linear programming
Linear Programming 3 describes a broad class of optimization tasks in which both the optimization criterion and the constraints are linear functions. Linear Programming consists of three parts: A set of
More informationLP-Modelling. dr.ir. C.A.J. Hurkens Technische Universiteit Eindhoven. January 30, 2008
LP-Modelling dr.ir. C.A.J. Hurkens Technische Universiteit Eindhoven January 30, 2008 1 Linear and Integer Programming After a brief check with the backgrounds of the participants it seems that the following
More informationLinear programming and duality theory
Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables
More informationDiscrete Optimization. Lecture Notes 2
Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The
More informationPrimal Heuristics for Branch-and-Price Algorithms
Primal Heuristics for Branch-and-Price Algorithms Marco Lübbecke and Christian Puchert Abstract In this paper, we present several primal heuristics which we implemented in the branch-and-price solver GCG
More informationCS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 36
CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 36 CS 473: Algorithms, Spring 2018 LP Duality Lecture 20 April 3, 2018 Some of the
More informationGomory Reloaded. Matteo Fischetti, DEI, University of Padova (joint work with Domenico Salvagnin) 1 MIP 2010
Gomory Reloaded Matteo Fischetti, DEI, University of Padova (joint work with Domenico Salvagnin) 1 Cutting planes (cuts) We consider a general MIPs of the form min { c x : A x = b, x 0, x j integer for
More informationSCIP. 1 Introduction. 2 Model requirements. Contents. Stefan Vigerske, Humboldt University Berlin, Germany
SCIP Stefan Vigerske, Humboldt University Berlin, Germany Contents 1 Introduction.................................................. 673 2 Model requirements..............................................
More informationImproved Gomory Cuts for Primal Cutting Plane Algorithms
Improved Gomory Cuts for Primal Cutting Plane Algorithms S. Dey J-P. Richard Industrial Engineering Purdue University INFORMS, 2005 Outline 1 Motivation The Basic Idea Set up the Lifting Problem How to
More informationThe Simplex Algorithm
The Simplex Algorithm Uri Feige November 2011 1 The simplex algorithm The simplex algorithm was designed by Danzig in 1947. This write-up presents the main ideas involved. It is a slight update (mostly
More informationEllipsoid Algorithm :Algorithms in the Real World. Ellipsoid Algorithm. Reduction from general case
Ellipsoid Algorithm 15-853:Algorithms in the Real World Linear and Integer Programming II Ellipsoid algorithm Interior point methods First polynomial-time algorithm for linear programming (Khachian 79)
More informationDepartment of Mathematics Oleg Burdakov of 30 October Consider the following linear programming problem (LP):
Linköping University Optimization TAOP3(0) Department of Mathematics Examination Oleg Burdakov of 30 October 03 Assignment Consider the following linear programming problem (LP): max z = x + x s.t. x x
More informationNoncommercial Software for Mixed-Integer Linear Programming
Noncommercial Software for Mixed-Integer Linear Programming J. T. Linderoth T. K. Ralphs December 23, 2004 Abstract We present an overview of noncommercial software tools for the solution of mixed-integer
More informationDEGENERACY AND THE FUNDAMENTAL THEOREM
DEGENERACY AND THE FUNDAMENTAL THEOREM The Standard Simplex Method in Matrix Notation: we start with the standard form of the linear program in matrix notation: (SLP) m n we assume (SLP) is feasible, and
More informationSolving the Euclidean Steiner Tree Problem in n-space
Solving the Euclidean Steiner Tree Problem in n-space Marcia Fampa (UFRJ), Jon Lee (U. Michigan), and Wendel Melo (UFRJ) January 2015 Marcia Fampa, Jon Lee, Wendel Melo Solving the Euclidean Steiner Tree
More informationThe Simplex Algorithm for LP, and an Open Problem
The Simplex Algorithm for LP, and an Open Problem Linear Programming: General Formulation Inputs: real-valued m x n matrix A, and vectors c in R n and b in R m Output: n-dimensional vector x There is one
More informationM2 ORO: Advanced Integer Programming. Part IV. Solving MILP (1) easy IP. Outline. Sophie Demassey. October 10, 2011
M2 ORO: Advanced Integer Programming Sophie Demassey Part IV Solving MILP (1) Mines Nantes - TASC - INRIA/LINA CNRS UMR 6241 sophie.demassey@mines-nantes.fr October 10, 2011 Université de Nantes / M2 ORO
More informationWireless frequency auctions: Mixed Integer Programs and Dantzig-Wolfe decomposition
Wireless frequency auctions: Mixed Integer Programs and Dantzig-Wolfe decomposition Laszlo Ladanyi (IBM T.J. Watson Research Center) joint work with Marta Eso (The Hotchkiss School) David Jensen (IBM T.J.
More informationParallelizing the dual revised simplex method
Parallelizing the dual revised simplex method Qi Huangfu 1 Julian Hall 2 1 FICO 2 School of Mathematics, University of Edinburgh Birmingham 9 September 2016 Overview Background Two parallel schemes Single
More informationAdvanced Operations Research Techniques IE316. Quiz 2 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 2 Review Dr. Ted Ralphs IE316 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material
More information