3.4 Relaxations and bounds
|
|
- Eleanore Sparks
- 5 years ago
- Views:
Transcription
1 3.4 Relaxations and bounds Consider a generic Discrete Optimization problem z = min{c(x) : x X} with an optimal solution x X. In general, the algorithms generate not only a decreasing sequence of upper bounds u 1 >... > u k z but also an increasing sequence of lower bounds l 1 <... < l k z. They terminate when (u k l k ) ε. Primal bounds For minimization problems, any feasible solution x X yields an upper bound u = c(x) on the optimal value, namely u z. In some cases, even finding a feasible solution may be challenging (NP-hard). Dual bounds To obtain lower bounds for minimization problems, we consider a relaxation of the problem. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
2 Quality guarantee: If x k is the best feasible solution found so far and l k the best dual bound, the termination criterion (c(x k ) l k ) ε, for a given ε > 0, guarantees that (c(x k ) z ) ε. For maximization problems, the primal (dual) bounds are lower (upper) bounds. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
3 Definition: Given a problem (P) z = min{c(x) : x X R n }, a problem (RP) z = min{ c(x) : x X R n } is a relaxation of P if X X c(x) c(x) for each x X. Proposition: If RP is a relaxation of P, z z. Proof: Let x be an optimal solution of P, then x X X and c(x ) c(x ) = z. Since x X, we have z c(x ). Proposition: Let x RP be an optimal solution of RP. If x RP is feasible for P (x RP X) and c(x RP ) = c(x RP ), then x RP is also optimal for P. We aim at a tradeoff between the bound quality (how tight is the upper/lower bound w.r.t. the optimal value z ) and the computational load needed to solve the relaxation. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
4 3.4.1 Different types of relaxations 1) Linear programming relaxation Definition: Given an arbitrary MILP (ILP) problem z ILP = min c t 1 x +ct 2 y A 1x +A 2y b x 0,y 0, integer its linear (programming) relaxation is the following LP problem: z LP = min c t 1 x +ct 2 y A 1x +A 2y b x 0,y 0 where the integrality constraints on the variables y j are omitted. Recall that the definition of the best formulation for a MILP (ILP) is closely related to that of linear relaxation: the stronger the formulation, the tighter the dual bound z LP provided by its linear relaxation. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
5 2) Relaxation by elimination A straightforward way to obtain a relaxation of a MILP problem is just to delete one ore more constraints. Examples: 1) Asymmetric TSP Delete the subtour elimination (cut-set) constraints and just keep the assignment constraints. 2) Multi-dimensional binary knapsack problem max s.t. n j=1 p jx j n j=1 w ijx j W i i {1,2,...,m} (1) x j {0,1} j {1,2,...,n} (2) If we delete all but one constraints, we obtain a standard binary knapsack problem. (Very) weak relaxations. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
6 3) Surrogate relaxation Idea: Take a linear combination of some constraints with multipliers λ i 0, and replace these constraints with the resulting (surrogate) constraint. Example: Multiple binary knapsack problem z mkp = max m n i=1 j=1 p jx ij s.t. n j=1 w jx ij W i i {1,2,...,m} (3) m i=1 x ij 1 j {1,2,...,n} (4) x ij {0,1} i, j (5) Given m knapsacks of capacities W i, select m disjoint subsets of items (one for each knapsack) so as to mazimize the total profit, while satisfying the capacity constraints. A surrogate relaxation: z S(λ) = max s.t. m n i=1 j=1 p jx ij m i=1 λ n i j=1 w jx ij m i=1 λ iw i (6) m i=1 x ij 1 j {1,2,...,n} (7) x ij {0,1} i, j (8) a knapsack problem with m copies of each item j (i-th copy has weight λ i w j and profit p j ) and at most one copy of each item can be selected. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
7 z S(λ) = max s.t. m n i=1 j=1 p jx ij m n i=1 j=1 (λ iw j )x ij m i=1 λ iw i (9) m i=1 x ij 1 j {1,2,...,n} (10) x ij {0,1} i, j (11) Since for each item j a copy i with smallest λ i is more convenient, the problem is a standard binary knapsack problem with capacity W = m i=1 λ iw i. To look for a multiplier vector λ 0 providing the tightest (smallest) upper bound, we may solve the surrogate dual problem: min λ 0 z S(λ). Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
8 4) Lagrangian relaxation Often the linear relaxation and the relaxation by elimination provide weak bounds (e.g., when deleting the connectivity constraints in TSP, and demand constraints in UFL). Idea: Eliminate the difficult constraints and add, for each one of them, a term in the objective function with a multiplier u which penalizes its violation and that is 0 (max problem) for all feasible solutions. Example: Multiple binary knapsack problem z mkp = max s.t. m n i=1 j=1 p jx ij n j=1 w jx ij W i i {1,2,...,m} (12) m i=1 x ij 1 j {1,2,...,n} (13) x ij {0,1} i, j (14) Lagrangian relaxation of the cardinality constraints (13): z L(u) = max m n i=1 j=1 p jx ij + n j=1 u j(1 m i=1 x ij) s.t. n j=1 w jx ij W i i {1,2,...,m} (15) x ij {0,1} i, j (16) with multipliers u j 0 for all i, so that z L(u) is an upper bound for z mkp, for every u 0. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
9 Since m n n m p j x ij + u j (1 x ij ) = i=1 j=1 j=1 i=1 m n n (p j u j )x ij + u j, i=1 j=1 j=1 in the Lagrangian subproblem each item j has profit p j = p j u j, weight w j and can be inserted in several knapsacks. The relaxed problem is thus equivalent to m independent binary knapsack problems. For i = 1,2,...,m, we have z i = max n j=1 p jx j s.t. n j=1 w jx j W i (17) and z L(u) = m i=1 z i + n j=1 u j x j {0,1} j {1,2,...,n} (18) To find a multiplier vector u providing the tightest Lagrangian bound, we can solve the Lagrangian dual problem: min u 0 z L(u). Lagrangian relaxation will be discussed in detail in Section 3.6. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
10 Comparison between the previous four types of relaxations (linear, elimination, surrogate and Lagrangian)? Do some of them dominate others? Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
11 5) Combinatorial relaxations i) Asymmetric TSP Given a directed graph G = (V,A) with V = {1,2,...,n} and a cost c ij R associated to each arc (i,j) A, determine a Hamiltonian circuit of minimum total cost. The set of all the assignments, namely X = {x {0,1} A : i:(i,j) A x ij = 1 j, is a relaxation of the set of all the Hamiltonian circuits. j:(i,j) A x ij = 1 i}, ii) Knapsack problem The set n X = {x Z n + : a j x j b } j=1 is a relaxation of X = {x Z n + : n j=1 a jx j b}. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
12 iii) Symmetric TSP Definition: Given an undirected graph G = (V,E) with V = {1,...,n}, a 1-tree di G is a subgraph which contains two edges incident to node 1, together with the edges of a spanning tree with respect to the nodes {2,...,n}. Example: Observation: The set of all the 1-trees is a relaxation of the set of all the Hamiltonian cycles. To find a minimum cost 1-tree: determine a minimum cost spanning tree on nodes {2,...,n} by using a greedy algorithm (Kruskal, Prim,...), select two edges incident in node 1 with smallest cost. Kruskal algorithm for minimum spanning tree: Consider the edges in the order of non decreasing cost. At each step, discard the edge if it creates a cycle with the previously selected edges. Stop when the selected edges cover all the nodes. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
13 3.4.2 Heuristics for primal bounds 1) Greedy methods A feasible solution is constructed piece by piece from scratch. At each step, a piece is selected among the available ones that yield the best local profit, and previous choices are not reconsidered. Example 1: Knapsack problem z ILP = max 16x 1 +22x 2 +12x 3 +8x 4 s.t. 5x 1 +7x 2 +4x 3 +3x 4 14 x 1,...,x 4 {0,1} The variables (items) are ordered by non-increasing profit/volume ratios (p j /a j ) : x 1 x 2 x 3 x 4 p j a j p j /a j Consider items in that order and select (x j = 1) those that do not violate the (residual) capacity constraint, skip the others (x j = 0). Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
14 Feasible solution obtained with the greedy procedure: x = (1,1,0,0) with z greedy = 38. Optimal integer solution: x = (0,1,1,1) with z ILP = 42. Clearly z greedy z ILP. Worst case? Example 2: Symmetric TSP with complete graph Nearest neighbor heuristic: Start from any node, at each step insert the closest node not yet visited, come back to the starting node. Complexity: O(n 2 ), where n is the number of nodes. See TSP algorithms in action at Empirical performance: on TSPLIB(rary) instances it yields tours whose average cost is about 1.26 times that of optimal tours. Worst-case performance: there are instances for which it finds tours that are arbitrarily worse than the optimal tours. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
15 2) Local search methods Consider a generic optimization problem min c(x) x X and try to iteratively improve a current feasible solution. Define, for any feasible solution x, a neighborhood N(x), i.e., a subset of nearby feasible solutions. Start from an initial feasible solution x 0. At each iteration: - Find a best solution x in the neighborhood N(x k ) of the current solution x k. - If c(x ) < c(x k ) then x k+1 = x, otherwise return x k which is a local minimum w.r.t. the neighborhood N(x k ) under consideration. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
16 Example: 2-opt heuristic for Symmetric TSP Given G = (V,E) and a current tour (Hamiltonian cycle) H E. For any given pair of nonadjacent edges e 1 and e 2, try to delete them and replace them by the two unique alternative edges that recombining the two paths into a new tour H. The 2-opt neighborhood N(H) contains all the tours that can be obtained by such a 2-interchange. If c(h ) < c(h) for such a 2-interchange then H becomes the current solution, otherwise H is a local minimum w.r.t the 2-opt neighborhood. Complexity: O(n 2 ) with n = V. Also k-opt for k = 3, with complexity O(n 3 ). Empirical performance: on TSPLIB instances 2-opt provides tours about 1.06 times the optimum, while 3-opt tours 1.04 times the optimum. For step-by-step application see Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
17 Metaheuristics To try to escape from local optima and improve upon local search heuristics, there are metaheuristics such as Simulated Annealing, Tabu Search or Genetic algorithms. Simulated Annealing: Allow occasional moves that yield feasible solutions with worse objective function values. The probability of performing such worsening moves depends on the difference in objective function value. Tabu Search: Allow moves to the best neighboring solution of the current feasible solution even if it has a worse value. Cycling (reconsidering previously examined feasible solutions) is avoided by forbidding to undo recent moves (by making them tabu) for a certain number of iterations. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17
3.10 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 information3.7 Cutting plane methods
3.7 Cutting plane methods Generic ILP problem min{ c t x : x X = {x Z n + : Ax b} } with m n matrix A and n 1 vector b of rationals. According to Meyer s theorem: There exists an ideal formulation: conv(x
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 informationChapter 3: Discrete Optimization Integer Programming
Chapter 3: Discrete Optimization Integer Programming Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Sito web: http://home.deib.polimi.it/amaldi/ott-13-14.shtml A.A. 2013-14 Edoardo
More information3.3 Easy ILP problems and totally unimodular matrices
3.3 Easy ILP problems and totally unimodular matrices Consider a generic ILP problem expressed in standard form where A Z m n with n m, and b Z m. min{c t x : Ax = b, x Z n +} (1) P(b) = {x R n : Ax =
More informationDiscrete (and Continuous) Optimization WI4 131
Discrete (and Continuous) Optimization WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek e-mail: C.Roos@ewi.tudelft.nl
More informationChapter 3: Discrete Optimization Integer Programming
Chapter 3: Discrete Optimization Integer Programming Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Website: http://home.deib.polimi.it/amaldi/opt-16-17.shtml Academic year 2016-17
More information3.7 Strong valid inequalities for structured ILP problems
3.7 Strong valid inequalities for structured ILP problems By studying the problem structure, we can derive strong valid inequalities yielding better approximations of conv(x ) and hence tighter bounds.
More information3.8 Strong valid inequalities
3.8 Strong valid inequalities By studying the problem structure, we can derive strong valid inequalities which lead to better approximations of the ideal formulation conv(x ) and hence to tighter bounds.
More informationWeek Cuts, Branch & Bound, and Lagrangean Relaxation
Week 11 1 Integer Linear Programming This week we will discuss solution methods for solving integer linear programming problems. I will skip the part on complexity theory, Section 11.8, although this is
More informationOptimization Exercise Set n.5 :
Optimization Exercise Set n.5 : Prepared by S. Coniglio translated by O. Jabali 2016/2017 1 5.1 Airport location In air transportation, usually there is not a direct connection between every pair of airports.
More informationDiscrete Optimization 2010 Lecture 8 Lagrangian Relaxation / P, N P and co-n P
Discrete Optimization 2010 Lecture 8 Lagrangian Relaxation / P, N P and co-n P Marc Uetz University of Twente m.uetz@utwente.nl Lecture 8: sheet 1 / 32 Marc Uetz Discrete Optimization Outline 1 Lagrangian
More informationTravelling Salesman Problem
Travelling Salesman Problem Fabio Furini November 10th, 2014 Travelling Salesman Problem 1 Outline 1 Traveling Salesman Problem Separation Travelling Salesman Problem 2 (Asymmetric) Traveling Salesman
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 informationAlgorithm Design Strategies V
Algorithm Design Strategies V Joaquim Madeira Version 0.0 October 2016 U. Aveiro, October 2016 1 Overview The 0-1 Knapsack Problem Revisited The Fractional Knapsack Problem Greedy Algorithms Example Coin
More informationIntroduction 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
More informationOptimization Exercise Set n. 4 :
Optimization Exercise Set n. 4 : Prepared by S. Coniglio and E. Amaldi translated by O. Jabali 2018/2019 1 4.1 Airport location In air transportation, usually there is not a direct connection between every
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 information5 Integer Linear Programming (ILP) E. Amaldi Foundations of Operations Research Politecnico di Milano 1
5 Integer Linear Programming (ILP) E. Amaldi Foundations of Operations Research Politecnico di Milano 1 Definition: An Integer Linear Programming problem is an optimization problem of the form (ILP) min
More informationand to estimate the quality of feasible solutions I A new way to derive dual bounds:
Lagrangian Relaxations and Duality I Recall: I Relaxations provide dual bounds for the problem I So do feasible solutions of dual problems I Having tight dual bounds is important in algorithms (B&B), and
More informationDual bounds: can t get any better than...
Bounds, relaxations and duality Given an optimization problem z max{c(x) x 2 }, how does one find z, or prove that a feasible solution x? is optimal or close to optimal? I Search for a lower and upper
More informationMVE165/MMG630, Applied Optimization Lecture 6 Integer linear programming: models and applications; complexity. Ann-Brith Strömberg
MVE165/MMG630, Integer linear programming: models and applications; complexity Ann-Brith Strömberg 2011 04 01 Modelling with integer variables (Ch. 13.1) Variables Linear programming (LP) uses continuous
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 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 informationEXACT ALGORITHMS FOR THE ATSP
EXACT ALGORITHMS FOR THE ATSP Branch-and-Bound Algorithms: Little-Murty-Sweeney-Karel (Operations Research, ); Bellmore-Malone (Operations Research, ); Garfinkel (Operations Research, ); Smith-Srinivasan-Thompson
More informationPart III: Traveling salesman problems
Transportation Logistics Part III: Traveling salesman problems c R.F. Hartl, S.N. Parragh 1/282 Motivation Motivation Why do we study the TSP? c R.F. Hartl, S.N. Parragh 2/282 Motivation Motivation Why
More informationNew Integer Programming Formulations of the Generalized Travelling Salesman Problem
American Journal of Applied Sciences 4 (11): 932-937, 2007 ISSN 1546-9239 2007 Science Publications New Integer Programming Formulations of the Generalized Travelling Salesman Problem Petrica C. Pop Department
More informationDiscrete (and Continuous) Optimization Solutions of Exercises 2 WI4 131
Discrete (and Continuous) Optimization Solutions of Exercises 2 WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek
More informationArtificial Intelligence Heuristic Search Methods
Artificial Intelligence Heuristic Search Methods Chung-Ang University, Jaesung Lee The original version of this content is created by School of Mathematics, University of Birmingham professor Sandor Zoltan
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 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 informationDetermine the size of an instance of the minimum spanning tree problem.
3.1 Algorithm complexity Consider two alternative algorithms A and B for solving a given problem. Suppose A is O(n 2 ) and B is O(2 n ), where n is the size of the instance. Let n A 0 be the size of the
More informationDiscrete Optimization 2010 Lecture 7 Introduction to Integer Programming
Discrete Optimization 2010 Lecture 7 Introduction to Integer Programming Marc Uetz University of Twente m.uetz@utwente.nl Lecture 8: sheet 1 / 32 Marc Uetz Discrete Optimization Outline 1 Intro: The Matching
More informationComputational Integer Programming. Lecture 2: Modeling and Formulation. Dr. Ted Ralphs
Computational Integer Programming Lecture 2: Modeling and Formulation Dr. Ted Ralphs Computational MILP Lecture 2 1 Reading for This Lecture N&W Sections I.1.1-I.1.6 Wolsey Chapter 1 CCZ Chapter 2 Computational
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 informationOptimisation and Operations Research
Optimisation and Operations Research Lecture 11: Integer Programming Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/ School of Mathematical
More informationUnit 1A: Computational Complexity
Unit 1A: Computational Complexity Course contents: Computational complexity NP-completeness Algorithmic Paradigms Readings Chapters 3, 4, and 5 Unit 1A 1 O: Upper Bounding Function Def: f(n)= O(g(n)) if
More informationBounds on the Traveling Salesman Problem
Bounds on the Traveling Salesman Problem Sean Zachary Roberson Texas A&M University MATH 613, Graph Theory A common routing problem is as follows: given a collection of stops (for example, towns, stations,
More informationSection Notes 8. Integer Programming II. Applied Math 121. Week of April 5, expand your knowledge of big M s and logical constraints.
Section Notes 8 Integer Programming II Applied Math 121 Week of April 5, 2010 Goals for the week understand IP relaxations be able to determine the relative strength of formulations understand the branch
More informationConnectedness of Efficient Solutions in Multiple. Objective Combinatorial Optimization
Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization Jochen Gorski Kathrin Klamroth Stefan Ruzika Communicated by H. Benson Abstract Connectedness of efficient solutions
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 informationThe Core Concept for the Multidimensional Knapsack Problem
The Core Concept for the Multidimensional Knapsack Problem Jakob Puchinger 1, Günther R. Raidl 1, and Ulrich Pferschy 2 1 Institute of Computer Graphics and Algorithms Vienna University of Technology,
More informationIn the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight.
In the original knapsack problem, the value of the contents of the knapsack is maximized subject to a single capacity constraint, for example weight. In the multi-dimensional knapsack problem, additional
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationMaximum Flow Problem (Ford and Fulkerson, 1956)
Maximum Flow Problem (Ford and Fulkerson, 196) In this problem we find the maximum flow possible in a directed connected network with arc capacities. There is unlimited quantity available in the given
More informationTheoretical Computer Science
Theoretical Computer Science 411 (010) 417 44 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: wwwelseviercom/locate/tcs Resource allocation with time intervals
More informationPart III: Traveling salesman problems
Transportation Logistics Part III: Traveling salesman problems c R.F. Hartl, S.N. Parragh 1/74 Motivation Motivation Why do we study the TSP? it easy to formulate it is a difficult problem many significant
More informationThe Steiner Network Problem
The Steiner Network Problem Pekka Orponen T-79.7001 Postgraduate Course on Theoretical Computer Science 7.4.2008 Outline 1. The Steiner Network Problem Linear programming formulation LP relaxation 2. The
More informationBBM402-Lecture 20: LP Duality
BBM402-Lecture 20: LP Duality Lecturer: Lale Özkahya Resources for the presentation: https://courses.engr.illinois.edu/cs473/fa2016/lectures.html An easy LP? which is compact form for max cx subject to
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 informationApproximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs
Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs Haim Kaplan Tel-Aviv University, Israel haimk@post.tau.ac.il Nira Shafrir Tel-Aviv University, Israel shafrirn@post.tau.ac.il
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 P and NP P: The family of problems that can be solved quickly in polynomial time.
More informationConnectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization
Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization Jochen Gorski Kathrin Klamroth Stefan Ruzika Abstract Connectedness of efficient solutions is a powerful property in
More informationNetwork Design and Game Theory Spring 2008 Lecture 6
Network Design and Game Theory Spring 2008 Lecture 6 Guest Lecturer: Aaron Archer Instructor: Mohammad T. Hajiaghayi Scribe: Fengming Wang March 3, 2008 1 Overview We study the Primal-dual, Lagrangian
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 informationCombinatorial optimization problems
Combinatorial optimization problems Heuristic Algorithms Giovanni Righini University of Milan Department of Computer Science (Crema) Optimization In general an optimization problem can be formulated as:
More informationCombinatorial Optimization
Combinatorial Optimization Problem set 8: solutions 1. Fix constants a R and b > 1. For n N, let f(n) = n a and g(n) = b n. Prove that f(n) = o ( g(n) ). Solution. First we observe that g(n) 0 for all
More informationCOT 6936: Topics in Algorithms! Giri Narasimhan. ECS 254A / EC 2443; Phone: x3748
COT 6936: Topics in Algorithms! Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu https://moodle.cis.fiu.edu/v2.1/course/view.php?id=612 Gaussian Elimination! Solving a system of simultaneous
More informationHands-on Tutorial on Optimization F. Eberle, R. Hoeksma, and N. Megow September 26, Branch & Bound
Hands-on Tutorial on Optimization F. Eberle, R. Hoeksma, and N. Megow September 6, 8 Branh & Bound Branh & Bound: A General Framework for ILPs Introdued in the 96 s by Land and Doig Based on two priniple
More informationLecture 11 October 7, 2013
CS 4: Advanced Algorithms Fall 03 Prof. Jelani Nelson Lecture October 7, 03 Scribe: David Ding Overview In the last lecture we talked about set cover: Sets S,..., S m {,..., n}. S has cost c S. Goal: Cover
More informationCS675: Convex and Combinatorial Optimization Fall 2014 Combinatorial Problems as Linear Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2014 Combinatorial Problems as Linear Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source Shortest
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 informationCS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Fall 2016 Combinatorial Problems as Linear and Convex Programs Instructor: Shaddin Dughmi Outline 1 Introduction 2 Shortest Path 3 Algorithms for Single-Source
More informationHill climbing: Simulated annealing and Tabu search
Hill climbing: Simulated annealing and Tabu search Heuristic algorithms Giovanni Righini University of Milan Department of Computer Science (Crema) Hill climbing Instead of repeating local search, it is
More informationRelaxations and Bounds. 6.1 Optimality and Relaxations. Suppose that we are given an IP. z = max c T x : x X,
6 Relaxations and Bounds 6.1 Optimality and Relaxations Suppose that we are given an IP z = max c T x : x X, where X = x : Ax b,x Z n and a vector x X which is a candidate for optimality. Is there a way
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 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 information4. Duality Duality 4.1 Duality of LPs and the duality theorem. min c T x x R n, c R n. s.t. ai Tx = b i i M a i R n
2 4. Duality of LPs and the duality theorem... 22 4.2 Complementary slackness... 23 4.3 The shortest path problem and its dual... 24 4.4 Farkas' Lemma... 25 4.5 Dual information in the tableau... 26 4.6
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 information2001 Dennis L. Bricker Dept. of Industrial Engineering The University of Iowa. Reducing dimensionality of DP page 1
2001 Dennis L. Bricker Dept. of Industrial Engineering The University of Iowa Reducing dimensionality of DP page 1 Consider a knapsack with a weight capacity of 15 and a volume capacity of 12. Item # Value
More informationBranch-and-Bound. Leo Liberti. LIX, École Polytechnique, France. INF , Lecture p. 1
Branch-and-Bound Leo Liberti LIX, École Polytechnique, France INF431 2011, Lecture p. 1 Reminders INF431 2011, Lecture p. 2 Problems Decision problem: a question admitting a YES/NO answer Example HAMILTONIAN
More informationGRAPH ALGORITHMS Week 7 (13 Nov - 18 Nov 2017)
GRAPH ALGORITHMS Week 7 (13 Nov - 18 Nov 2017) C. Croitoru croitoru@info.uaic.ro FII November 12, 2017 1 / 33 OUTLINE Matchings Analytical Formulation of the Maximum Matching Problem Perfect Matchings
More informationData Structures in Java
Data Structures in Java Lecture 21: Introduction to NP-Completeness 12/9/2015 Daniel Bauer Algorithms and Problem Solving Purpose of algorithms: find solutions to problems. Data Structures provide ways
More informationZebo Peng Embedded Systems Laboratory IDA, Linköping University
TDTS 01 Lecture 8 Optimization Heuristics for Synthesis Zebo Peng Embedded Systems Laboratory IDA, Linköping University Lecture 8 Optimization problems Heuristic techniques Simulated annealing Genetic
More informationExact and Heuristic Algorithms for the Symmetric and Asymmetric Vehicle Routing Problem with Backhauls
Exact and Heuristic Algorithms for the Symmetric and Asymmetric Vehicle Routing Problem with Backhauls Paolo Toth, Daniele Vigo ECCO IX - Dublin 1996 Exact and Heuristic Algorithms for VRPB 1 Vehicle Routing
More informationWeek 8. 1 LP is easy: the Ellipsoid Method
Week 8 1 LP is easy: the Ellipsoid Method In 1979 Khachyan proved that LP is solvable in polynomial time by a method of shrinking ellipsoids. The running time is polynomial in the number of variables n,
More informationInteger Linear Programming (ILP)
Integer Linear Programming (ILP) Zdeněk Hanzálek, Přemysl Šůcha hanzalek@fel.cvut.cz CTU in Prague March 8, 2017 Z. Hanzálek (CTU) Integer Linear Programming (ILP) March 8, 2017 1 / 43 Table of contents
More informationFINAL EXAM PRACTICE PROBLEMS CMSC 451 (Spring 2016)
FINAL EXAM PRACTICE PROBLEMS CMSC 451 (Spring 2016) The final exam will be on Thursday, May 12, from 8:00 10:00 am, at our regular class location (CSI 2117). It will be closed-book and closed-notes, except
More informationDecomposition-based Methods for Large-scale Discrete Optimization p.1
Decomposition-based Methods for Large-scale Discrete Optimization Matthew V Galati Ted K Ralphs Department of Industrial and Systems Engineering Lehigh University, Bethlehem, PA, USA Départment de Mathématiques
More informationApproximation Algorithms for Re-optimization
Approximation Algorithms for Re-optimization DRAFT PLEASE DO NOT CITE Dean Alderucci Table of Contents 1.Introduction... 2 2.Overview of the Current State of Re-Optimization Research... 3 2.1.General Results
More informationAlgorithms. NP -Complete Problems. Dong Kyue Kim Hanyang University
Algorithms NP -Complete Problems Dong Kyue Kim Hanyang University dqkim@hanyang.ac.kr The Class P Definition 13.2 Polynomially bounded An algorithm is said to be polynomially bounded if its worst-case
More informationInteger Linear Programming Modeling
DM554/DM545 Linear and Lecture 9 Integer Linear Programming Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. Assignment Problem Knapsack Problem
More informationAnt Colony Optimization: an introduction. Daniel Chivilikhin
Ant Colony Optimization: an introduction Daniel Chivilikhin 03.04.2013 Outline 1. Biological inspiration of ACO 2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic 4. ACO for the Traveling
More informationOptimisation and Operations Research
Optimisation and Operations Research Lecture 15: The Greedy Heuristic Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/ School of
More informationThis means that we can assume each list ) is
This means that we can assume each list ) is of the form ),, ( )with < and Since the sizes of the items are integers, there are at most +1pairs in each list Furthermore, if we let = be the maximum possible
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 informationTechnische Universität München, Zentrum Mathematik Lehrstuhl für Angewandte Geometrie und Diskrete Mathematik. Combinatorial Optimization (MA 4502)
Technische Universität München, Zentrum Mathematik Lehrstuhl für Angewandte Geometrie und Diskrete Mathematik Combinatorial Optimization (MA 4502) Dr. Michael Ritter Problem Sheet 1 Homework Problems Exercise
More informationLinear Programming. Scheduling problems
Linear Programming Scheduling problems Linear programming (LP) ( )., 1, for 0 min 1 1 1 1 1 11 1 1 n i x b x a x a b x a x a x c x c x z i m n mn m n n n n! = + + + + + + = Extreme points x ={x 1,,x n
More informationThe Multidimensional Knapsack Problem: Structure and Algorithms
The Multidimensional Knapsack Problem: Structure and Algorithms Jakob Puchinger NICTA Victoria Laboratory Department of Computer Science & Software Engineering University of Melbourne, Australia jakobp@csse.unimelb.edu.au
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 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 informationMaximum flow problem
Maximum flow problem 7000 Network flows Network Directed graph G = (V, E) Source node s V, sink node t V Edge capacities: cap : E R 0 Flow: f : E R 0 satisfying 1. Flow conservation constraints e:target(e)=v
More informationThe Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006
The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006 1 Simplex solves LP by starting at a Basic Feasible Solution (BFS) and moving from BFS to BFS, always improving the objective function,
More informationA tabu search algorithm for the minmax regret minimum spanning tree problem with interval data
Noname manuscript No. (will be inserted by the editor) A tabu search algorithm for the minmax regret minimum spanning tree problem with interval data Adam Kasperski Mariusz Makuchowski Pawe l Zieliński
More informationa 1 a 2 a 3 a 4 v i c i c(a 1, a 3 ) = 3
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 17 March 30th 1 Overview In the previous lecture, we saw examples of combinatorial problems: the Maximal Matching problem and the Minimum
More information(tree searching technique) (Boolean formulas) satisfying assignment: (X 1, X 2 )
Algorithms Chapter 5: The Tree Searching Strategy - Examples 1 / 11 Chapter 5: The Tree Searching Strategy 1. Ex 5.1Determine the satisfiability of the following Boolean formulas by depth-first search
More information7. Lecture notes on the ellipsoid algorithm
Massachusetts Institute of Technology Michel X. Goemans 18.433: Combinatorial Optimization 7. Lecture notes on the ellipsoid algorithm The simplex algorithm was the first algorithm proposed for linear
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 information1 T 1 = where 1 is the all-ones vector. For the upper bound, let v 1 be the eigenvector corresponding. u:(u,v) E v 1(u)
CME 305: Discrete Mathematics and Algorithms Instructor: Reza Zadeh (rezab@stanford.edu) Final Review Session 03/20/17 1. Let G = (V, E) be an unweighted, undirected graph. Let λ 1 be the maximum eigenvalue
More information15.081J/6.251J Introduction to Mathematical Programming. Lecture 24: Discrete Optimization
15.081J/6.251J Introduction to Mathematical Programming Lecture 24: Discrete Optimization 1 Outline Modeling with integer variables Slide 1 What is a good formulation? Theme: The Power of Formulations
More information