DOCSLIB.ORG

Cutting Plane and Subgradient Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cutting Plane and Subgradient Methods Cutting Plane and Subgradient Methods John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA October 12, 2009 Collaborators: Luc Basescu, Brian Borchers, Mohammad Oskoorouchi, Srini Ramaswamy, Kartik Sivaramakrishnan, Mike Todd Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 1 / 72 Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 2 / 72 Interior Point Cutting Plane and Column Generation Methods Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 3 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 4 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction The Traveling Salesman Problem TSP Webpage Robert Bosch Bill Cook, Georgia Tech Oberlin College Lower bounds determined using cutting planes and branch-and-cut. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 5 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction MIP Formulation Want to solve the integer programming problem T maxy2IRm b y subject to AT y ≤ c yi integer for i 2 I Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 6 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ≥ a0 y ≤ c0 x 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ≥ a0 y ≤ c0 x 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ; ≥ a0 y ≤ c0 x x0 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ; ≥ a0 y ≤ c0 x x0 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration fy 2 IRm : AT y ≤ cg PP PP PPP rrr P rrrrr H HH Soln to LP relaxation rrHrrrr H t HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration fy 2 IRm : AT y ≤ cg PP PP PPP rrr P Cutting plane T a1 y = c1 rrrrr H HH Soln to LP relaxation rrHrrrr H t HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T fy 2 IR : A y ≤ c; a1 y ≤ c1g PP PP PPP rrr P Soln to LP relaxation rrrrr t H HH rrHrrrr H HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T fy 2 IR : A y ≤ c; a1 y ≤ c1g Cutting plane PP PP aT y = c PPP2 2 rrr P Soln to LP relaxation rrrrr t H HH rrHrrrr H HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T T fy 2 IR : A y ≤ c; a1 y ≤ c1; a2 y ≤ b2g PP PP PPP rrr P rrrrr H Solution to LP relaxation HH rrHrrrr d H HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T T fy 2 IR : A y ≤ c; a1 y ≤ c1; a2 y ≤ b2g PP PP PPP rrr P rrrrr H Solution to LP relaxation HH rrHrrrr d H Solution is integral, HH H so optimal to IP rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 9 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Valid constraints for MaxCut Find the maximum cut in a graph. Find ground state of Ising spin glass. Vertices $ spins: “Up” or “Down”. Edges $ interactions: ±1. t t t t t Interactions known. Deduce spins. t t t t t 1 u; v opposite spins y = t t t t t uv 0 u; v same spin t t t t t y $ incidence vector of cut. t t t t t Valid constraint: Any cut and any cycle intersect f e t tg in an even number of edges. h ye +yf +yg −yh ≤ 2 t t Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 10 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Valid constraints for MaxCut Find the maximum cut in a graph. Find ground state of Ising spin glass. Vertices $ spins: “Up” or “Down”. Edges $ interactions: ±1. t t t t t Interactions known. Deduce spins. t t t t t 1 u; v opposite spins y = t t t t t uv 0 u; v same spin t t t t t y $ incidence vector of cut. t t t t t Valid constraint: Any cut and any cycle intersect f e t tg in an even number of edges. h ye +yf +yg −yh ≤ 2 t t Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 10 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2. Can typically solve 100x100 grids in about 5 minutes with be = ±1. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 11 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2. Can typically solve 100x100 grids in about 5 minutes with be = ±1. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 11 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2. Can typically solve 100x100 grids in about 5 minutes with be = ±1. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 11 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2.
Load more

Recommended publications
  • Hyperheuristics in Logistics Kassem Danach
    Hyperheuristics in Logistics Kassem Danach To cite this version: Kassem Danach. Hyperheuristics in Logistics. Combinatorics [math.CO]. Ecole Centrale de Lille, 2016. English. NNT : 2016ECLI0025. tel-01485160 HAL Id: tel-01485160 https://tel.archives-ouvertes.fr/tel-01485160 Submitted on 8 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No d’ordre: 315 Centrale Lille THÈSE présentée en vue d’obtenir le grade de DOCTEUR en Automatique, Génie Informatique, Traitement du Signal et des Images par Kassem Danach DOCTORAT DELIVRE PAR CENTRALE LILLE Hyperheuristiques pour des problèmes d’optimisation en logistique Hyperheuristics in Logistics Soutenue le 21 decembre 2016 devant le jury d’examen: President: Pr. Laetitia Jourdan Université de Lille 1, France Rapporteurs: Pr. Adnan Yassine Université du Havre, France Dr. Reza Abdi University of Bradford, United Kingdom Examinateurs: Pr. Saïd Hanafi Université de Valenciennes, France Dr. Abbas Tarhini Lebanese American University, Lebanon Dr. Rahimeh Neamatian Monemin University Road, United Kingdom Directeur de thèse: Pr. Frédéric Semet Ecole Centrale de Lille, France Co-encadrant: Dr. Shahin Gelareh Université de l’ Artois, France Invited Professor: Dr. Wissam Khalil Université Libanais, Lebanon Thèse préparée dans le Laboratoire CRYStAL École Doctorale SPI 072 (EC Lille) 2 Acknowledgements Firstly, I would like to express my sincere gratitude to my advisor Prof.
    [Show full text]
  • Cocircuits of Vector Matroids
    RICE UNIVERSITY Cocircuits of Vector Matroids by John David Arellano A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Master of Arts APPROVED, THESIS COMMITTEE: ?~ Ill::f;Cks, Chair Associate Professor of Computational and Applied Mathematics ~~~~ Richard A. Tapia Maxfield-Oshman Professor of Engineering University Professor of Computational and Applied Mathematics Wotao Yin Assistant Professor of Computational and Applied Mathematics Houston, Texas August, 2011 ABSTRACT Cocircuits of Vector Matroids by John David Arellano In this thesis, I present a set covering problem (SCP) formulation of the matroid cogirth problem, finding the cardinality of the smallest cocircuit of a matroid. Ad­ dressing the matroid cogirth problem can lead to significantly enhancing the design process of sensor networks. The solution to the matroid cogirth problem provides the degree of redundancy of the corresponding sensor network, and allows for the evalu­ ation of the quality of the network. I provide an introduction to matroids, and their relation to the degree of redundancy problem. I also discuss existing methods devel­ oped to solve the matroid cogirth problem and the SCP. Computational results are provided to validate a branch-and-cut algorithm that addresses the SCP formulation. Acknowledgments I would like to thank my parents and family for the love and support throughout my graduate career. I would like to thank my advisor, Dr. Illya Hicks, and the rest of my committee for their guidance and support. I would also like to thank Dr. Maria Cristina Villalobos for her guidance and support during my undergraduate career. A thanks also goes to Nabor Reyna, Dr.
    [Show full text]
  • Generic Branch-Cut-And-Price
    Generic Branch-Cut-and-Price Diplomarbeit bei PD Dr. M. L¨ubbecke vorgelegt von Gerald Gamrath 1 Fachbereich Mathematik der Technischen Universit¨atBerlin Berlin, 16. M¨arz2010 1Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin, gamrath@zib.de 2 Contents Acknowledgments iii 1 Introduction 1 1.1 Definitions . .3 1.2 A Brief History of Branch-and-Price . .6 2 Dantzig-Wolfe Decomposition for MIPs 9 2.1 The Convexification Approach . 11 2.2 The Discretization Approach . 13 2.3 Quality of the Relaxation . 21 3 Extending SCIP to a Generic Branch-Cut-and-Price Solver 25 3.1 SCIP|a MIP Solver . 25 3.2 GCG|a Generic Branch-Cut-and-Price Solver . 27 3.3 Computational Environment . 35 4 Solving the Master Problem 39 4.1 Basics in Column Generation . 39 4.1.1 Reduced Cost Pricing . 42 4.1.2 Farkas Pricing . 43 4.1.3 Finiteness and Correctness . 44 4.2 Solving the Dantzig-Wolfe Master Problem . 45 4.3 Implementation Details . 48 4.3.1 Farkas Pricing . 49 4.3.2 Reduced Cost Pricing . 52 4.3.3 Making Use of Bounds . 54 4.4 Computational Results . 58 4.4.1 Farkas Pricing . 59 4.4.2 Reduced Cost Pricing . 65 5 Branching 71 5.1 Branching on Original Variables . 73 5.2 Branching on Variables of the Extended Problem . 77 5.3 Branching on Aggregated Variables . 78 5.4 Ryan and Foster Branching . 79 i ii Contents 5.5 Other Branching Rules . 82 5.6 Implementation Details . 85 5.6.1 Branching on Original Variables . 87 5.6.2 Ryan and Foster Branching .
    [Show full text]
  • Branch-And-Price: Column Generation for Solving Huge Integer Programs ,Y
    BranchandPrice Column Generation for y Solving Huge Integer Programs Cynthia Barnhart Ellis L Johnson George L Nemhauser Martin WPSavelsb ergh Pamela H Vance School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta GA Abstract We discuss formulations of integer programs with a huge number of variables and their solution by column generation metho ds ie implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a no de of the branch andb ound tree We present classes of mo dels for whichthisapproach decomp oses the problem provides tighter LP relaxations and eliminates symmetryWethen discuss computational issues and implementation of column generation branchand b ound algorithms including sp ecial branching rules and ecientways to solvethe LP relaxation February Revised May Revised January Intro duction The successful solution of largescale mixed integer programming MIP problems re quires formulations whose linear programming LP relaxations give a go o d approxima Currently at MIT Currently at Auburn University This research has b een supp orted by the following grants and contracts NSF SES NSF and AFORS DDM NSF DDM NSF DMI and IBM MHV y An abbreviated and unrefereed version of this pap er app eared in Barnhart et al tion to the convex hull of feasible solutions In the last decade a great deal of attention has b een given to the branchandcut approach to solving MIPs Homan and Padb erg and Nemhauser and Wolsey give general exp ositions of this metho dology The basic
    [Show full text]
  • IEEE Paper Template in A4 (V1)
    I.J. Information Technology and Computer Science, 2015, 07, 28-34 Published Online June 2015 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijitcs.2015.07.04 Great Deluge Algorithm for the Linear Ordering Problem: The Case of Tanzanian Input-Output Table Amos Mathias Department of Science Mathematics and Technology Education, University of Dodoma, Box 523, Dodoma, Tanzania Email: ms_mathias@yahoo.co.uk Allen R. Mushi Department of Mathematics, University of Dar es salaam, Box 35062, DSM, Tanzania Email: allenmushi66@gmail.com Abstract—Given a weighted complete digraph, the Linear Since then, it has received considerable attention by Ordering Problem (LOP) consists of finding and acyclic many researchers and hence becoming a problem of tournament with maximum weight. It is sometimes referred to interest to date. LOP is found in a number of applications, as triangulation problem or permutation problem depending on such as Triangulation of Input-Output matrices in the context of its application. This study introduces an economics, Archaeological seriation, Minimizing total algorithm for LOP and applied for triangulation of Tanzanian Input-Output tables. The algorithm development process uses weighted completion time in one-machine scheduling, Great Deluge heuristic method. It is implemented using C++ Aggregation of Individual preferences, Grötschel, Junger programming language and tested on a personal computer with and Reinelt [5], ordering of teams in sports tournaments, 2.40GHZ speed processor. The algorithm has been able to Mushi [1] as well as Machine translation, Tromble [6]. triangulate the Tanzanian input-output tables of size 79×79 A number of algorithms for the LOP solution have within a reasonable time (1.17 seconds).
    [Show full text]
  • Experiments with the Fixed-Parameter Approach for Two
    Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 9, no. 1, pp. 149–163 (2005) Experiments with the Fixed-Parameter Approach for Two-Layer Planarization Matthew Suderman Sue Whitesides School of Computer Science McGill University http://www.cs.mcgill.ca/ msuder@cs.mcgill.ca sue@cs.mcgill.ca Abstract We present computational results of an implementation based on the fixed parameter tractability (FPT) approach for biplanarizing graphs. These results show that the implementation can efficiently find minimum biplanarizing sets containing up to about 18 edges, thus making it com- parable to previous integer linear programming approaches. We show how our implementation slightly improves the theoretical running time to O(6bpr(G) + |G|) for any input graph G. Finally, we explain how our experimental work predicts how performance on sparse graphs may be improved. Article Type Communicated by Submitted Revised regular paper G. Liotta February 2004 July 2005 Research supported by an NSERC operating grant and an FQRNT scholarship. M. Suderman et al., Experiments with FPT , JGAA, 9(1) 149–163 (2005) 150 1 Introduction A layered drawing of a graph G is a 2-dimensional drawing of G in which each vertex is placed on one of several parallel lines called layers, and each edge is drawn as a straight line between its end-vertices. In this paper, we consider drawings on two layers in which the end-vertices of each edge lie on different layers. These drawings have applications in visualization [1, 10], DNA mapping [17], and VLSI layout [12]; a recent survey [15] gives more details.
    [Show full text]
  • 1 Date: September 15, 1998 File
    1 date: September 15, 1998 file: mitche2 BRANCH-AND-CUT ALGORITHMS Branch-and-cut methods for general integer FOR INTEGER PROGRAMMING, programming problems are also of great interest Branch-and-cut (see, for example, the papers [4, 7, 11, 16, 17, 22, 28, 30]). It is usually not possible to efficiently Branch-and-cut methods are exact algorithms solve a general integer programming problem us- for integer programming problems. They consist ing just a cutting plane approach, and it is there- of a combination of a cutting plane method fore necessary to also to branch, resulting in a with a branch-and-bound algorithm.These branch-and-cut approach. A pure branch-and- methods work by solving a sequence of lin- bound approach can be sped up considerably by ear programming relaxations of the integer pro- the employment of a cutting plane scheme, ei- gramming problem. Cutting plane methods im- ther just at the top of the tree, or at every node prove the relaxation of the problem to more of the tree. closely approximate the integer programming For general problems, the specialized facets problem, and branch-and-bound algorithms pro- used when solving a specific combinatorial opti- ceed by a sophisticated divide and conquer ap- mization problem are not available. Some use- proach to solve problems. The material in this ful families of general inequalities have been entry builds on the material contained in the developed; these include cuts based on knap- entries on cutting plane and branch-and-bound sack problems [17, 22, 23], Gomory cutting methods. planes [19, 20, 5, 12], lift and project cut- Perhaps the best known branch-and-cut algo- ting planes [29, 33, 3, 4], and Fenchel cutting rithms are those that have been used to solve planes [9].
    [Show full text]
  • Branch-And-Cut and Branch-And-Cut-And-Price Algorithms for Solving Vehicle Routing Problems
    Downloaded from orbit.dtu.dk on: Oct 05, 2021 Branch-and-cut and Branch-and-Cut-and-Price Algorithms for Solving Vehicle Routing Problems Jepsen, Mads Kehlet Publication date: 2011 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Jepsen, M. K. (2011). Branch-and-cut and Branch-and-Cut-and-Price Algorithms for Solving Vehicle Routing Problems. Technical University of Denmark. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. PHD Thesis Branch-and-cut and Branch-and-Cut-and-Price Algorithms for Solving Vehicle Routing Problems Author: Mads Kehlet Jepsen Supervisor: Professor David Pisinger min cexe e E X2 s:t: x = 2 i V e 8 2 c e δ(i) 2X x = 2 K e j j e δ(0) 2X x 2r(S) S V ; S 2 e ≥ 8 ⊆ c j j ≥ e δ(S) 2X x 0; 1; 2 e δ(0) e 2 f g 8 2 x 0; 1 e E δ(0) e 2 f g 8 2 n min cpλp p P X2 s.t α λ = 1 i V ijp p 8 2 c p P (i;j) δ+(i) X2 X2 λ 0; 1 p P p 2 f g 8 2 min c x p y e e − i i e E i N X2 X2 x = 2y i V e i 8 2 e δ(i) 2X y0 = 1 x 2y i S; S V ; S 2 e ≥ i 8 2 8 ⊆ c j j ≥ e δ(S) 2X d y Q i i ≤ i N X2 x 0; 1 e E e 2 f g 8 2 y 0; 1 i V: i 2 f g 8 2 8.
    [Show full text]
  • Branch-And-Cut for Nonlinear Power Systems Problems by Chen Chen A
    Branch-and-Cut for Nonlinear Power Systems Problems by Chen Chen A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering / Industrial Engineering and Operations Research in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Shmuel S. Oren, Chair Professor Alper Atamt¨urk Professor Kameshwar Poolla Fall 2015 Branch-and-Cut for Nonlinear Power Systems Problems Copyright 2015 by Chen Chen 1 Abstract Branch-and-Cut for Nonlinear Power Systems Problems by Chen Chen Doctor of Philosophy in Engineering / Industrial Engineering and Operations Research University of California, Berkeley Professor Shmuel S. Oren, Chair This dissertation is concerned with the design of branch-and-cut algorithms for a variety of nonconvex nonlinear problems pertaining to power systems operations and planning. By understanding the structure of specific problems, we can lever- age powerful commercial optimization solvers designed for convex optimization and mixed-integer programs. The bulk of the work concerns the Alternating Current Optimal Power Flow (ACOPF) problem. The ACOPF problem is to find a mini- mum cost generation dispatch that will yield flows that can satisfy demand as well as various engineering constraints. A standard formulation can be posed as a non- convex Quadratically Constrained Quadratic Program with complex variables. We develop a novel spatial branch-and-bound algorithm for generic nonconvex QCQP with bounded complex variables that relies on a semidefinite programming (SDP) relaxation strengthened with linear valid inequalities. ACOPF-specific domain reduc- tion or bound tightening techniques are also introduced to improve the algorithm's convergence rate.
    [Show full text]
  • A Hyper-Matheuristic Methodology for Mixed Integer Linear Optimization Problems
    Optimization Online (2020) manuscript 7351, 1–20 A Hyper-Matheuristic Methodology for Mixed Integer Linear Optimization Problems Mart´ın Gonzalez´ a,∗, Jose Juan Lopez-Esp´ ´ına, Juan Aparicioa and El-Ghazali Talbib aCenter of Operations Research, Miguel Hernandez´ University, Elche, Alicante, Spain bUniversity of Lille, France E-mail: martin.gonzaleze@umh.es [Mart´ın Gonzalez]´ Abstract Metaheuristics and exact methods are the most common tools used to solve Mixed Integer Linear Optimization Problems (MILPs). These problems are usually NP-hard problems, being intractable to obtain optimal solutions in reasonable time for large scale instances of the problem. In this work, we propose a MILP-based decomposition in which the main problem is divided in two hierarchical subproblems according to the complexity of the latter. In this paper, we investigate a decomposition separating the discrete and continuous variables, treating each subproblem with different optimization methods. A metaheuristic focuses on the discrete variable search space while the exact method takes into account the continuous variables. Hence, the metaheuristic uses an indirect representation which encode an incomplete solution for the problem, while the exact method is applied to decode the solution and generate a complete solution. The experimental results show that the solutions obtained applying the proposed MILP-based decomposition are better than those obtained solving the problem globally using a metaheuristic method. Moreover, a novel Hyper-Matheuristic scheme is introduced in which the best algorithm selection is found for a set of cooperative metaheuristics and exact optimization algorithms. Keywords: Hyper-Matheuristic, Metaheuristics, Exact Methods, Mixed Integer Problems, MILP Decomposition. 1. Introduction Mixed Integer Linear Optimization (MILP) deals with mathematical optimization problems involving two families of variables: discrete and continuous variables.
    [Show full text]
  • Cutting Plane Methods and Subgradient Methods
    INFORMS—New Orleans 2005 c 2005 INFORMS | isbn 0000-0000 doi 10.1287/educ.1053.0000 53 Cutting Plane Methods and Subgradient Methods John E. Mitchell Mathematical Sciences, Rensselaer Polytechnic Institute, Troy NY 12180, USA. mitchj@rpi.edu Abstract Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of optimization problems. We discuss the use of interior point methods in a column generation scheme. Semidefinite programming relaxations of combinatorial optimization problems are often tighter than linear programming relaxations. We describe some research in using SDP relaxations to find exact solutions to combinatorial optimization problems. Semidefinite programs are expensive to solve directly, so we also consider cutting surface approaches to solving them. Finally, we look at recent smoothing techniques for solving nonsmooth optimization problems using a subgradient approach; these methods have some links to cutting surface approaches. Keywords interior point column generation, cutting planes, cutting surfaces, semidefinite pro- gramming, subgradients 1. Introduction In the 25 years since the publication of Karmarkar’s seminal paper [71], interior point algo- rithms have become the method of choice for large scale linear programming problems. They have also had a great impact beyond linear programming: the extension of the theory of interior point methods for linear programming to more general conic programming [109] has led to the explosion of interest in semidefinite programming and also second-order cone programming.
    [Show full text]
  • Cutting Planes for Mixed-Integer Programming: Theory and Practice
    Cutting Planes for Mixed-Integer Programming: Theory and Practice Oktay G¨unl¨uk Math Sciences, IBM Research April 2018 { ORF523, Princeton Mathematical optimization 1 • A generic mathematical optimization problem: min : f(x) subject to: gi(x) ≤ 0 i = 1; : : : ; m x 2 X • Computationally tractable cases: n { If f(x) and all gi(x) are linear, and X = R+ ) LP n1 n2 { If f(x) and all gi(x) are linear, and X = Z+ × R+ ) MILP n { If f(x) is quadratic and all gi(x) are linear, and X = R+ ) QP n { If f(x) and all gi(x) are quadratic, and X = R+ ) QCQP • Only LP can be solved in polynomial time. Even Box QP is hard! min : xT Qx subject to: 1 ≥ x ≥ 0 Mixed-integer programming 2 • A generic Mixed Integer Linear Program has the form: T minfc x : Ax ≥ b; x ≥ 0; xj integer; j 2 Ig where matrix A does not necessarily have a special structure. • A very large number of practical problems can be modeled in this form: { Production planning, { Airline scheduling (routing, staffing, etc. ) { Telecommunication network design, { Classroom scheduling, { Combinatorial auctions, { ... • In theory, MIP is NP-hard: not much hope for efficient algorithms. • But in practice, even very large MIPs can be solved to optimality in reasonable time. Mixed-integer programming 3 • A generic Mixed Integer Linear Program has the form: T minfc x : Ax ≥ b; x ≥ 0; xj integer; j 2 Ig where matrix A does not necessarily have a special structure. cT x x∗ LP relaxation conv(IP ) Overview of the talk 4 • Introduction { Mixed-integer programming, branch-and-cut • Commercial Software (Cplex)
    [Show full text]