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Improved Branch and Bound Algorithms for Integer IMPROVED BRANCH AND BOUND ALGORITHMS FOR INTEGER PROGRAMMING By Brian Borchers A Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Ma jor Sub ject Mathematical Sciences Approved by the Examining Committee Professor John E Mitchell Thesis Adviser Professer Joseph G Ecker Member Professor George Hab etler Member Professor Carlton E Lemke Member Professor John Mittenthal Member Rensselaer Polytechnic Institute Troy New York June For Graduation August IMPROVED BRANCH AND BOUND ALGORITHMS FOR INTEGER PROGRAMMING By Brian Borchers An Abstract of a Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Ma jor Sub ject Mathematical Sciences The original of the complete thesis is on le in the Rensselaer Polytechnic Institute Library Approved by the Examining Committee Professor John E Mitchell Thesis Adviser Professer Joseph G Ecker Member Professor George Hab etler Member Professor Carlton E Lemke Member Professor John Mittenthal Member Rensselaer Polytechnic Institute Troy New York June For Graduation August CONTENTS LIST OF TABLES iv LIST OF FIGURES vii ACKNOWLEDGEMENT ix ABSTRACT x Intro duction and Historical Review The Branch and Bound Metho d Interior Point Metho ds for Linear Programming Using the DualAne Metho d in a Branch and Bound Co de for Mixed Integer Linear Programming Problems The DualAne Metho d The DualAne Metho d for Problems with Upp er Bounds Starting and Stopping the DualAne Metho d Detecting Fractional Variables The Branch and Bound Algorithm Branching Strategies Computational Details Sample Problems Computational Results Using the PrimalDual Metho d in a Branch and Bound Co de for Mixed Integer Linear Programming Problems The PrimalDual Metho d The PrimalDual Metho d and Complementary Slackness Detecting Fractional Variables Warm starting the PrimalDual Metho d Implementing the PrimalDual Metho d The Exp erimental Branch and Bound Co de More Sample Problems ii Computational Results Solving Mixed Integer Nonlinear Programming Problems Solving the Subproblems Heuristics for Detecting Fractional Solutions Generating Lower Bounds The Branch and Bound Algorithm Computational Results Discussion and Conclusions LITERATURE CITED APPENDICES A Benchmarks for the PrimalDual Co de B A Small Network Design Problem iii LIST OF TABLES Table Sample problem statistics Table OSL results for problem set one Table Branch and b ound results for problem set one Table Fixed order branch and b ound results for problem set one Table OSL results for problem set two Table Branch and b ound results for problem set two Table Fixed order branch and b ound results for problem set two Table OSL results for problem set three Table Branch and b ound results for problem set three Table Fixed order branch and b ound results for problem set three Table OSL results for problem set four Table Branch and b ound results for problem set four Table Fixed order branch and b ound results for problem set four Table Problem set one without early branching Table Problem set two without early branching Table Problem set three without early branching Table Problem set four without early branching Table Problem set one without warm start Table Problem set two without warm start Table Problem set three without warm start Table Problem set four without warm start Table Sample problem statistics Table Simplex iterations p er second in EKKSSLV and EKKMSLV iv Table OSL results for problem set one Table Branch and b ound results for problem set one Table Problem set one without early branching Table Problem set one without warm start Table OSL results for problem set two Table Branch and b ound results for problem set two Table Problem set two without early branching Table Problem set two without warm start Table OSL results for problem set three Table Branch and b ound results for problem set three Table Problem set three without early branching Table Problem set three without warm start Table OSL results for problem set four Table Branch and b ound results for problem set four Table Problem set four without early branching Table Problem set four without warm start T able OSL results for problem set ve Table Branch and b ound results for problem set ve Table Problem set ve without early branching Table Problem set ve without warm start Table OSL results for problem sets eight through ten Table Branch and b ound results for problem sets eight through ten Table Problem set eight through ten without early branching Table Problems eight through ten without warm start Table OSL results for problem set six Table Branch and b ound results for problem set six v Table OSL results for problem set seven Table Branch and b ound results for problem set seven Table Branch and b ound results for p erturb ed problem set six Table OSL results for p erturb ed problem set six Table Characteristics of the sample problems Table Computational results for the co de without heuristics Table Computational results for the co de with heuristics Table A OSL simplex solutions of the LP relaxations Table A OSL primaldual solutions of the LP relaxations Table A Exp erimental primaldual solutions of the LP relaxations Table A Exp erimental dualane solutions of the LP relaxations Table B Flows b etween cities Table B Distances b etween cities vi LIST OF FIGURES Figure A branch and b ound example Figure The branch and b ound algorithm Figure The revised branch and b ound algorithm Figure Lower b ounds for a sample problem Figure Estimates of x for a sample problem Figure The branch and b ound algorithm Figure Subproblems solved relative to OSL Figure CPU time relativetoOSL Figure Subproblems solved relative to OSL Figure Iterations p er subproblem Figure CPU time relativetoOSL Figure The central tra jectory Figure The branch and b ound algorithm Figure Subproblems solved relative to OSL Figure Iterations p er subproblem Figure CPU time p er subproblem relativetoOSL Figure CPU times relativetoOSL Figure The branch and b ound algorithm for MINLP Figure A LP results for problem sets one through ten Figure A Problem b efore minimum degree ordering Figure A Problem after minimum degree ordering Figure A The Cholesky factors for problem Figure A Problem b efore minimum degree ordering vii Figure A Problem after minimum degree ordering Figure A The Cholesky factors for problem Figure A Problem b efore ordering Figure A Problem after minimum degree ordering Figure A The Cholesky factors for problem viii ACKNOWLEDGEMENT Iwould liketotake this opp ortunitytothankmy advisor John Mitchell for in tro ducing me to the whole eld of interior p oint metho ds and sup ervising the work that wentinto this thesis Iwould also like to thank the memb ers of the thesis committeeJoeEcker George Hab etler Carlton E Lemke and John Mittenthal My ancee Suzanne Blo o dgo o d provided moral supp ort and pro ofread the text of the thesis for sp elling errors Our cat Dustmop kept me company during long nights sp ent at the terminal This researchwas partially supp orted by ONR Grantnumb er NJ ix ABSTRACT This thesis is an investigation into improved branch and b ound algorithms for linear and nonlinear mixed integer programming Heuristics are develop ed for determining when the solution to a subproblem will involve fractional variables When this situation is detected the algorithm branches early creating new subproblems with the fractional variables xed at integer values In this wayweavoid the work of solving the current subproblem to optimality Branch and b ound co des for mixed integer linear programming problems have traditionally used the simplex metho d to solve the linear programming subprob lems that arise Early detection of fractional variables is dicult with the simplex metho d b ecause a variable might b ecome basic and thus fractional at the last it eration of the simplex metho d In contrast with an interior p oint metho d variables converge steadily to their optimal values This makes it relatively easy to detect fractional variables To b e useful in a branch and b ound algorithm an interior p oint metho d must generate lower b ounds which are used to fathom subproblems and primal solution estimates which the heuristics use to detect a fractional solution Most interior p oint metho ds satisfy these requirements but we concentrate on the dualane metho d and the primaldual path following metho d We describ e exp erimental co des based on b oth metho ds Computational results are provided for a numb er of randomly generated problems as well as for problems taken from the literature These results indicate that the early branching idea is very eective but more work is needed h and b ound co des comp etitive with simplex based to make the exp erimental branc branch and b ound
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