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Computational experience with an interior point algorithm on the satisfiability problem
 Ann. Oper. Res
, 1990
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An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems
, 1993
"... are addressed. Two algorithms are developed for certain feasibility versions of the SDP, and the rst of these is shown to have polynomial time complexity when the dimension of the matrix map involved is xed. The second algorithm is a globally convergent Newtonlike method applied to a leastsquares ..."
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Cited by 39 (4 self)
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are addressed. Two algorithms are developed for certain feasibility versions of the SDP, and the rst of these is shown to have polynomial time complexity when the dimension of the matrix map involved is xed. The second algorithm is a globally convergent Newtonlike method applied to a leastsquares penalty function. The problem of characterizing and identifying quadratic maps with convex images is analyzed from both structural and complexity theoretic points of view. Then a study is made of the geometry of a class of convex sets called spectrahedra, which are the feasible regions in semide nite programs. Finally, in Chapter 7, we develop some cutting plane techniques for MQP, based on eigenvalue inequalities. Acknowledgements I express my sincere gratitude to my thesis advisor Professor Alan Goldman for his support, ideas and encouragement. My special thanks to Professors Laszlo Lovasz and James Renegar for sparing their time generously and giving me very useful suggestions. I thank the warm and friendly Professors Dan Naiman and Ed Scheinerman for making my four year long stay at Johns Hopkins a very pleasant one. I am indebted to Prof. JongShi Pang and Prof. Roger Horn for giving a patient ear to many of my enthusiastic ideas and o ering suggestions. I also thank Prof. ShihPing Han for being a wonderful teacher, and Prof. Leslie Hall for being a patient second reader of my thesis. My interest in Multiquadratic Programming was initiated during my internship at AT&T Bell Laboratories in the summer of 1990, and I am obliged to Dr. Narendra Karmarkar for arranging this internship. I thank Dr. Farid Alizadeh, Dr. Florian Jarre, Profs. Raphael Loewy, Michael Overton and Stephen Vavasis for patiently answering my questions and sending me some literature. My heartfelt appreciation is due my parents, Lakshmi Pathy and Satyavathi, my sister,
A Continuous Approach to Inductive Inference
 Mathematical Programming
, 1992
"... In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g ..."
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Cited by 39 (2 self)
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In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g using outputs obtained by applying a limited number of random inputs to the hidden function. Given this inputoutput sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used...
An Algorithm for Large Scale 01 Integer Programming With Application to Airline Crew Scheduling
, 1995
"... We present an approximation algorithm for solving large 01 integer programming problems where A is 01 and where b is integer. The method can be viewed as a dual coordinate search for solving the LPrelaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working t ..."
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Cited by 37 (5 self)
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We present an approximation algorithm for solving large 01 integer programming problems where A is 01 and where b is integer. The method can be viewed as a dual coordinate search for solving the LPrelaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems ...
Solving RealWorld Linear Ordering Problems . . .
, 1995
"... Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear prog ..."
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Cited by 23 (8 self)
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Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear programming relaxations. A point which is a good warm start for a simplexbased cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some realworld problems; the algorithm appears to be competitive with a simplexbased cutting plane algorithm.
Orbital branching
 in Proceedings of IPCO XII
, 2007
"... Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branchandbound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an ar ..."
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Cited by 19 (3 self)
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Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branchandbound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an arbitrary constraint. Many important families of integer programs are structured such that small instances from the family are embedded in larger instances. This structural information can be exploited to define a group of strong constraints on which to base the orbital branching disjunction. The symmetric nature of the problems is further exploited by enumerating nonisomorphic solutions to instances of the small family and using these solutions to create a collection of typically easytosolve integer programs. The solution of each integer program in the collection is equivalent to solving the original large instance. The effectiveness of this methodology is demonstrated by computing the optimal incidence width of Steiner Triple Systems and minimum cardinality covering designs.
Optimization Algorithms for the MinimumCost Satisfiability Problem
"... Given a Boolean satisfiability (Sat) problem whose variables have nonnegative weights, the minimumcost satisfiability (MinCostSat) problem finds a satisfying truth assignment that minimizes a weighted sum of the truth values of the variables. Many NPoptimization problems are either special cases ..."
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Cited by 17 (2 self)
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Given a Boolean satisfiability (Sat) problem whose variables have nonnegative weights, the minimumcost satisfiability (MinCostSat) problem finds a satisfying truth assignment that minimizes a weighted sum of the truth values of the variables. Many NPoptimization problems are either special cases of MinCostSat or can be transformed into MinCostSat efficiently. However, in the past, these problems have been largely considered in isolation. In this dissertation, we (1) classify existing MinCostSat problems, (2) study factors affecting the performance of MinCostSat solvers, (3) propose algorithms for MinCostSat problems, and (4) implement and validate the performance of stateoftheart solvers for special cases of MinCostSat, including set and binate covering, MaxSat, and grouppartial MaxSat. We categorize MinCostSat problems as either native or nonnative. Nonnative problems can only be transformed into MinCostSat by adding slack variables. These problems include the MaxSat, partial MaxSat, and grouppartial MaxSat problems which have applications ranging from course assignment to FPGA detailed routing. Native problems are various subcases of MinCostSat. We further divide these into two
Generalizations Of The Trust Region Problem
 OPTIMIZATION METHODS AND SOFTWARE
, 1993
"... The trust region problem requires the global minimum of a general quadratic function subject to an ellipsoidal constraint. The development of algorithms for the solution of this problem has found applications in nonlinear and combinatorial optimization. In this paper we generalize the trust region p ..."
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Cited by 16 (0 self)
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The trust region problem requires the global minimum of a general quadratic function subject to an ellipsoidal constraint. The development of algorithms for the solution of this problem has found applications in nonlinear and combinatorial optimization. In this paper we generalize the trust region problem by allowing a general quadratic constraint. The main results are a characterization of the global minimizer of the generalized trust region problem, and the development of an algorithm that finds an approximate global minimizer in a finite number of iterations.
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 16 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
An Interior Point Approach to Boolean Vector Function Synthesis
 In Proceedings of the 36th MSCAS
, 1993
"... The Boolean vector function synthesis problem can be stated as follows: Given a truth table with n input variables and m output variables, synthesize a Boolean vector function that describes the table. In this paper we describe a new formulation of the Boolean vector function synthesis problem as a ..."
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Cited by 13 (1 self)
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The Boolean vector function synthesis problem can be stated as follows: Given a truth table with n input variables and m output variables, synthesize a Boolean vector function that describes the table. In this paper we describe a new formulation of the Boolean vector function synthesis problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. Preliminary computational results are presented. Introduction The Boolean Vector Function Synthesis Problem has applications in logic, artificial intelligence, machine learning, and digital integrated circuit design. In this paper, we describe a Satisfiability Problem formulation of the Boolean Vector Function Synthesis Problem. This formulation can be approached with a wide range of algorithms. In this paper, preliminary computational results are presented using an interior point algorit...