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Numerical experience with lower bounds for MIQP branchandbound
, 1995
"... The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branchandbound framework is considered. It is shown how lower bounds can be computed efficiently during the branchandbound process. Improved lower bounds such as the ones derived in this paper can reduc ..."
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Cited by 46 (0 self)
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The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branchandbound framework is considered. It is shown how lower bounds can be computed efficiently during the branchandbound process. Improved lower bounds such as the ones derived in this paper can reduce the number of QP problems that have to be solved. The branchandbound approach is also shown to be superior to other approaches to solving MIQP problems. Numerical experience is presented which supports these conclusions. Key words : Integer Programming, Mixed Integer Quadratic Programming, BranchandBound AMS subject classification: 90C10, 90C11, 90C20 1 Introduction One of the most successful methods for solving mixedinteger nonlinear problems is branchandbound. Land and Doig [16] first introduced a branchandbound algorithm for the travelling salesman problem. Dakin [3] introduced the now common branching dichotomy and was the first to realize that it is possible to so...
Some Generalizations Of The CrissCross Method For Quadratic Programming
 MATH. OPER. UND STAT. SER. OPTIMIZATION
, 1992
"... Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite criss ..."
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Cited by 13 (8 self)
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Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite crisscross method, based on leastindex resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic crisscross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.
BranchandCut for Combinatorial Optimization Problems without Auxiliary Binary Variables
 KNOWLEDGE ENGINEERING REVIEW
, 2001
"... Many optimization problems involve combinatorial constraints on continuous variables. An example of a combinatorial constraint is that at most one variable in a group of nonnegative variables may be positive. Traditionally, in the mathematical programming community, such problems have been modeled a ..."
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Cited by 9 (3 self)
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Many optimization problems involve combinatorial constraints on continuous variables. An example of a combinatorial constraint is that at most one variable in a group of nonnegative variables may be positive. Traditionally, in the mathematical programming community, such problems have been modeled as mixedinteger programs by introducing auxiliary binary variables and additional constraints. Because the number of variables and constraints becomes larger and the combinatorial structure is not used to advantage, these mixedinteger programming models may not be solved satisfactorily, except for small instances. Traditionally, constraint programming approaches to such problems keep and use the combinatorial structure, but do not use linear programming bounds in the search for an optimal solution. Here we present a branchandcut approach that considers the combinatorial constraints without the introduction of binary variables. We review the development of this approach and show how strong constraints can be derived using ideas from polyhedral combinatorics. To illustrate the ideas, we present a production scheduling model that arises in the manufacture of fiber optic cables.
Further Development on the Interior Algorithm for Convex Quadratic Programming
 Dept. of EngineeringEconomic Systems, Stanford University
, 1987
"... The interior trust region algorithm for convex quadratic programming is further developed. This development is motivated by the barrier function and the "center" pathfollowing methods, which create a sequence of primal and dual interior feasible points converging to the optimal solution. At each it ..."
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Cited by 8 (1 self)
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The interior trust region algorithm for convex quadratic programming is further developed. This development is motivated by the barrier function and the "center" pathfollowing methods, which create a sequence of primal and dual interior feasible points converging to the optimal solution. At each iteration, the gap between the primal and dual objective values (or the complementary slackness value) is reduced at a global convergence ratio (1 \Gamma 1 4 p n ), where n is the number of variables in the convex QP problem. A safeguard line search technique is also developed to relax the smallstepsize restriction in the original path following algorithm. Key words: Convex Quadratic Programming, Primal and Dual, Complementarity Slackness, Polynomial Interior Algorithm. Abbreviated title: Interior Algorithm for Convex Quadratic Programming Since Karmarkar proposed the new polynomial algorithm (Karmarkar [19]), several developments have been made to the growing literature on interior a...
Analyse und Restrukturierung eines Verfahrens zur direkten Lösung von OptimalSteuerungsproblemen (The Theory of MUSCOD in a Nutshell)
, 1995
"... MUSCOD (MU ltiple Shooting COde for Direct Optimal Control) is the implementation of an algorithm for the direct solution of optimal control problems. The method is based on multiple shooting combined with a sequential quadratic programming (SQP) technique; its original version was developed in the ..."
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Cited by 4 (0 self)
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MUSCOD (MU ltiple Shooting COde for Direct Optimal Control) is the implementation of an algorithm for the direct solution of optimal control problems. The method is based on multiple shooting combined with a sequential quadratic programming (SQP) technique; its original version was developed in the early 1980s by Plitt under the supervision of Bock [Plitt81, Bock84]. The following report is intended to describe the basic aspects of the underlying theory in a concise but readable form. Such a description is not yet available: the paper by Bock and Plitt [Bock84] gives a good overview of the method, but it leaves out too many important details to be a complete reference, while the diploma thesis by Plitt [Plitt81], on the other hand, presents a fairly complete description, but is rather difficult to read. Throughout the present document, emphasis is given to a clear presentation of the concepts upon which MUSCOD is based. An effort has been made to properly reflect the structure of the a...
A Polynomial Method of Weighted Centers for Convex Quadratic Programming
 Journal of Information & Optimization Sciences
, 1991
"... A generalization of the weighted central pathfollowing method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central pathfollowing method for linear programming and the interior point methods for convex quadratic programming. B ..."
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Cited by 3 (2 self)
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A generalization of the weighted central pathfollowing method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central pathfollowing method for linear programming and the interior point methods for convex quadratic programming. By means of the linear approximation of the weighted logarithmic barrier function and weighted inscribed ellipsoids, `weighted' trajectories are defined. Each strictly feasible primal dual point pair define such a weighted trajectory. The algorithm can start in any strictly feasible primaldual point pair that defines a weighted trajectory, which is followed through the algorithm. This algorithm has the nice feature, that it is not necessary to start the algorithm close to the central path and so additional transformations are not needed. In return, the theoretical complexity of our algorithm is dependent on the position of the starting point. Polynomiality is proved under the usual mild cond...
A Polyhedral Study of the Cardinality Constrained Knapsack Problem
 in W.J. Cook and A.S. Schulz (Eds.), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science
, 2003
"... A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas as finance, location, and scheduling. Traditionally, cardinality constraints are mode ..."
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Cited by 2 (0 self)
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A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 01 variables and additional constraints that relate the continuous and the 01 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 01 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facetdefining inequalities. We present three families of nontrivial facetdefining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 01 variables over the traditional MIP approach for this class of problems.
Generalized Outer Approximation
"... This article deals with the solution of Mixed Integer Nonlinear Programming (MINLP) problems of the form P ..."
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Cited by 2 (0 self)
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This article deals with the solution of Mixed Integer Nonlinear Programming (MINLP) problems of the form P
A BranchandPrice Approach for the Maximum Weight Independent Set Problem
, 2005
"... The maximum weight independent set problem (MWISP) is one of the most wellknown and wellstudied problems in combinatorial optimization. This paper presents a novel approach to solve MWISP exactly by decomposing the original graph into vertexinduced subgraphs. The approach solves MWISP for the or ..."
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Cited by 1 (0 self)
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The maximum weight independent set problem (MWISP) is one of the most wellknown and wellstudied problems in combinatorial optimization. This paper presents a novel approach to solve MWISP exactly by decomposing the original graph into vertexinduced subgraphs. The approach solves MWISP for the original graph by solving MWISP on the subgraphs in order to generate columns for a branchandprice framework. The authors investigate different implementation techniques that can be associated with the approach and offer computational results to identify the strengths and weaknesses of each implementation technique.