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16
Karmarkar's Algorithm and Combinatorial Optimization Problems
, 1988
"... Branch-and-cut methods are very successful techniques for solving a wide variety of integer programming problems, and they can provide a guarantee of optimality. We describe how a branch-and-cut method can be tailored to a specific integer programming problem, and how families of general cutting pla ..."
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Cited by 44 (5 self)
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Branch-and-cut methods are very successful techniques for solving a wide variety of integer programming problems, and they can provide a guarantee of optimality. We describe how a branch-and-cut method can be tailored to a specific integer programming problem, and how families of general cutting planes can be used to solve a wide variety of problems. Other important aspects of successful implementations are discussed in this chapter. The area of branch-and-cut algorithms is constantly evolving, and it promises to become even more important with the exploitation of faster computers and parallel computing. 1
Polynomial interior point cutting plane methods
- Optimization Methods and Software
, 2003
"... Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approxim ..."
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Cited by 21 (7 self)
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Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Long-step versions of the algorithms for solving convex optimization problems are presented. 1
Restoration of Services in Interdependent Infrastructure Systems: A Network Flows Approach
- Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute
, 2003
"... Abstract — Modern society depends on the operations of civil infrastructure systems, such as transportation, energy, telecommunications and water. Clearly, disruption of any of these systems would present a significant detriment to daily living. However, these systems have become so interconnected, ..."
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Cited by 20 (3 self)
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Abstract — Modern society depends on the operations of civil infrastructure systems, such as transportation, energy, telecommunications and water. Clearly, disruption of any of these systems would present a significant detriment to daily living. However, these systems have become so interconnected, one relying on another, that disruption of one may lead to disruptions in all. The focus of this research is on developing techniques which can be used to respond to events that have the capability to impact interdependent infrastructure systems. As discussed in the paper, infrastructure interdependencies occur when, due to either geographical proximity or shared operations, an impact on one infrastructure system affects one or more other infrastructure systems. The approach is to model the salient elements of these systems and provide decision makers with a means to manipulate the set of models, i.e. a decision support system. 1
A Semidefinite Programming Approach to the Quadratic Knapsack Problem
, 2000
"... In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibi ..."
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Cited by 19 (1 self)
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In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.
Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems
, 1995
"... Branch and cut methods for integer programming problems solve a sequence of linear programming problems. Traditionally, these linear programming relaxations have been solved using the simplex method. The reduced costs available at the optimal solution to a relaxation may make it possible to fix var ..."
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Cited by 11 (6 self)
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Branch and cut methods for integer programming problems solve a sequence of linear programming problems. Traditionally, these linear programming relaxations have been solved using the simplex method. The reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one. If the solution to a relaxation is fractional, additional constraints can be generated which cut off the solution to the relaxation, but do not cut off any feasible integer points. Gomory cutting planes and other classes of cutting planes are generated from the final tableau. In this paper, we consider using an interior point method to solve the linear programming relaxations. We show that it is still possible to generate Gomory cuts and other cuts without having to recreate a tableau, and we also show how variables can be fixed without using the optimal reduced costs. The procedures we develop do not require that the current relaxation be solved to optimality; this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation.
Computational Experience of an Interior-Point SQP Algorithm in a Parallel Branch-and-Bound Framework
"... An interior-point algorithm within a parallel branch-and-bound framework for solving nonlinear mixed integer programs is described. The nonlinear programming relaxations at each node are solved using an interior point SQP method. In contrast to solving the relaxation to optimality at each tree node ..."
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Cited by 11 (3 self)
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An interior-point algorithm within a parallel branch-and-bound framework for solving nonlinear mixed integer programs is described. The nonlinear programming relaxations at each node are solved using an interior point SQP method. In contrast to solving the relaxation to optimality at each tree node, the relaxation is only solved to near-optimality. Analogous to employing advanced bases in simplex-based linear MIP solvers, a “dynamic” collection of warmstart vectors is kept to provide “advanced warmstarts” at each branch-and-bound node. The code has the capability to run in both shared-memory and distributed-memory parallel environments. Preliminary computational results on various classes of linear mixed integer programs and quadratic portfolio problems are presented.
Interior point and semidefinite approaches in combinatorial optimization
, 2005
"... Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient ..."
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Cited by 8 (4 self)
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Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primal-dual interior-point methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to non-convex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include non-convex potential reduction methods, interior point cutting plane methods, primal-dual IPMs and first-order algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.
Computational experience of an interior point algorithm in a parallel branch-and-cut framework
- IN PROCEEDINGS FOR SIAM CONFERENCE ON PARALLEL PROCESSING FOR SCIENTIFIC COMPUTING
, 1997
"... An interior-point algorithm within a branch-and-bound framework for solving nonlinear mixed integer programs is described. In contrast to solving the relaxation to optimality at each tree node, the relaxation is only solved to near-optimality. Analogous to using advanced bases for warmstart solution ..."
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Cited by 4 (0 self)
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An interior-point algorithm within a branch-and-bound framework for solving nonlinear mixed integer programs is described. In contrast to solving the relaxation to optimality at each tree node, the relaxation is only solved to near-optimality. Analogous to using advanced bases for warmstart solutions in the case of linear MIP, a "dynamic" collection of warmstart vectors is kept. Computational results on various classes of nonlinear mixed integer programs are presented.
RECENT DEVELOPMENTS IN INTERIOR-POINT METHODS
"... The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such asconvex quadrat ..."
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Cited by 3 (1 self)
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The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such asconvex quadratic programming, semidefi nite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. Interior-point methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear programs. We review some core developments in the area.
