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13
Optimal Homologous Cycles, Total Unimodularity, and Linear Programming
, 2010
"... Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer (Z) coefficients, we show the following (Theorem 5.2): For a fin ..."
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Cited by 29 (11 self)
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Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer (Z) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex K of dimension greater than p, the boundary matrix [∂p+1] is totally unimodular if and only if Hp(L, L0) is torsionfree, for all pure subcomplexes L0, L in K of dimensions p and p + 1 respectively, where L0 ⊂ L. Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain integer solution. Thus the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NPhard under Z2 coefficients which, being a field, is in general easier to deal with. One consequence of our result, among others, is that one can compute in polynomial time an optimal 2cycle in a given homology class for any finite simplicial complex embedded in R 3. Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles.
A Computational View of InteriorPoint Methods for Linear Programming
 IN: ADVANCES IN LINEAR AND INTEGER PROGRAMMING
, 1994
"... Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing te ..."
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Cited by 16 (10 self)
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Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing techniques, the initialization approaches, the methods of computing search directions (and lying behind them linear algebra techniques), centering strategies and methods of stepsize selection. Several reasons for the manifestations of numerical difficulties like e.g.: the primal degeneracy of optimal solutions or the lack of feasible solutions are explained in a comprehensive way. A motivation for obtaining an optimal basis is given and a practicable algorithm to perform this task is presented. Advantages of different methods to perform postoptimal analysis (applicable to interior point optimal solutions) are discussed. Important questions that still remain open in the implementations of i...
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.
A Study of Preconditioners for Network Interior Point Methods
 Computational Optimization and Applications
, 2002
"... We study and compare preconditioners available for network interior point methods. We derive upper bounds for the condition number of the preconditioned matrices used in the solution of systems of linear equations defining the algorithm search directions. The preconditioners... ..."
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Cited by 15 (0 self)
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We study and compare preconditioners available for network interior point methods. We derive upper bounds for the condition number of the preconditioned matrices used in the solution of systems of linear equations defining the algorithm search directions. The preconditioners...
Methodology and a Modular Tool for Multiple Criteria Analysis of LP Models
 WORKING PAPER WP94102, INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS, LAXENBURG
, 1994
"... ..."
A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds
 MATH. OF COMPUTATION
, 1997
"... We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combine ..."
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Cited by 11 (1 self)
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We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combined in a Lagrangian barrier function. A sequence of such functions are approximately minimized within the domain defined by the simple bounds. Global convergence of the sequence of generated iterates to a firstorder stationary point for the original problem is established. Furthermore, possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero. This paper is a companion to previous work of ours on augmented Lagrangian methods.
A Stable PrimalDual Approach for Linear Programming
"... This paper studies a primaldual interior/exteriorpoint pathfollowing approach for linearprogramming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primaldual optimality equations Fu(x, y, z) = 0. Under nonde ..."
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Cited by 3 (1 self)
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This paper studies a primaldual interior/exteriorpoint pathfollowing approach for linearprogramming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primaldual optimality equations Fu(x, y, z) = 0. Under nondegeneracy assumptions, this nonlinear system is wellposed,i.e. it has a nonsingular Jacobian at optimality and is not necessarily illconditioned as the iterates approach optimality. We use a simple preprocessing step to eliminate boththe primal and dual feasibility equations. This results in a single bilinear equation that maintains the wellposedness property. We then apply both a direct solution techniqueas well as a preconditioned conjugate gradient method (PCG), within an inexact Newton framework, directly on the linearized equations. This is done without forming the usualnormal equations, NEQ, or augmented system. Sparsity is maintained. The work of aniteration for the PCG approach consists almost entirely in the (approximate) solution of this wellposed linearized system. Therefore, improvements depend on efficient preconditioning.
Numerical Stability in Linear Programming and Semidefinite Programming
, 2006
"... We study numerical stability for interiorpoint methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms. ..."
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Cited by 1 (1 self)
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We study numerical stability for interiorpoint methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms.
Nondegeneracy of Polyhedra and Linear Programs
, 1995
"... This paper deals with nondegeneracy of polyhedra and linear programming (LP) problems. We allow for the possibility that the polyhedra and the feasible polyhedra of the LP problems under consideration be nonpointed. (A polyhedron is pointed if it has a vertex.) With respect to a given polyhedron, w ..."
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This paper deals with nondegeneracy of polyhedra and linear programming (LP) problems. We allow for the possibility that the polyhedra and the feasible polyhedra of the LP problems under consideration be nonpointed. (A polyhedron is pointed if it has a vertex.) With respect to a given polyhedron, we consider two notions of nondegeneracy and then provide several equivalent characterizations for each of them. With respect to LP problems, we study the notion of constant cost nondegeneracy first introduced by Tsuchiya [25] under a different name, namely dual nondegeneracy. (We do not follow this terminology since the term dual nondegeneracy is already used to refer to a related but different type of nondegeneracy.) We show two main results about constant cost nondegeneracy of an LP problem. The first one shows that constant cost nondegeneracy of an LP problem is equivalent to the condition that the union of all minimal faces of the feasible polyhedron be equal to the set of feasible point...