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  • Articles  (35)
  • Linear programming
  • 1995-1999  (35)
  • Mathematics  (35)
  • Technology
  • 1
    Electronic Resource
    Electronic Resource
    Primal—dual methods for linear programming (1995)
    Springer
    Mathematical programming 70 (1995), S. 251-277 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; Barrier methods ; Interior-point methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. After a preliminary discussion of the convergence of the (primal) projected Newton barrier method, three types of barrier method are analyzed. These methods may be categorized as primal, dual and primal—dual, and may be derived from the application of Newton's method to different variants of the same system of nonlinear equations. A fourth variant of the same equations leads to a new primal—dual method. In each of the methods discussed, convergence is demonstrated without the need for a nondegeneracy assumption or a transformation that makes the provision of a feasible point trivial. In particular, convergence is established for a primal—dual algorithm that allows a different step in the primal and dual variables and does not require primal and dual feasibility. Finally, a new method for treating free variables is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585940
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  • 2
    Electronic Resource
    Electronic Resource
    The fleet assignment problem: Solving a large-scale integer program (1995)
    Springer
    Mathematical programming 70 (1995), S. 211-232 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; Mixed-integer programming ; Large-scale optimization ; Airline fleet assignment
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585938
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  • 3
    Electronic Resource
    Electronic Resource
    Partitioning mathematical programs for parallel solution (1998)
    Springer
    Mathematical programming 80 (1998), S. 35-61 
    Details
    ISSN: 1436-4646
    Keywords: Graph partitioning ; Linear programming ; Bundle method ; Parallel optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes heuristics for partitioning a generalM × N matrix into arrowhead form. Such heuristics are useful for decomposing large, constrained, optimization problems into forms that are amenable to parallel processing. The heuristics presented can be easily implemented using publicly available graph partitioning algorithms. The application of such techniques for solving large linear programs is described. Extensive computational results on the effectiveness of our partitioning procedures and their usefulness for parallel optimization are presented. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582130
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  • 4
    Electronic Resource
    Electronic Resource
    Linear programming, complexity theory and elementary functional analysis (1995)
    Springer
    Mathematical programming 70 (1995), S. 279-351 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; Complexity theory ; Interior-point methods ; Semi-definite programming ; Condition numbers ; Convex programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We propose analyzing interior-point methods using notions of problem-instance size which are direct generalizations of the condition number of a matrix. The notions pertain to linear programming quite generally; the underlying vector spaces are not required to be finite-dimensional and, more importantly, the cones defining nonnegativity are not required to be polyhedral. Thus, for example, the notions are appropriate in the context of semi-definite programming. We prove various theorems to demonstrate how the notions can be used in analyzing interior-point methods. These theorems assume little more than that the interiors of the cones (defining nonnegativity) are the domains of self-concordant barrier functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585941
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  • 5
    Electronic Resource
    Electronic Resource
    Presolving in linear programming (1995)
    Springer
    Mathematical programming 71 (1995), S. 221-245 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; Presolving ; Interior-point methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Most modern linear programming solvers analyze the LP problem before submitting it to optimization. Some examples are the solvers WHIZARD (Tomlin and Welch, 1983), OB1 (Lustig et al., 1994), OSL (Forrest and Tomlin, 1992), Sciconic (1990) and CPLEX (Bixby, 1994). The purpose of the presolve phase is to reduce the problem size and to discover whether the problem is unbounded or infeasible. In this paper we present a comprehensive survey of presolve methods. Moreover, we discuss the restoration procedure in detail, i.e., the procedure that undoes the presolve. Computational results on the NETLIB problems (Gay, 1985) are reported to illustrate the efficiency of the presolve methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01586000
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  • 6
    Electronic Resource
    Electronic Resource
    On the choice of parameters for power-series interior point algorithms in linear programming (1995)
    Springer
    Mathematical programming 68 (1995), S. 49-71 
    Details
    ISSN: 1436-4646
    Keywords: Power-series interior point algorithms: Parameter transformations ; Best parameter ; k-parameter ; Truncated power-series approximation ; Higher-order derivatives ; Linear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we study higher-order interior point algorithms, especially power-series algorithms, for solving linear programming problems. Since higher-order differentials are not parameter-invariant, it is important to choose a suitable parameter for a power-series algorithm. We propose a parameter transformation to obtain a good choice of parameter, called ak-parameter, for general truncated powerseries approximations. We give a method to find ak-parameter. This method is applied to two powerseries interior point algorithms, which are built on a primal—dual algorithm and a dual algorithm, respectively. Computational results indicate that these higher-order power-series algorithms accelerate convergence compared to first-order algorithms by reducing the number of iterations. Also they demonstrate the efficiency of thek-parameter transformation to amend an unsuitable parameter in power-series algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585757
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  • 7
    Electronic Resource
    Electronic Resource
    Superlinear primal-dual affine scaling algorithms for LCP (1995)
    Springer
    Mathematical programming 69 (1995), S. 311-333 
    Details
    ISSN: 1436-4646
    Keywords: Interior-point methods ; Primal-dual affine scaling ; Linear programming ; Linear complementarity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We describe an interior-point algorithm for monotone linear complementarity problems in which primal-dual affine scaling is used to generate the search directions. The algorithm is shown to have global and superlinear convergence with Q-order up to (but not including) two. The technique is shown to be consistent with a potential-reduction algorithm, yielding the first potential-reduction algorithm that is both globally and superlinearly convergent.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585562
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  • 8
    Electronic Resource
    Electronic Resource
    Minimizing average completion time in the presence of release dates (1998)
    Springer
    Mathematical programming 82 (1998), S. 199-223 
    Details
    ISSN: 1436-4646
    Keywords: Scheduling ; Preemptive scheduling ; Average weighted completion time ; Approximation algorithms ; Relaxations ; Linear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time. In this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs. In contrast to the problem of minimizing average completion time when all jobs are available at time 0, all the problems that we consider are NP-hard, and essentially nothing was known about constructing good approximations in polynomial time. We give the first constant-factor approximation algorithms for several variants of the single and parallel machine models. Many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both. Many of the algorithms generalize to the minimization of averageweighted completion time as well. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585872
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  • 9
    Electronic Resource
    Electronic Resource
    A simplified homogeneous and self-dual linear programming algorithm and its implementation (1996)
    Springer
    Annals of operations research 62 (1996), S. 151-171 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; homogeneous and self-dual linear feasibility model ; predictor-corrector algorithm ; implementation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We present a simplification and generalization of the recent homogeneous and self-dual linear programming (LP) algorithm. The algorithm does not use any Big-M initial point and achieves $$O(\sqrt {nL} )$$ -iteration complexity, wheren andL are the number of variables and the length of data of the LP problem. It also detects LP infeasibility based on a provable criterion. Its preliminary implementation with a simple predictor and corrector technique results in an efficient computer code in practice. In contrast to other interior-point methods, our code solves NETLIB problems, feasible or infeasible, starting simply fromx=e (primal variables),y=0 (dual variables),z=e (dual slack variables), wheree is the vector of all ones. We describe our computational experience in solving these problems, and compare our results with OB1.60, a state-of-the-art implementation of interior-point algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02206815
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  • 10
    Electronic Resource
    Electronic Resource
    An infeasible-interior-point algorithm using projections onto a convex set (1996)
    Springer
    Annals of operations research 62 (1996), S. 59-80 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; infeasible-interior-point method ; polynomiality ; projection ; 90C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We present a new class of primal-dual infeasible-interior-point methods for solving linear programs. Unlike other infeasible-interior-point algorithms, the iterates generated by our methods lie in general position in the positive orthant of ℝ2 and are not restricted to some linear manifold. Our methods comprise the following features: At each step, a projection is used to “recenter” the variables to the domainx i s i ≥μ. The projections are separable into two-dimensional orthogonal projections on a convex set, and thus they are seasy to implement. The use of orthogonal projections allows that a full Newton step can be taken at each iteration, even if the result violates the nonnegativity condition. We prove that a short step version of our method converges in polynomial time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02206811
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