ALBERT

All Library Books, journals and Electronic Records Telegrafenberg

X
feed icon rss

Your email was sent successfully. Check your inbox.

An error occurred while sending the email. Please try again.

Proceed reservation?

Export
Filter
  • Articles  (19)
  • Linear programming  (19)
  • 1990-1994  (19)
  • 1993  (19)
  • Mathematics  (19)
  • 1
    Electronic Resource
    Electronic Resource
    Near boundary behavior of primal—dual potential reduction algorithms for linear programming (1993)
    Springer
    Mathematical programming 58 (1993), S. 243-255 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; interior point algorithm ; primal—dual potential function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper is concerned with selection of theρ-parameter in the primal—dual potential reduction algorithm for linear programming. Chosen from [n + $$\sqrt n $$ , ∞), the level ofρ determines the relative importance placed on the centering vs. the Newton directions. Intuitively, it would seem that as the iterate drifts away from the central path towards the boundary of the positive orthant,ρ must be set close ton + $$\sqrt n $$ . This increases the relative importance of the centering direction and thus helps to ensure polynomial convergence. In this paper, we show that this is unnecessary. We find for any iterate thatρ can be sometimes chosen in a wide range [n + $$\sqrt n $$ , ∞) while still guaranteeing the currently best convergence rate of O( $$\sqrt n $$ L) iterations. This finding is encouraging since in practice large values ofρ have resulted in fast convergence rates. Our finding partially complements the recent result of Zhang, Tapia and Dennis (1990) concerning the local convergence rate of the algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581269
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Combining phase I and phase II in a potential reduction algorithm for linear programming (1993)
    Springer
    Mathematical programming 59 (1993), S. 133-150 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; interior-point methods ; combined phase I—phase II
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality. The algorithm is closely related to a similar method of Anstreicher. The new features are that we use a two-dimensional programming problem to derive better lower bounds than Anstreicher, that our direction-finding subproblem treats phase I and phase II more symmetrically, and that we do not need an initial lower bound. Our method also allows for the generation of a feasible solution (so that phase I is terminated) during the course of the iterations, and we describe two ways to encourage this behavior.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581241
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    A quadratically convergent O( $$\sqrt n $$ L)-iteration algorithm for linear programming (1993)
    Springer
    Mathematical programming 59 (1993), S. 151-162 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; primal and dual ; superlinear and quadratic convergence ; polynomiality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( $$\sqrt n $$ L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581242
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    A note on the prize collecting traveling salesman problem (1993)
    Springer
    Mathematical programming 59 (1993), S. 413-420 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; prize collecting ; rounding fractional solutions ; traveling salesman problem ; worst-case analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We study the version of the prize collecting traveling salesman problem, where the objective is to find a tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. We present an approximation algorithm with constant bound. The algorithm is based on Christofides' algorithm for the traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers, feasible for the original problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581256
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 5
    Electronic Resource
    Electronic Resource
    Strict monotonicity and improved complexity in the standard form projective algorithm for linear programming (1993)
    Springer
    Mathematical programming 62 (1993), S. 517-535 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; Karmarkar's algorithm ; Projective algorithm ; Standard form
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In a recent paper, Shaw and Goldfarb show that a version of the standard form projective algorithm can achieve $$O\left( {\sqrt {nL} } \right)$$ step complexity, as opposed to the O(nL) step complexity originally demonstrated for the algorithm. The analysis of Shaw and Goldfarb shows that the algorithm, using a constant, fixed steplength, approximately follows the central trajectory. In this paper we show that simple modifications of the projective algorithm obtain the same complexity improvement, while permitting a linesearch of the potential function on each step. An essential component is the addition of a single constraint, motivated by Shaw and Goldfarb's analysis, which makes the standard form algorithm strictly monotone in the true objective.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585181
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 6
    Electronic Resource
    Electronic Resource
    Determination of optimal vertices from feasible solutions in unimodular linear programming (1993)
    Springer
    Mathematical programming 59 (1993), S. 23-31 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; duality theorem ; unimodular ; totally unimodular ; interior point methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we consider a linear programming problem with the underlying matrix unimodular, and the other data integer. Given arbitrary near optimum feasible solutions to the primal and the dual problems, we obtain conditions under which statements can be made about the value of certain variables in optimal vertices. Such results have applications to the problem of determining the stopping criterion in interior point methods like the primal—dual affine scaling method and the path following methods for linear programming.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581235
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 7
    Electronic Resource
    Electronic Resource
    A strictly improving linear programming Phase I algorithm (1993)
    Springer
    Annals of operations research 46-47 (1993), S. 409-430 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; Phase I ; nonlinear programming ; least squares ; quadratic programming ; strict improvement ; degeneracy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Instead of trying to recognize and avoid degenerate steps in the simplex method (as some variants do), we have developed a new Phase I algorithm that is impervious to degeneracy. The new algorithm solves a non-negative least-squares problem in order to find a Phase I solution. In each iteration, a simple two-variable least-squares subproblem is used to select an incoming column to augment a set of independent columns (called “basic”) to get a strictly better fit to the right-hand side. Although this is analogous in many ways to the simplex method, it can be proved that strict improvement is attained at each iteration, even in the presence of degeneracy. Thus cycling cannot occur, and convergence is guaranteed. This algorithm is closely related to a number of existing algorithms proposed for non-negative least-squares and quadratic programs. When used on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase I algorithm were almost 3.5 times faster than a particular implementation of the simplex method; on some problems, it was over 10 times faster. Best results were generally seen on the more degenerate problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02023107
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 8
    Electronic Resource
    Electronic Resource
    Degeneracy in interior point methods for linear programming: a survey (1993)
    Springer
    Annals of operations research 46-47 (1993), S. 107-138 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; interior point methods ; degeneracy ; polynomial algorithms ; global and local convergence ; basis recovery ; numerical performance ; sensitivity analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The publication of Karmarkar's paper has resulted in intense research activity into Interior Point Methods (IPMs) for linear programming. Degeneracy is present in most real-life problems and has always been an important issue in linear programming, especially in the Simplex method. Degeneracy is also an important issue in IPMs. However, the difficulties are different in the two methods. In this paper, we survey the various theoretical and practical issues related to degeneracy in IPMs for linear programming. We survey results, which, for the most part, have already appeared in the literature. Roughly speaking, we shall deal with the effect of degeneracy on the following: the convergence of IPMs, the trajectories followed by the algorithms, numerical performance, and finding basic solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02096259
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 9
    Electronic Resource
    Electronic Resource
    Studies of lexicography in the generalized network simplex method (1993)
    Springer
    Annals of operations research 46-47 (1993), S. 235-248 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; generalized networks ; simplex method ; degeneracy ; lexicography ; cycling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper introduces an analytical approach for studying lexicography in generalized network problems. The equations obtained can help us to understand and to extend the existing theory. First, it is verified that all nonzero elements have the same sign in each row vector of a basis inverse for a generalized network (GN) problem with positive multipliers. However, this property does not necessarily hold when there exist negative multipliers. Second, we developed a strategy to select the dropping arc in the GN simplex algorithm when addressing GN problems with positive andnegative multipliers. This strategy is also based on lexicography and requires performing some comparisons. However, the values to be compared are already known since they can be obtained as a by-product of the calculations necessary to compute the basis representation of the entering arc. Consequently, the computational effort per pivot step isO(n) in the worst case. This worst case effort is the same as that required by the strongly convergent rules for selecting the dropping arc in the method of strong convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02023097
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 10
    Electronic Resource
    Electronic Resource
    Pivot rules for linear programming: A survey on recent theoretical developments (1993)
    Springer
    Annals of operations research 46-47 (1993), S. 203-233 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; simplex method ; pivot rules ; cycling ; recursion ; minimal index rule ; parametric programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The purpose of this paper is to discuss the various pivot rules of the simplex method and its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with finiteness properties of simplex type pivot rules. Well known classical results concerning the simplex method are not considered in this survey, but the connection between the new pivot methods and the classical ones, if there is any, is discussed. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first and largest class is the class of essentially combinatorial pivot rules including minimal index type rules and recursive rules. These rules only use labeling and signs of the variables. The second class contains those pivot rules which can actually be considered as variants or generalizations or specializations of Lemke's method, and so they are closely related to parametric programming. The last class has the common feature that the rules all have close connections to certain interior point methods. Finally, we mention some open problems for future research.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02096264
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
Close ⊗
This website uses cookies and the analysis tool Matomo. More information can be found here...