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  • 1
    Electronic Resource
    Electronic Resource
    Sequential location-allocation problems on chains and trees with probabilistic link demands (1985)
    Springer
    Mathematical programming 32 (1985), S. 249-277 
    Details
    ISSN: 1436-4646
    Keywords: Minisum Locations ; Location-Allocation ; Network Location ; Dynamic Location
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we consider multiperiod minisum location problems on networks in which demands can occur continuously on links according to a uniform probability density function. In addition, demands may change dynamically over time periods and at most one facility can be located per time period. Two types of networks are considered in conjunction with three behavioral strategies. The first type of network discussed is a chain graph. A myopic strategy and long-range strategy for locatingp-facilities are considered, as is a discounted present worth strategy for locating two facilities. Although these problems are generally nonconvex, effective methods are developed to readily identify all local and global minima. This analysis forms the basis for similar problems on tree graphs. In particular, we construct algorithms for the 3-facility myopic problem and the 2-facility long-range and discounted cost problems on a tree graph. Extensions and suggestions for further research on problems involving more general networks are provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582049
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Facet inequalities from simple disjunctions in cutting plane theory (1986)
    Springer
    Mathematical programming 34 (1986), S. 72-83 
    Details
    ISSN: 1436-4646
    Keywords: Cutting Planes ; Disjunctive Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The duality between facets of the convex hull of disjunctive sets and the extreme points of reverse polars of these sets is utilized to establish simple rules for the derivation of all facet cuts for simple disjunctions, namely, elementary disjunctions in nonnegative variables. These rules generalize the cut generation procedure underlying polyhedral convexity cuts with negative edge extensions. The latter are also shown to possess some interesting properties with respect to a biextremal problem that maximizes the distance, from the origin, of the nearest point feasible to the cut. A computationally inexpensive procedure is given to generate facet cuts for simple disjunctions which are dominant with respect to any specified preemptive ordering of variables.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582164
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Mixed-integer bilinear programming problems (1993)
    Springer
    Mathematical programming 59 (1993), S. 279-305 
    Details
    ISSN: 1436-4646
    Keywords: Bilinear program ; linearization ; cutting planes ; Lagrangian relaxation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper addresses a class of problems called mixed-integer bilinear programming problems. These problems are identical to the well known bilinear programming problems with the exception that one set of variables is restricted to be binary valued, and they arise in various production, location—allocation, and distribution application contexts. We first identify some special cases of this problem which are relatively more readily solvable, even though their continuous relaxations are still nonconvex. For the more general case, we employ a linearization technique and design a composite Lagrangian relaxation-implicit enumeration-cutting plane algorithm. Extensive computational experience is provided to test the efficacy of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581249
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  • 4
    Electronic Resource
    Electronic Resource
    A leader-follower model and analysis for a two-stage network of oligopolies (1992)
    Springer
    Annals of operations research 34 (1992), S. 37-72 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper is concerned with the existence, uniqueness and computation of leader-follower equilibrium solutions for an industry involved with two major stages of production. We assume that there exists one set of firms performing the first stage of production, which produces a semi-finished product. This semi-finished product is converted to a final good by a second set of firms performing the second stage of production. Furthermore, also competing in the final product market is a third set of firms, which are vertically integrated through the two stages of production and which are assumed to lead the second set of firms by explicitly considering the reaction or response of these latter firms to their own outputs. We model such an industry as a two-stage network of oligopolies, and define equilibrium solutions based on assumed market structures. Our analysis examines the existence and uniqueness of such equilibrium solutions, characterizes the nature of the production strategies of the various firms at an equilibrium, and prescribes algorithms to compute such solutions. This provides the machinery required to perform sensitivity analyses for studying the effects of various mergers or integrations on individual firm profits, and on the industry outputs and prices at equilibrium. The presentation is self-contained, and does not necessarily require any significant prior preparation in economic theory on the part of the reader.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02098172
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes (1990)
    Springer
    Annals of operations research 25 (1990), S. 197-209 
    Details
    ISSN: 1572-9338
    Keywords: Convex envelopes ; bilinear functions ; bilinear programming problems ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper, we constructively derive an explicit characterization of the convex envelope of a bilinear function over a special type of polytope in ℝ2. Our motivation stems from the use of such functions for deriving strengthened lower bounds within the context of a branch-and-bound algorithm for solving bilinear programming problems. For the case of polytopes with no edges having finite positive slopes, that is polytopes with “downward” sloping edges (which we call D-polytopes), we obtain a direct, explicit characterization of the convex envelope. This case subsumes the analysis of Al-Khayyal and Falk (1983) for constructing the convex envelope of a bilinear function over a rectangle in ℝ2. For non-D-polytopes, the analysis is more complex. We propose three strategies for this case based on (i) encasing the region in a D-polytope, (ii) employing a discretization technique, and (iii) providing an explicit characterization over a triangle along with a triangular decomposition approach. The analysis is illustrated using numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283695
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Unrelated machine scheduling with time-window and machine downtime constraints: An application to a naval battle-group problem (1994)
    Springer
    Annals of operations research 50 (1994), S. 339-365 
    Details
    ISSN: 1572-9338
    Keywords: Unrelated machine scheduling ; time-window constraints ; Lagrangian relaxation ; strong integer programming formulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper deals with an unrelated machine scheduling problem of minimizing the total weighted flow time, subject to time-window job availability and machine downtime side constraints. We present a zero-one integer programming formulation of this problem. The linear programming relaxation of this formulation affords a tight lower bound and often generates an integer optimal solution for the problem. By exploiting the special structures inherent in the formulation, we develop some classes of strong valid inequalities that can be used to tighten the initial formulation, as well as to provide cutting planes in the context of a branch-and-cut procedure. A major computational bottleneck is the solution of the underlying linear programming relaxation because of the extremely high degree of degeneracy inherent in the formulation. In order to overcome this difficulty, we employ a Lagrangian dual formulation to generate lower and upper bounds and to drive the branch-and-bound algorithm. As a practical instance of the unrelated machine scheduling problem, we describe a combinatorial naval defense problem. This problem seeks to schedule a set of illuminators (passive homing devices) in order to strike a given set of targets using surface-to-air missiles in a naval battle-group engagement scenario. We present computational results for this problem using suitable realistic data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02085647
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    A finitely convergent algorithm for bilinear programming problems using polar cuts and disjunctive face cuts (1980)
    Springer
    Mathematical programming 19 (1980), S. 14-31 
    Details
    ISSN: 1436-4646
    Keywords: Bilinear Programming ; Polar Cuts ; Disjunctive Cuts ; Cutting Plane
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A finite algorithm is presented in this study for solving Bilinear programs. This is accomplished by developing a suitable cutting plane which deletes at least a face of a polyhedral set. At an extreme point, a polar cut using negative edge extensions is used. At other points, disjunctive cuts are adopted. Computational experience on test problems in the literature is provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581626
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    A mathematical programming approach for determining oligopolistic market equilibrium (1982)
    Springer
    Mathematical programming 24 (1982), S. 92-106 
    Details
    ISSN: 1436-4646
    Keywords: Oligopoly ; Nash Equilibria ; Fixed Points
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract During the past several years it has become increasingly common to use mathematical programming methods for deriving economic equilibria of supply and demand. Well-defined approaches exist for the case of a single firm (monopoly) and for the case of many firms (perfect competition). In this paper a certain family of convex programs is formulated to determine equilibria for the case of a few firms (oligopoly). Solutions to this family of convex programs are shown to be Nash equilibria in the formal sense ofN person games. This equivalence leads to a mathematical programming-based algorithm for determining an oligopolistic market equilibrium.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585096
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    A class of convergent primal-dual subgradient algorithms for decomposable convex programs (1986)
    Springer
    Mathematical programming 35 (1986), S. 279-297 
    Details
    ISSN: 1436-4646
    Keywords: Nondifferentiable optimization ; subgradient optimization ; penalty functions ; Lagrangian duals
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we develop a primal-dual subgradient algorithm for preferably decomposable, generally nondifferentiable, convex programming problems, under usual regularity conditions. The algorithm employs a Lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties, in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal-dual optimal solutions. Several classical types of penalty functions are shown to satisfy these specified properties. A geometric convergence rate is established for the algorithm under some additional assumptions. This approach has three principal advantages. Firstly, both primal and dual solutions are available which prove to be useful in several contexts. Secondly, the choice of step sizes, which plays an important role in subgradient optimization, is guided more determinably in this method via primal and dual information. Thirdly, typical subgradient algorithms suffer from the lack of an appropriate stopping criterion, and so the quality of the solution obtained after a finite number of steps is usually unknown. In contrast, by using the primal-dual gap, the proposed algorithm possesses a natural stopping criterion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580881
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Capacitated, balanced, sequential location-allocation problems on chains and trees (1990)
    Springer
    Mathematical programming 49 (1990), S. 381-396 
    Details
    ISSN: 1436-4646
    Keywords: Minisum locations ; location—allocation problems ; network location ; dynamic location ; sequential location
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper considers finite horizon, multiperiod, sequential, minisum location-allocation problems on chain graphs and tree networks. The demand has both deterministic and probabilistic components, and increases dynamically from period to period. The problem is to locate one additionalcapacitated facility in each of thep specified periods, and to determine the service allocations of the facilities, in order to optimally satisfy the demand on the network. In this context, two types of objective criteria or location strategies are addressed. The first is a myopic strategy in which the present period cost is minimized sequentially for each period, and the second is a discounted present worth strategy. For the chain graph, we analyze ap-facility problem under both these criteria, while for the tree graph, we analyze a 3-facility myopic problem, and a 2-facility discounted present worth problem. All these problems are nonconvex, and we specify a finite set of candidate solutions which may be compared in order to determine a global optimal solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01588799
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    Paper (German National Licenses)
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