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  • 1
    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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  • 2
    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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    Paper (German National Licenses)
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  • 3
    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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  • 4
    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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  • 5
    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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  • 6
    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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  • 7
    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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  • 8
    Electronic Resource
    Electronic Resource
    A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems (1989)
    Springer
    Applied mathematics & optimization 20 (1989), S. 193-221 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a primal-dual conjugate subgradient algorithm for solving convex programming problems. The motivation, however, is to employ it for solving specially structured or decomposable linear programming problems. The algorithm coordinates a primal penalty function and a Lagrangian dual function, in order to generate a (geometrically) convergent sequence of primal and dual iterates. Several refinements are discussed to improve the performance of the algorithm. These are tested on some network problems, with side constraints and variables, faced by the Freight Equipment Management Program of the Association of American Railroads, and suggestions are made for implementation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01447654
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    Solving Euclidean Distance Multifacility Location Problems Using Conjugate Subgradient and Line-Search Methods (1999)
    Springer
    Computational optimization and applications 14 (1999), S. 275-291 
    Details
    ISSN: 1573-2894
    Keywords: Euclidean multifacility location problem ; nondifferentiable optimization ; conjugate subgradient methods ; variable target value method ; inexact line-search
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Subgradient methods are popular for solving nondifferentiable optimization problems because of their relative ease in implementation, but are not always robust and require a careful design of strategies in order to yield an effective procedure for any given class of problems. In this paper, we present an approach for solving the Euclidean distance multifacility location problem (EMFLP) using conjugate or deflected subgradient based algorithms along with suitable line-search strategies. The subgradient deflection method considered is the Average Direction Strategy (ADS) imbedded within the Variable Target Value Method (VTVM). We also investigate the generation of two types of subgradients to be employed in conjunction with ADS. The first type is a simple valid subgradient that assigns zero values to contributions corresponding to the nondifferentiable terms in the objective function, and so, the subgradient is composed by summing the contributions corresponding to the differentiable terms alone. The second type expends more effort to derive a low-norm member of the subdifferential in order to enhance the prospect of obtaining a descent direction. Furthermore, a special Newton-based line-search that exploits the nondifferentiability of the problem is also designed to be implemented in the developed algorithm in order to study its impact on the convergence behavior. Various combinations of the above strategies are composed and evaluated on a set of test problems. The results show that a modification of the VTVM method along with the first or a certain combination of the two subgradient generation strategies, and the use of a suitable line-search technique, provides promising results. An alternative block-halving step-size strategy used within VTVM in conjunction with the proposed line-search method yields a competitive second choice performance.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026440104273
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique (1992)
    Springer
    Journal of global optimization 2 (1992), S. 101-112 
    Details
    ISSN: 1573-2916
    Keywords: Reformulation ; linearization ; polynomial programs ; multilinear programs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00121304
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    Paper (German National Licenses)
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