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  • 1
    Electronic Resource
    Electronic Resource
    An elementary survey of general duality theory in mathematical programming (1981)
    Springer
    Mathematical programming 21 (1981), S. 241-261 
    Details
    ISSN: 1436-4646
    Keywords: Dual problem ; Duality Theory ; Optimality Conditions ; Price Functions ; Nonlinear Programming ; Integer Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming. The idea of replacing dual variables (prices) by price functions, suggested by Everett and developed by Gould, is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming. The dual problem adopted has a (traditional) economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality. In addition we attempt to make precise the relationship between primal, dual and saddlepoint results in both the traditional Lagrangean and the more general duality theories and to see the implications of passing from prices to price functions. Finally, and perhaps surprisingly, it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584248
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  • 2
    Electronic Resource
    Electronic Resource
    On the use of penumbras in blocking and antiblocking theory (1982)
    Springer
    Mathematical programming 22 (1982), S. 71-81 
    Details
    ISSN: 1436-4646
    Keywords: Blocking ; Antiblocking ; Polarity ; Penumbra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract E. Johnson has recently shown that the concept of a penumbra leads to a simple geometric description of the blocker and antiblocker of a given set. Here we develop some basic results on penumbras which permit us to slightly generalize and simplify results on their relationship to blocking and antiblocking theory. In addition, motivated by the obvious symmetry of our results, we examine the effect of reversing the blocking and antiblocking operations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581026
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  • 3
    Electronic Resource
    Electronic Resource
    Sequential convexification in reverse convex and disjunctive programming (1989)
    Springer
    Mathematical programming 44 (1989), S. 337-350 
    Details
    ISSN: 1436-4646
    Keywords: Disjunctive programming ; reverse convex programming ; facial disjunctive sets ; convex hull generation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper is about a property of certain combinatorial structures, called sequential convexifiability, shown by Balas (1974, 1979) to hold for facial disjunctive programs. Sequential convexifiability means that the convex hull of a nonconvex set defined by a collection of constraints can be generated by imposing the constraints one by one, sequentially, and generating each time the convex hull of the resulting set. Here we extend the class of problems considered to disjunctive programs with infinitely many terms, also known as reverse convex programs, and give necessary and sufficient conditions for the solution sets of such problems to be sequentially convexifiable. We point out important classes of problems in addition to facial disjunctive programs (for instance, reverse convex programs with equations only) for which the conditions are always satisfied. Finally, we give examples of disjunctive programs for which the conditions are violated, and so the procedure breaks down.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01587096
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  • 4
    Electronic Resource
    Electronic Resource
    Blocking and antiblocking sets (1974)
    Springer
    Mathematical programming 6 (1974), S. 157-166 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The purpose of this paper is to study a generalization of the concept of blocking and antiblocking polyhedra that has been introduced by D.R. Fulkerson. The polyhedra studied by Fulkerson are restricted to the nonnegative orthant in R m . The present generalization considers sets restricted to a cone. This leads to two polarity correspondences, which are related to the Minkowski polarity for convex sets.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580233
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  • 5
    Electronic Resource
    Electronic Resource
    Decomposition in global optimization (1991)
    Springer
    Journal of global optimization 1 (1991), S. 131-144 
    Details
    ISSN: 1573-2916
    Keywords: Decomposition ; general mathematical programming ; general duality ; hierarchical systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this article is to propose a simple framework for the various decomposition schemes in mathematical programming. Special instances are discussed. Particular attention is devoted to the general mathematical programming problem with two sets of variables. An economic interpretation in the context of hierarchical planning is done for the suggested decomposition procedure. The framework is based on general duality theory in mathematical programming and thus focussing on approaches leading to global optimality.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00119987
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  • 6
    Electronic Resource
    Electronic Resource
    Augmented Lagrangian and Tchebycheff Approaches in Multiple Objective Programming (1999)
    Springer
    Journal of global optimization 14 (1999), S. 251-266 
    Details
    ISSN: 1573-2916
    Keywords: Augmented Lagrangian ; Efficient solutions ; Multiple objective programs ; Tchebycheff scalarization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Relationships between the Tchebycheff scalarization and the augmented Lagrange multiplier technique are examined in the framework of general multiple objective programs (MOPs). It is shown that under certain conditions the Tchebycheff method can be represented as a quadratic weighted-sums scalarization of the MOP, that is, given weight values in the former, the coefficients of the latter can be found so that the same efficient point is selected. Analysis for concave and linear MOPs is included. Resulting applications in multiple criteria decision making are also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008314306344
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  • 7
    Electronic Resource
    Electronic Resource
    On the nucleolus of NTU-games defined by multiple objective linear programs (1996)
    Springer
    Mathematical methods of operations research 43 (1996), S. 337-352 
    Details
    ISSN: 1432-5217
    Keywords: Multiple Objective Linear Programs ; Nucleolus ; NTU-games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this article we derive a class of cooperative games with non-transferable utility from multiple objective linear programs. This is done in order to introduce the nucleolus, a solution concept from cooperative game theory, as a solution to multiple objective linear problems. We show that the nucleolus of such a game is a singleton, which is characterized by inclusion in the least core and the reduced game property. Furthermore the nucleolus satisfies efficiency, anonymity and strategic equivalence. We also present a polynomially bounded algorithm for computation of the nucleolus. Letn be the number of objective functions. The nucleolus is obtained by solving at most2n linear programs. Initially the ideal point is computed by solvingn linear programs. Then a sequence of at mostn linear programs is solved, and the nucleolus is obtained as the unique solution of the last program.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01194552
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  • 8
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    Column generation with free replicability in DEA (2009)
    Elsevier
    Details
    Publication Date: 2009-10-01
    Print ISSN: 0305-0483
    Electronic ISSN: 1873-5274
    Topics: Economics
    Published by Elsevier
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  • 9
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    Unbiased approximation in multicriteria optimization (2003)
    Springer
    Details
    Publication Date: 2003-01-01
    Print ISSN: 1432-2994
    Electronic ISSN: 1432-5217
    Topics: Mathematics , Economics
    Published by Springer
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  • 10
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    Dynamic productivity improvement in a model with multiple processes (2001)
    Springer
    Details
    Publication Date: 2001-12-01
    Print ISSN: 1432-2994
    Electronic ISSN: 1432-5217
    Topics: Mathematics , Economics
    Published by Springer
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