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  • 1
    Electronic Resource
    Electronic Resource
    Novel approaches to the discrimination problem (1992)
    Springer
    Mathematical methods of operations research 36 (1992), S. 517-545 
    Details
    ISSN: 1432-5217
    Keywords: Discriminant analysis ; Quadratic programming ; Complexity ; Integer programming ; Bilevel programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We consider the problem of determining a hyperplane that separates, as “well” as possible, two finite sets of points inR n . We analyze two criteria for judging the quality of a candidate hyperplane (i) the maximal distance of a misclassified point to the hyperplane (ii) the number of misclassified points. In each case, we investigate the computational complexity of the corresponding mathematical programs, give equivalent formulations, suggest solution algorithms and present preliminary numerical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01416243
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  • 2
    Electronic Resource
    Electronic Resource
    Descent approaches for quadratic bilevel programming (1994)
    Springer
    Journal of optimization theory and applications 81 (1994), S. 379-399 
    Details
    ISSN: 1573-2878
    Keywords: Bilevel programming ; nonconvex and nondifferentiable optimization ; quadratic programming ; computational complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The bilevel programming problem involves two optimization problems where the data of the first one is implicitly determined by the solution of the second. In this paper, we introduce two descent methods for a special instance of bilevel programs where the inner problem is strictly convex quadratic. The first algorithm is based on pivot steps and may not guarantee local optimality. A modified steepest descent algorithm is presented to overcome this drawback. New rules for computing exact stepsizes are introduced and a hybrid approach that combines both strategies is discussed. It is proved that checking local optimality in bilevel programming is a NP-hard problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02191670
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  • 3
    Electronic Resource
    Electronic Resource
    A hybrid Tabu-ascent algorithm for the linear Bilevel Programming Problem (1996)
    Springer
    Journal of global optimization 8 (1996), S. 217-233 
    Details
    ISSN: 1573-2916
    Keywords: Bilevel programming ; adaptive search methods ; combinatorial optimization ; Tabu Search
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The linear Bilevel Programming Problem (BLP) is an instance of a linear hierarchical decision process where the lower level constraint set is dependent on decisions taken at the upper level. In this paper we propose to solve this NP-hard problem using an adaptive search method related to the Tabu Search metaheuristic. Numerical results on large scale linear BLPs are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00121266
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  • 4
    Electronic Resource
    Electronic Resource
    Links Between Linear Bilevel and Mixed 0–1 Programming Problems (1997)
    Springer
    Journal of optimization theory and applications 93 (1997), S. 273-300 
    Details
    ISSN: 1573-2878
    Keywords: Bilevel programming ; mixed 0–1 programming ; embedded algorithms ; branch-and-bound methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study links between the linear bilevel and linear mixed 0–1 programming problems. A new reformulation of the linear mixed 0–1 programming problem into a linear bilevel programming one, which does not require the introduction of a large finite constant, is presented. We show that solving a linear mixed 0–1 problem by a classical branch-and-bound algorithm is equivalent in a strong sense to solving its bilevel reformulation by a bilevel branch-and-bound algorithm. The mixed 0–1 algorithm is embedded in the bilevel algorithm through the aforementioned reformulation; i.e., when applied to any mixed 0–1 instance and its bilevel reformulation, they generate sequences of subproblems which are identical via the reformulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022645805569
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