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  • 1
    Electronic Resource
    Electronic Resource
    A note on functions whose local minima are global (1982)
    Springer
    Journal of optimization theory and applications 36 (1982), S. 457-463 
    Details
    ISSN: 1573-2878
    Keywords: Global optimality ; nonconvex programming ; mathematical programming ; nonlinear optimization ; generalized convexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note, we introduce a new class of generalized convex functions and show that a real functionf which is continuous on a compact convex subsetM of ℝ n and whose set of global minimizers onM is arcwise-connected has the property that every local minimum is global if, and only if,f belongs to that class of functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934358
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  • 2
    Electronic Resource
    Electronic Resource
    Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization (1988)
    Springer
    Journal of optimization theory and applications 58 (1988), S. 11-37 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; concave minimization ; DC-programming ; Lipschitzian optimization ; branch-and-bound methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939768
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  • 3
    Electronic Resource
    Electronic Resource
    DC Programming: Overview (1999)
    Springer
    Journal of optimization theory and applications 103 (1999), S. 1-43 
    Details
    ISSN: 1573-2878
    Keywords: DC functions ; DC programming ; global optimization ; nonconvex programming ; optimality conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021765131316
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  • 4
    Electronic Resource
    Electronic Resource
    A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization (1986)
    Springer
    Journal of optimization theory and applications 51 (1986), S. 271-291 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; nonlinear optimization ; branch-and-bound methods ; concave minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Based on a review of existing algorithms, a general branch-and-bound concept in global optimization is presented. A sufficient and necessary convergence condition is established, and a broad class of realizations is derived that include existing and several new approaches for concave minimization problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939825
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  • 5
    Electronic Resource
    Electronic Resource
    On consistency of bounding operations in deterministic global optimization (1989)
    Springer
    Journal of optimization theory and applications 61 (1989), S. 143-146 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; branch-and-bound methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence φq → φ in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940850
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