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  • 1
    Electronic Resource
    Electronic Resource
    Optimality conditions in mathematical programming and composite optimization (1994)
    Springer
    Mathematical programming 67 (1994), S. 225-245 
    Details
    ISSN: 1436-4646
    Keywords: Composite functions ; Compound derivatives ; Compound tangent sets ; Fermat rule ; Mathematical programming ; Multipliers ; Optimality conditions ; Second order conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat's rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki (1988) and Cominetti (1989); but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. We point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. Our conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral we recover the classical conditions of Ben-Tal—Zowe (1982) and Cominetti (1990).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582222
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Metrically well-set minimization problems (1992)
    Springer
    Applied mathematics & optimization 26 (1992), S. 273-285 
    Details
    ISSN: 1432-0606
    Keywords: Sensitivity of solutions ; Well-posedness ; Mathematical programming in abstract spaces ; 90C31 ; 49A27
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A concept of well-posedness, or more exactly of stability in a metric sense, is introduced for minimization problems on metric spaces generalizing the notion due to Tykhonov to situations in which there is no uniqueness of solutions. It is compared with other concepts, in particular to a variant of the notion after Hadamard reformulated via a metric semicontinuity approach. Concrete criteria of well-posedness are presented, e.g., for convex minimization problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01371085
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  • 3
    Electronic Resource
    Electronic Resource
    On the Minimization of Difference Functions (1998)
    Springer
    Journal of global optimization 12 (1998), S. 373-382 
    Details
    ISSN: 1573-2916
    Keywords: Critical point ; d.c. Functions ; Difference of convex functions ; Generalized derivatives ; Minimization ; Optimality conditions ; Palais-Smale condition ; Subdifferentials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Simple necessary optimality conditions are formulated for a function f of the form f _ g–h, where g and h are nonsmooth functions. Related sufficient conditions are given for local minimization and global minimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008233531797
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  • 4
    Electronic Resource
    Electronic Resource
    Miscellaneous incidences of convergence theories in optimization and nonlinear analysis I: Behavior of solutions (1994)
    Springer
    Set-valued analysis 2 (1994), S. 259-274 
    Details
    ISSN: 1572-932X
    Keywords: 54C60 ; 52A41 ; 49J45 ; 49J52 ; 54E55 ; Approximate minimization ; bounded hemiconvergence ; marginal function ; minimizer ; stable function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We examine some connections between convergence theories and optimization. In particular we study the Lipschitzian character of the infimal value function with respect to variations of the objective function. We also study the approximate solution multifunction for convex and nonconvex objective functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01027105
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  • 5
    Electronic Resource
    Electronic Resource
    Compactness Properties, Openness Criteria and Coderivatives (1998)
    Springer
    Set-valued analysis 6 (1998), S. 363-380 
    Details
    ISSN: 1572-932X
    Keywords: coderivative ; coderivatively compact ; compactness ; injectivity ; normal compactness ; openness ; nonsmooth analysis ; subdifferential
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the local surjectivity and openness properties of mappings and correspondences by using coderivatives. We recall local criteria but concentrate on point criteria. Our study relies on a compactness condition which is of independent interest, for instance, for the study of the behavior of the injectivity constant of a linear map or of a convex process under stabilization procedures. Several known criteria for openness are shown to be a consequences of this new compactness condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008746317549
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  • 6
    Electronic Resource
    Electronic Resource
    Metric Estimates for the Calculus of Multimappings (1997)
    Springer
    Set-valued analysis 5 (1997), S. 291-308 
    Details
    ISSN: 1572-932X
    Keywords: conization ; contingent ; continuity ; derivative ; estimates ; hemicontinuity ; hemidistance ; gap ; multifunction ; multimapping ; normal cone ; pseudo-Lipschitz ; semicontinuity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper completes previous results of Penot and Jourani dealing with continuity properties of multimappings obtained via intersections or inverse images. A key role is played by estimates of metric regularity type. These estimates have consequences on the calculus of generalized derivatives and on the computation of normal cones. The considered multimappings are not supposed to have convex values, but when this property holds, simple qualification conditions ensure these estimates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008625212506
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  • 7
    Electronic Resource
    Electronic Resource
    Cooperative behavior of functions, relations and sets (1998)
    Springer
    Mathematical methods of operations research 48 (1998), S. 229-246 
    Details
    ISSN: 1432-5217
    Keywords: Key words: Alliedness ; coderivative ; homotone ; normal ; normal compactness ; quasi-homotone ; subdifferential ; synergy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract. One of the main difficulties in nonsmooth analysis is to devise calculus rules. It is our purpose here to show that a certain cooperative behavior between functions (resp. sets, resp. multifunctions) yields calculus rules for subdifferentials (resp. normal cones, resp. coderivatives). In previous contributions, the qualification conditions ensuring calculus rules were given in a non symmetric way. The new conditions can be combined easily and encompass various criteria. We also address the important question of the extension of calculus rules from the Lipschitz case to the general case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s001860050025
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  • 8
    Electronic Resource
    Electronic Resource
    Applications sous-linéaires à valeurs dans un espace de fonctions continues (1984)
    Springer
    Annali di matematica pura ed applicata 136 (1984), S. 133-151 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We investigate the possibility of representing a convex mapping f with values in some space C(T) of continuous functions on a compact space T, as a supremum of continuous affine mappings, when the domain D of f is some convex subset of a Banach space X. Various limiting examples are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01773381
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  • 9
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    Softness, sleekness and regularity properties in nonsmooth analysis (2008)
    Elsevier
    Details
    Publication Date: 2008-05-01
    Print ISSN: 0362-546X
    Electronic ISSN: 1873-5215
    Topics: Mathematics
    Published by Elsevier
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  • 10
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    Second-order moderate derivatives (1994)
    Elsevier
    Details
    Publication Date: 1994-04-01
    Print ISSN: 0362-546X
    Electronic ISSN: 1873-5215
    Topics: Mathematics
    Published by Elsevier
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