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  • 1
    Electronic Resource
    Electronic Resource
    Fractional programming without differentiability (1976)
    Springer
    Mathematical programming 11 (1976), S. 283-290 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The notion of quasi-differentiability is examined and related to fractional programming. Necessary and sufficient conditions are given and various other properties of quasi-differentiable functions are discussed. Differentiability is not assumed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580396
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  • 2
    Electronic Resource
    Electronic Resource
    A note on the existence of subgradients (1982)
    Springer
    Mathematical programming 24 (1982), S. 225-228 
    Details
    ISSN: 1436-4646
    Keywords: Convex Function ; Subgradient ; Max-formula
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We describe an apparently novel way of constructing the subgradient of a convex function defined on a finite dimensional vector space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585105
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  • 3
    Electronic Resource
    Electronic Resource
    A simple constraint qualification in infinite dimensional programming (1986)
    Springer
    Mathematical programming 35 (1986), S. 83-96 
    Details
    ISSN: 1436-4646
    Keywords: Infinite dimensional linear programming ; semi-infinite programming ; constraint qualification ; optimality conditions ; dual program
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A new, simple, constraint qualification for infinite dimensional programs with linear programming type constraints is used to derive the dual program; see Theorem 3.1. Applications include a proof of the explicit solution of the best interpolation problem presented in [8].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589443
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  • 4
    Electronic Resource
    Electronic Resource
    Partially finite convex programming, Part I: Quasi relative interiors and duality theory (1992)
    Springer
    Mathematical programming 57 (1992), S. 15-48 
    Details
    ISSN: 1436-4646
    Keywords: Primary 90C25, 49B27 ; Secondary 90C48, 52A07, 65K05 ; Convex programming ; duality ; constraint qualification ; Fenchel duality ; semi-infinite programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581072
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  • 5
    Electronic Resource
    Electronic Resource
    Partially finite convex programming, Part II: Explicit lattice models (1992)
    Springer
    Mathematical programming 57 (1992), S. 49-83 
    Details
    ISSN: 1436-4646
    Keywords: Primary 90C25, 49B27 ; Secondary 90C48, 52A07, 65K05 ; Convex programming ; duality ; constraint qualification ; semi-infinite programming ; constrained approximation ; spectral estimation ; transportation problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L 1 approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581073
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  • 6
    Electronic Resource
    Electronic Resource
    A note on perfect duality and limiting lagrangeans (1980)
    Springer
    Mathematical programming 18 (1980), S. 330-337 
    Details
    ISSN: 1436-4646
    Keywords: Limiting Lagrangean ; Duality Gaps ; Perfect Duality ; Convex Conjugates ; Convex Duality ; Semi-infinite Programs ; Closure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this note we derive and extend the substance of recent results on Perfect Duality and Limiting Lagrangeans by using standard convex analysis and convex duality theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01588327
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  • 7
    Electronic Resource
    Electronic Resource
    Direct theorems in semi-infinite convex programming (1981)
    Springer
    Mathematical programming 21 (1981), S. 301-318 
    Details
    ISSN: 1436-4646
    Keywords: Semi-infinite Programs ; Open Helly-type Theorems ; Finite Subprograms ; Convex Programs ; Multi-criteria Programs ; Quasi-convex Programs ; Quasi-differentiable Programs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We show that a semi-infinite quasi-convex program with certain regularity conditions possesses finitely constrained subprograms with the same optimal value. This result is applied to various problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584251
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  • 8
    Electronic Resource
    Electronic Resource
    Approximate subgradients and coderivatives in R n (1996)
    Springer
    Set-valued analysis 4 (1996), S. 375-398 
    Details
    ISSN: 1572-932X
    Keywords: Primary 49J52 ; Secondary 26A27 ; 26B12 ; 49J50 ; 52A20 ; subgradient ; coderivative ; generalized Jacobian ; Lipschitz function ; bump function ; gauge ; nowhere dense set ; Lebesgue measure ; disconnectedness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00436112
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  • 9
    Electronic Resource
    Electronic Resource
    On the convergence of von Neumann's alternating projection algorithm for two sets (1993)
    Springer
    Set-valued analysis 1 (1993), S. 185-212 
    Details
    ISSN: 1572-932X
    Keywords: Primary 47H09, 65J05 ; secondary 41A25, 41A50, 46C99, 47N10, 49M45, 65Kxx, 90C25 ; Algorithm ; von Neumann's algorithm ; method ; alternating method ; iterative method ; projection ; cyclic projections ; successive projections ; Hilbert space ; convex sets ; linear convergence ; norm convergence ; weak convergence ; open mapping theorem ; multifunctions ; convex relations ; convex feasibility problem ; least-squares approximation ; angle between two subspaces
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a Hilbert space. Our research is formulated within the framework of Fejér monotonicity, convex and set-valued analysis. We also discuss the case of finitely many sets.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01027691
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  • 10
    Electronic Resource
    Electronic Resource
    A very rapidly convergent product expansion for π (1983)
    Springer
    BIT 23 (1983), S. 538-540 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933626
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