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  • 1
    Electronic Resource
    Electronic Resource
    Sublinear upper bounds for stochastic programs with recourse (1989)
    Springer
    Mathematical programming 43 (1989), S. 131-149 
    Details
    ISSN: 1436-4646
    Keywords: Stochastic programming ; sublinear function ; simple recourse problem ; recourse model ; duality ; approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582286
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  • 2
    Electronic Resource
    Electronic Resource
    On decision rules in stochastic programming (1974)
    Springer
    Mathematical programming 7 (1974), S. 117-143 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The paper surveys the basic results and nonresults for decision rules in stochastic programming. It exhibits some of the difficulties encountered when trying to restrict the class of acceptable rules to those possessing specific functional forms. A liberal dosage of examples is provided which illustrate various cases. The treatment is unified by making use of the equivalence of various formulations which have appeared in the literature. An appendix is devoted to the P-model for stochastic programs with chance constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585511
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  • 3
    Electronic Resource
    Electronic Resource
    Quantitative stability of variational systems: III.ε-approximate solutions (1993)
    Springer
    Mathematical programming 61 (1993), S. 197-214 
    Details
    ISSN: 1436-4646
    Keywords: Epi-convergence ; epi-distance ; stability ; convex optimization ; approximate solutions ; subgradients ; level sets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We prove that theε-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar property is obtained for the distance between the level sets of extended real valued functions. We also show that these properties imply that theε-subgradient mapping is Lipschitz continuous.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582147
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  • 4
    Electronic Resource
    Electronic Resource
    Convergence of set-valued mappings: Equi-outer semicontinuity (1996)
    Springer
    Set-valued analysis 4 (1996), S. 333-360 
    Details
    ISSN: 1572-932X
    Keywords: 54C60 ; 54B20 ; 49J40 ; 49J52 ; set-valued mappings ; epi-convergence ; multifunction ; equi-continuity ; equi-semicontinuity ; Arzelà-Ascolì theorem ; maximal monotone ; operators ; differential inclusions ; closed convex processes ; sublinear mappings ; subgradient mappings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The concept of equi-outer semicontinuity allows us to relate the pointwise and the graphical convergence of set-valued-mappings. One of the main results is a compactness criterion that extends the classical Arzelà-Ascolì theorem for continuous functions to this new setting; it also leads to the exploration of the notion of continuous convergence. Equi-lower semicontinuity of functions is related to the outer semicontinuity of epigraphical mappings. Finally, some examples involving set-valued mappings are re-examined in terms of the concepts introduced here.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00436110
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  • 5
    Electronic Resource
    Electronic Resource
    Modeling and solution strategies for unconstrained stochastic optimization problems (1984)
    Springer
    Annals of operations research 1 (1984), S. 3-22 
    Details
    ISSN: 1572-9338
    Keywords: Stochastic optimization ; subgradient ; stochastic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We review some modeling alternatives for handling risk in decision-making processes for unconstrained stochastic optimization problems. Solution strategies are discussed and compared.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01874449
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  • 6
    Electronic Resource
    Electronic Resource
    Production planning via scenario modelling (1993)
    Springer
    Annals of operations research 43 (1993), S. 309-335 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02025089
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  • 7
    Electronic Resource
    Electronic Resource
    Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems (1995)
    Springer
    Annals of operations research 56 (1995), S. 189-208 
    Details
    ISSN: 1572-9338
    Keywords: Epi-distance ; large deviations ; stochastic programs ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Several exponential bounds are derived by means of the theory of large deviations for the convergence of approximate solutions of stochastic optimization problems. The basic results show that the solutions obtained by replacing the original distribution by an empirical distribution provides an effective tool for solving stochastic programming problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031707
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  • 8
    Electronic Resource
    Electronic Resource
    Preface (1995)
    Springer
    Annals of operations research 59 (1995), S. v 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031740
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  • 9
    Electronic Resource
    Electronic Resource
    Glivenko-Cantelli type theorems: An application of the convergence theory of stochastic suprema (1991)
    Springer
    Annals of operations research 30 (1991), S. 157-168 
    Details
    ISSN: 1572-9338
    Keywords: Glivenko ; Cantelli lemma ; narrow (weak, weak) ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The uniform convergence of empirical processes on certain classes of sets follows from the convergence theory for random lower semicontinuous functions studies in the context of stochastic optimization. In the process, a richer class of sets for which one can prove this type of result is exhibited.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02204814
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  • 10
    Electronic Resource
    Electronic Resource
    Statistical estimation from an optimization viewpoint (1999)
    Springer
    Annals of operations research 85 (1999), S. 79-101 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Statistics and Optimization have been closely linked from the very outset. The search fora “best” estimator (least squares, maximum likelihood, etc.) certainly relies on optimizationtools. On the other hand, Statistics has often provided the motivation for the development ofalgorithmic procedures for certain classes of optimization problems. However, it is onlyrelatively recently, more specifically in connection with the development of an approximationand sampling theory for stochastic programming problems, that the full connectionhas come to light. This in turn suggests a more comprehensive approach to the formulationof statistical estimation questions. This expository paper reviews some of the features ofthis approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018934214007
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