Peter W. Glynn · Wiley StatsRef: Statistics Reference Online · 2014
Abstract Uncertainty is present in virtually all decision‐making environments. In some applications settings, one may safely choose to ignore the presence of uncertainty because its influence on the optimal decision is likely to be small. In other settings, one would ideally like to explicitly incorporate the presence of uncertainty into a mathematical formulation of the problem, because it is a priori clear that an optimal solution will be significantly influenced by the presence of uncertainty. Loosely speaking, stochastic optimization is the set of mathematical and computational tools that have been developed in order to characterize or compute optimal solutions to mathematical models involving such stochastic elements. From a mathematical viewpoint, stochastic optimization has two main subdisciplines. The first is that of stochastic control. Stochastic control is concerned with optimization problems in which the optimizing solution is ‘infinite dimensional’ (or of very high dimensionality) in character. The second subdiscipline involves finite‐dimensional stochastic optimization. Much of the corresponding literature focuses on intelligent ‘search’ methods for numerically computing the optimal setting of a (finite‐dimensional) vector of decision variables.
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Lauren A. Hannah · Elsevier eBooks · 2015
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Ali H. Sayed · Cambridge University Press eBooks · 2022
This extraordinary three-volume work, written in an engaging and rigorous style by a world authority in the field, provides an accessible, comprehensive introduction to the full spectrum of mathematical and statistical techniques underpinning contemporary methods in data-driven learning and inference. This first volume, Foundations, introduces core topics in inference and learning, such as matrix theory, linear algebra, random variables, convex optimization and stochastic optimization, and prepares students for studying their practical application in later volumes. A consistent structure and pedagogy is employed throughout this volume to reinforce student understanding, with over 600 end-of-chapter problems (including solutions for instructors), 100 figures, 180 solved examples, datasets and downloadable Matlab code. Supported by sister volumes Inference and Learning, and unique in its scale and depth, this textbook sequence is ideal for early-career researchers and graduate students across many courses in signal processing, machine learning, statistical analysis, data science and inference.
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Wang, Tianyang, author · Digital Collections of Colorado (Colorado State University) · 2020
The course includes a set of materials collected for the OER project funded by the Colorado OER Council Grant (AY 2019-20).
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Bruno Hideki Fukushima-Kimura, Yoshinori Kamijima, Kazushi Kawamura et al. · Transactions of the Institute of Systems Control and Information Engineers · 2023
The topic we address in this paper concerns the minimization of a Hamiltonian function for an Ising model through the application of simulated annealing algorithms based on (single-site) Glauber dynamics and stochastic cellular automata (SCA). Some rigorous results are presented in order to justify the application of simulated annealing for a particular kind of SCA. After that, we compare the SCA algorithm and its variation, namely the ε-SCA algorithm, studied in this paper with the Glauber dynamics by analyzing their accuracy in obtaining optimal solutions for the max-cut problem on Erdös-Rényi random graphs, the traveling salesman problem (TSP), and the minimization of Gaussian and Bernoulli spin glass Hamiltonians. We observed that the SCA performed better than the Glauber dynamics in some special cases, while the ε-SCA showed the highest performance in all scenarios.
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Marco Cavazzuti · 2012
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Edward Anderson · Business risk management · 2013
In this chapter the authors use the methods of optimization to determine how to make good decisions in a stochastic environment and they consider more complex problems. There will be a need to set up a model of what is happening and then to analyze this model. Because of this additional complexity, the focus will switch back to the normative, rather than the descriptive, so they will ask what managers should do, rather than what they will do. For most of the models that they deal with in the chapter it is necessary to carry out an analysis using some computational tool like a spreadsheet. Now they turn to the solution of stochastic optimization problems where the decision maker has to make a string of successive decisions with more information becoming available as time goes on. The aim is to use the Monte Carlo simulation approach in this multistage environment.
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Lecture notes in control and information sciences · 1986
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Peter W. Glynn · Encyclopedia of Actuarial Science · 2004
Abstract Uncertainty is present in virtually all decision‐making environments. In some applications settings, one may safely choose to ignore the presence of uncertainty because its influence on the optimal decision is likely to be small. In other settings, one would ideally like to explicitly incorporate the presence of uncertainty into a mathematical formulation of the problem, because it is a priori clear that an optimal solution will be significantly influenced by the presence of uncertainty. Loosely speaking, stochastic optimization is the set of mathematical and computational tools that have been developed in order to characterize or compute optimal solutions to mathematical models involving such stochastic elements. From a mathematical viewpoint, stochastic optimization has two main subdisciplines. The first is that of stochastic control. Stochastic control is concerned with optimization problems in which the optimizing solution is ‘infinite dimensional’ (or of very high dimensionality) in character. The second subdiscipline involves finite‐dimensional stochastic optimization. Much of the corresponding literature focuses on intelligent ‘search’ methods for numerically computing the optimal setting of a (finite‐dimensional) vector of decision variables.
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Anton J. Kleywegt, Alexander Shapiro · 2001
This chapter contains sections titled: Introduction Optimization Under Uncertainty Stochastic Programming Dynamic Programming References
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Makoto Yamakawa, Makoto Ohsaki · 2023
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Schneider, Johannes J, Kirkpatrick, Scott · CERN Document Server (European Organization for Nuclear Research) · 2007
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Johannes Schneider, Scott Kirkpatrick · Scientific computation · 2006
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Georg Ch. Pflug · Encyclopedia of Optimization · 2001
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Ricardo Harboe · 1993
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K. Marti · 2015
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Jianqiang Cheng · theses.fr (ABES) · 2013
Dans cette thèse, nous étudions trois types de problèmes stochastiques : les problèmes avec contraintes probabilistes, les problèmes distributionnellement robustes et les problèmes avec recours. Les difficultés des problèmes stochastiques sont essentiellement liées aux problèmes de convexité du domaine des solutions, et du calcul de l espérance mathématique ou des probabilités qui nécessitent le calcul complexe d intégrales multiples. A cause de ces difficultés majeures, nous avons résolu les problèmes étudiées à l aide d approximations efficaces.Nous avons étudié deux types de problèmes stochastiques avec des contraintes en probabilités, i.e., les problèmes linéaires avec contraintes en probabilité jointes (LLPC) et les problèmes de maximisation de probabilités (MPP). Dans les deux cas, nous avons supposé que les variables aléatoires sont normalement distribués et les vecteurs lignes des matrices aléatoires sont indépendants. Nous avons résolu LLPC, qui est un problème généralement non convexe, à l aide de deux approximations basée sur les problèmes coniques de second ordre (SOCP). Sous certaines hypothèses faibles, les solutions optimales des deux SOCP sont respectivement les bornes inférieures et supérieures du problème du départ. En ce qui concerne MPP, nous avons étudié une variante du problème du plus court chemin stochastique contraint (SRCSP) qui consiste à maximiser la probabilité de la contrainte de ressources. Pour résoudre ce problème, nous avons proposé un algorithme de Branch and Bound pour calculer la solution optimale. Comme la relaxation linéaire n est pas convexe, nous avons proposé une approximation convexe efficace. Nous avons par la suite testé nos algorithmes pour tous les problèmes étudiés sur des instances aléatoires. Pour LLPC, notre approche est plus performante que celles de Bonferroni et de Jaganathan. Pour MPP, nos résultats numériques montrent que notre approche est là encore plus performante que l approximation des contraintes probabilistes individuellement.La deuxième famille de problèmes étudiés est celle relative aux problèmes distributionnellement robustes où une partie seulement de l information sur les variables aléatoires est connue à savoir les deux premiers moments. Nous avons montré que le problème de sac à dos stochastique (SKP) est un problème semi-défini positif (SDP) après relaxation SDP des contraintes binaires. Bien que ce résultat ne puisse être étendu au cas du problème multi-sac-à-dos (MKP), nous avons proposé deux approximations qui permettent d obtenir des bornes de bonne qualité pour la plupart des instances testées. Nos résultats numériques montrent que nos approximations sont là encore plus performantes que celles basées sur les inégalités de Bonferroni et celles plus récentes de Zymler. Ces résultats ont aussi montré la robustesse des solutions obtenues face aux fluctuations des distributions de probabilités. Nous avons aussi étudié une variante du problème du plus court chemin stochastique. Nous avons prouvé que ce problème peut se ramener au problème de plus court chemin déterministe sous certaine hypothèses. Pour résoudre ce problème, nous avons proposé une méthode de B&B où les bornes inférieures sont calculées à l aide de la méthode du gradient projeté stochastique. Des résultats numériques ont montré l efficacité de notre approche. Enfin, l ensemble des méthodes que nous avons proposées dans cette thèse peuvent s appliquer à une large famille de problèmes d optimisation stochastique avec variables entières.
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Gazioglu, S. · OpenMETU (Middle East Technical University) · 2007
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James Spall, H Arsham, N Baba et al. · Pure and applied mathematics · 1982
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Avijit Mukherjee · 2004
This dissertation presents dynamic stochastic optimization models for Air Traffic Flow Management (ATFM) that enables decisions to adapt to new information on evolving capacities of National Airspace System (NAS) resources. Uncertainty is represented by a set of capacity scenarios, each depicting a particular time-varying capacity profile of NAS resources. We use the concept of a scenario tree in which multiple scenarios are possible initially. Scenarios are eliminated as possibilities in a succession of branching points, until the specific scenario that will be realized on a particular day is known. Thus the scenario tree branching provides updated information on evolving scenarios, and allows ATFM decisions to be re-addressed and revised. First, we propose a dynamic stochastic model for a single airport ground holding problem (SAGHP) that can be used for planning Ground Delay Programs (GDPs) when there is uncertainty about future airport arrival capacities. Ground delays of non-departed flights can be revised based on updated information from scenario tree branching. The problem is formulated so that a wide range of objective functions, including non-linear delay cost functions and functions that reflect equity concerns can be optimized. Furthermore, the
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