08015nam 22004213 450 991086917950332120240703080304.09783031579882(CKB)32609617600041(MiAaPQ)EBC31507308(Au-PeEL)EBL31507308(EXLCZ)993260961760004120240703d2024 uy 0engur|||||||||||txtrdacontentcrdamediacrrdacarrierRisk-Averse Optimization and Control Theory and Methods1st ed.Cham :Springer International Publishing AG,2024.©2024.1 online resource (462 pages)Springer Series in Operations Research and Financial Engineering Series9783031579875 Intro -- Preface -- Introduction -- Contents -- 1 Elements of the Utility Theory -- 1.1 Preference Relations -- 1.2 Expected Utility Theory -- 1.2.1 The Prospect Space of Probability Measures -- 1.2.2 Affine Numerical Representation -- 1.2.3 Integral Representation. Utility Functions -- 1.2.4 Monotonicity and Risk Aversion -- 1.3 Dual Utility Theory -- 1.3.1 The Prospect Space of Quantile Functions -- 1.3.2 Affine Numerical Representation -- 1.3.3 Integral Representation with Rank Dependent Utility Functions -- 1.3.4 Choquet Integral Representation of Dual Utility -- 1.3.5 Risk Aversion -- 1.4 Utility Theory for Random Vectors -- 1.4.1 Expected Utility Theory for Random Vectors -- 1.4.1.1 Lotteries -- 1.4.1.2 Integral Numerical Representation -- 1.4.1.3 Monotonicity and Risk Aversion -- 1.4.2 Dual Utility Theory for Random Variables -- 1.4.2.1 Comonotonic Random Variables -- 1.4.2.2 Integral Numerical Representation -- 1.4.2.3 Risk Aversion -- 1.4.3 Certainty Equivalents -- 2 Measures of Risk -- 2.1 Foundations -- 2.1.1 Axiomatic Definition of Measures of Risk -- 2.1.2 Relation to Certainty Equivalents -- 2.2 Quantile-Based Measures of Risk -- 2.2.1 Value at Risk -- 2.2.2 Average Value at Risk -- 2.2.3 Inverse Average Value at Risk -- 2.2.4 Distortion Measures -- 2.3 Dual Representation -- 2.3.1 Conjugate Duality for Risk Measures -- 2.3.2 Subdifferentiation -- 2.4 Law-Invariant Risk Measures -- 2.4.1 Basic Properties -- 2.4.2 Kusuoka Representation -- 2.5 Risk Measures on the Spaces of Quantile Functions -- 2.5.1 Integrable Prospects -- 2.5.2 Bounded Prospects -- 2.5.3 Kusuoka Representation -- 2.6 Systemic Risk Measures -- 2.6.1 Dual Representation -- 2.6.2 Linear Scalarizations -- 2.6.3 Nonlinear Scalarization -- 2.7 Risk Forms -- 2.7.1 Risk Models with Variable Probability Measures -- 2.7.2 Dual and Kusuoka Representations.2.7.3 Risk Disintegration -- 2.7.4 Risk in Two-Stage Partially Observable Systems -- 2.7.4.1 Fixed Observation Distribution -- 2.7.4.2 Controlled Observation Distribution -- 2.7.5 Mini-Batch Risk Measures -- 3 Optimization of Measures of Risk -- 3.1 Differentiation of Composite Risk Functions -- 3.1.1 Composition with a Convex Operator -- 3.1.2 Composition with a Convex Integrand -- 3.1.3 Composition with a Convex Function -- 3.1.4 Composition with a Regular Lipschitz Continuous Function -- 3.1.5 Composition with a Regular Lipschitz Continuous Integrand -- 3.1.6 Composition with a Vector-Valued Integrand -- 3.1.7 Composition with a Semismooth Function -- 3.1.8 Conservativeness of the Subdifferential Field -- 3.2 Optimization of Composite Risk Functions -- 3.2.1 Composition with a Convex Function -- 3.2.2 Composition with a Lipschitz Continuous Function -- 3.3 Duality Theory -- 3.3.1 Duality Associated with Risk Measures -- 3.3.2 Duality Associated with Nonanticipativity Constraints -- 3.4 Stochastic Subgradient Methods -- 3.4.1 Stochastic Subgradients of Mini-Batch Risk Functions -- 3.4.2 The Basic Subgradient Method -- 3.4.2.1 The Convex Case -- 3.4.2.2 The Nonsmooth Nonconvex Case -- 3.4.3 The Method with Dual Averaging -- 4 Dynamic Risk Optimization -- 4.1 Dynamic Measures of Risk -- 4.1.1 Time Consistency -- 4.1.2 One-Step Conditional Risk Mappings -- 4.2 Dual Representation -- 4.2.1 Subdifferentiation of Conditional Risk Mappings -- 4.2.2 Dual Representation of Conditional Risk Mappings -- 4.2.3 Dual Representation of Dynamic Risk Measures -- 4.2.4 Scenario Tree Models -- 4.3 Multistage Risk Optimization Problems -- 4.3.1 Problem Formulation -- 4.3.2 Convexity and Subdifferentiation of Composite Dynamic Risk Measures -- 4.3.3 Differentiation of Causal Operators -- 4.4 Optimality Conditions -- 4.4.1 Subregular Recourse.4.4.2 Optimality Conditions for Problems with Built-In Nonanticipativity -- 4.4.3 Optimality Conditions for Problems with Nonanticipativity Constraints -- 4.4.4 Duality Associated with Nonanticipativity Constraints -- 4.4.5 Dynamic Programming Relations -- 4.5 Primal Decomposition Methods -- 4.5.1 The Value Function -- 4.5.2 The Risk-Averse Multi-Cut Method -- 4.5.3 Primal Multistage Decomposition -- 4.6 Scenario Decomposition -- 5 Optimization with Stochastic Dominance Constraints -- 5.1 Stochastic Orders -- 5.1.1 Definitions Based on Distribution Functions -- 5.1.2 Preference to Small Outcomes -- 5.1.3 Inverse Characterizations -- 5.1.4 Relations to the Expected Utility Theory -- 5.1.5 Relations to Distortions -- 5.1.6 Relations to Measures of Risk -- 5.1.7 Acceptance Sets -- 5.2 Optimization Problems and Optimality Conditions -- 5.2.1 The Convex Case -- 5.2.2 The Non-Convex Differentiable Case -- 5.3 Inverse Formulations -- 5.4 First-Order Stochastic Dominance Constraints -- 6 Multivariate and Sequential Stochastic Orders -- 6.1 Definition and Representations of Multivariate Orders -- 6.2 Optimality Conditions for Problems with Multivariate Stochastic Dominance Constraints -- 6.2.1 Optimality Conditions -- 6.2.2 Optimality Conditions in a Subgradient Form -- 6.2.3 Optimality Conditions for the Inverse Formulation -- 6.3 Sequential Orders and Time Consistency -- 6.4 Multistage Stochastic Optimization Problems with Sequential Stochastic Order Constraints -- 6.4.1 Optimality Conditions -- 6.4.2 Relations to Dynamic Risk Measures -- 7 Numerical Methods for Problems with Stochastic Dominance Constraints -- 7.1 Reformulations of the Second Order Dominance -- 7.2 Event Cut Methods -- 7.2.1 Shortfall Event Cut Method -- 7.2.2 Inverse Event Cut Method -- 7.3 Dual Methods -- 7.3.1 Progressive Bundle Method -- 7.3.2 Progressive Augmented Lagrangian Method.7.4 Exact Penalty Method -- 7.5 Methods for Problems with Multivariate DominanceConstraints -- 7.6 Two-Stage Problems with Stochastic Dominance Constraints -- 7.6.1 Decomposition Method with Inverse Event Cuts -- 7.6.2 Decomposition Method with Shortfall Event Cuts -- 8 Risk-Averse Control of Markov Systems -- 8.1 Finite Horizon Models -- 8.1.1 Risk Measures for Discrete-Time Processes -- 8.1.2 Risk Measures for Controlled Stochastic Processes -- 8.1.3 Markov Risk Measures -- 8.1.4 Dynamic Programming -- 8.1.5 Models with Random Costs -- 8.2 Infinite-Horizon Discounted Models -- 8.2.1 Discounted Measures of Risk -- 8.2.2 Dynamic Programming Equations. Value Iteration -- 8.2.3 Policy Iteration -- 8.2.4 Models with Transition-Dependent Costs -- 8.3 Total Risk Problems for Transient Models -- 8.3.1 Evaluation of a Stationary Markov Policy -- 8.3.2 Dynamic Programming Equations. Value Iteration -- 8.3.3 Policy Iteration -- 8.4 Partially Observable Systems -- 8.4.1 Partially Observable Markov Decision Processes -- 8.4.2 Markov Risk Measures -- 8.4.3 Dynamic Programming -- 8.5 Relation to Min-Max Markov Control Models -- Bibliographical Remarks -- References -- Index.Springer Series in Operations Research and Financial Engineering SeriesDentcheva Darinka731289Ruszczyński Andrzej731290MiAaPQMiAaPQMiAaPQ9910869179503321Risk-Averse Optimization and Control4170556UNINA