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Nonlinear Monte Carlo Methods with Polynomial Runtime for Bellman Equations of Discrete Time High-Dimensional Stochastic Optimal Control Problems
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-04 DOI: 10.1007/s00245-024-10213-7
Christian Beck, Arnulf Jentzen, Konrad Kleinberg, Thomas Kruse
{"title":"Nonlinear Monte Carlo Methods with Polynomial Runtime for Bellman Equations of Discrete Time High-Dimensional Stochastic Optimal Control Problems","authors":"Christian Beck,&nbsp;Arnulf Jentzen,&nbsp;Konrad Kleinberg,&nbsp;Thomas Kruse","doi":"10.1007/s00245-024-10213-7","DOIUrl":"10.1007/s00245-024-10213-7","url":null,"abstract":"<div><p>Discrete time <i>stochastic optimal control</i> problems and <i>Markov decision processes</i> (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical foundation of <i>reinforcement learning</i>. In this article we study the numerical approximation of MDPs with infinite time horizon, finite control set, and general state spaces. Our set-up in particular covers infinite-horizon optimal stopping problems of discrete time Markov processes. A key tool to solve MDPs are <i>Bellman equations</i> which characterize the value functions of the MDPs and determine the optimal control strategies. By combining ideas from the <i>full-history recursive multilevel Picard approximation method</i>, which was recently introduced to solve certain nonlinear partial differential equations, and ideas from <i>Q</i><i>-learning</i> we introduce a class of suitable <i>nonlinear Monte Carlo methods</i> and prove that the proposed methods do not suffer from the <i>curse of dimensionality</i> in the numerical approximation of the solutions of Bellman equations and the associated discrete time stochastic optimal control problems.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10213-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Feedback Stabilization of Convective Brinkman-Forchheimer Extended Darcy Equations
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-31 DOI: 10.1007/s00245-024-10217-3
Sagar Gautam, Kush Kinra, Manil T. Mohan
{"title":"Feedback Stabilization of Convective Brinkman-Forchheimer Extended Darcy Equations","authors":"Sagar Gautam,&nbsp;Kush Kinra,&nbsp;Manil T. Mohan","doi":"10.1007/s00245-024-10217-3","DOIUrl":"10.1007/s00245-024-10217-3","url":null,"abstract":"<div><p>In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a <i>d</i>-dimensional torus: </p><div><div><span>$$begin{aligned} frac{partial {varvec{y}}}{partial t}-mu Delta {varvec{y}}+({varvec{y}}cdot nabla ){varvec{y}}+alpha {varvec{y}}+beta vert {varvec{y}}vert ^{r-1}{varvec{y}}+gamma vert {varvec{y}}vert ^{q-1}{varvec{y}}+nabla p={varvec{g}}+{varvec{u}}, nabla cdot {varvec{y}}=0, end{aligned}$$</span></div></div><p>where <span>(din {2,3})</span>, <span>(mu ,alpha ,beta &gt;0)</span>, <span>(gamma in {mathbb {R}})</span>, <span>(r,qin [1,infty ))</span> with <span>(r&gt;qge 1)</span>. We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of <i>m</i>-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143110089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Steady Navier–Stokes Equations with Regularized Directional Do-Nothing Boundary Condition: Optimal Boundary Control for a Velocity Tracking Problem
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-27 DOI: 10.1007/s00245-024-10216-4
Pedro Nogueira, Ana L. Silvestre, Jorge Tiago
{"title":"Steady Navier–Stokes Equations with Regularized Directional Do-Nothing Boundary Condition: Optimal Boundary Control for a Velocity Tracking Problem","authors":"Pedro Nogueira,&nbsp;Ana L. Silvestre,&nbsp;Jorge Tiago","doi":"10.1007/s00245-024-10216-4","DOIUrl":"10.1007/s00245-024-10216-4","url":null,"abstract":"<div><p>We consider the steady Navier–Stokes equations with mixed boundary conditions, where a regularized directional do-nothing (RDDN) condition is defined on the Neumann boundary portion. An auxiliary Stokes reference flow, which also works as a lifting of the inhomogeneous Dirichlet boundary values, is used to define the RDDN condition. Our aim is to study the minimization of a velocity tracking cost functional with controls localized on a part of the boundary. We prove the existence of a solution for this optimal control problem and derive the corresponding first order necessary optimality conditions in terms of dual variables. All results are obtained under appropriate assumptions on the size of the data and the controls, which, however, are less restrictive when compared with the case of the classical do-nothing outflow condition. This is further confirmed by the numerical examples presented, which include scenarios where only noisy data is available.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10216-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Viscosity Solutions of a Class of Second Order Hamilton–Jacobi–Bellman Equations in the Wasserstein Space
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-27 DOI: 10.1007/s00245-025-10219-9
Hang Cheung, Ho Man Tai, Jinniao Qiu
{"title":"Viscosity Solutions of a Class of Second Order Hamilton–Jacobi–Bellman Equations in the Wasserstein Space","authors":"Hang Cheung,&nbsp;Ho Man Tai,&nbsp;Jinniao Qiu","doi":"10.1007/s00245-025-10219-9","DOIUrl":"10.1007/s00245-025-10219-9","url":null,"abstract":"<div><p>This paper is devoted to solving a class of second order Hamilton–Jacobi–Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise. The well-posedness of viscosity solution to the HJB equation under a new notion is established under general assumptions on the coefficients. Our approach adopts the smooth metric developed by Bayraktar et al. (Proc Am Math Soc 151(09):4089–4098, 2023) as our gauge function for the purpose of smooth variational principle used in the proof of comparison theorem. Further estimates and regularity of the metric, including a novel second order derivative estimate with respect to the measure variable, are derived in order to ensure the uniqueness and existence.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Stationarity Conditions and Constraint Qualifications for Multiobjective Optimization Problems with Cardinality Constraints
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-23 DOI: 10.1007/s00245-025-10224-y
Rohollah Garmanjani, Evelin H. M. Krulikovski, Alberto Ramos
{"title":"On Stationarity Conditions and Constraint Qualifications for Multiobjective Optimization Problems with Cardinality Constraints","authors":"Rohollah Garmanjani,&nbsp;Evelin H. M. Krulikovski,&nbsp;Alberto Ramos","doi":"10.1007/s00245-025-10224-y","DOIUrl":"10.1007/s00245-025-10224-y","url":null,"abstract":"<div><p>The purpose of this paper is to develop Pareto optimality conditions and constraint qualifications (CQs) for Multiobjective Programs with Cardinality Constraints (MOPCaC). In general, such problems are difficult to solve, not only because they involve a cardinality constraint that is neither continuous nor convex, but also because there may be a potential conflict between the various objective functions. Thus, we reformulate the MOPCaC based on the problem with continuous variables, namely the relaxed problem. Furthermore, we consider different notions of optimality (weak/strong Pareto optimal solutions). Thereby, we define new stationarity conditions that extend the classical Karush-Kuhn-Tucker (KKT) conditions of the scalar case. Moreover, we also introduce new CQs, based on the recently defined multiobjective normal cone, to ensure compliance with such stationarity conditions. Important statements are illustrated by examples.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exact Controllability to Nonnegative Trajectory for a Chemotaxis System 趋化系统的非负轨迹精确可控性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-22 DOI: 10.1007/s00245-025-10221-1
Qiang Tao, Muming Zhang
{"title":"Exact Controllability to Nonnegative Trajectory for a Chemotaxis System","authors":"Qiang Tao,&nbsp;Muming Zhang","doi":"10.1007/s00245-025-10221-1","DOIUrl":"10.1007/s00245-025-10221-1","url":null,"abstract":"<div><p>This paper studies the controllability for a Keller–Segel type chemotaxis model with singular sensitivity. Based on the Hopf–Cole transformation, a nonlinear parabolic system, which has first-order couplings, and the coupling coefficients are functions that depend on both time and space variables, is derived. Then, the controllability result is proved by a new global Carleman estimate for general coupled parabolic equations allowed to contain a convective term. Also, the global existence of nonnegative solution for the chemotaxis system is discussed.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fractional, Semilinear, and Sparse Optimal Control: A Priori Error Bounds 分数、半线性和稀疏最优控制：先验误差界
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-22 DOI: 10.1007/s00245-024-10200-y
Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero
{"title":"Fractional, Semilinear, and Sparse Optimal Control: A Priori Error Bounds","authors":"Francisco Bersetche,&nbsp;Francisco Fuica,&nbsp;Enrique Otárola,&nbsp;Daniel Quero","doi":"10.1007/s00245-024-10200-y","DOIUrl":"10.1007/s00245-024-10200-y","url":null,"abstract":"<div><p>In this work, we use the integral definition of the fractional Laplace operator and study a sparse optimal control problem involving a fractional, semilinear, and elliptic partial differential equation as state equation; control constraints are also considered. We establish the existence of optimal solutions and first and second order optimality conditions. We also analyze regularity properties for optimal variables. We propose and analyze two finite element strategies of discretization: a fully discrete scheme, where the control variable is discretized with piecewise constant functions, and a semidiscrete scheme, where the control variable is not discretized. For both discretization schemes, we analyze convergence properties and a priori error bounds.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Relationship Between Viscosity and Distribution Solutions for Nonlinear Neumann Type PDEs: The Probabilistic Approach 非线性Neumann型偏微分方程黏度与分布解的关系：概率方法
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-18 DOI: 10.1007/s00245-025-10222-0
Jiagang Ren, Shoutian Wang, Jing Wu
{"title":"On the Relationship Between Viscosity and Distribution Solutions for Nonlinear Neumann Type PDEs: The Probabilistic Approach","authors":"Jiagang Ren,&nbsp;Shoutian Wang,&nbsp;Jing Wu","doi":"10.1007/s00245-025-10222-0","DOIUrl":"10.1007/s00245-025-10222-0","url":null,"abstract":"<div><p>Based on probabilistic methods, we discuss the relationship between viscosity and distribution solutions for semi-linear partial differential equations (PDEs) with Neumann boundary conditions. We also extend the research to a type of nonlinear PDEs, which is completed through the well-posedness and continuity results of solutions to the corresponding forward-backward SDE.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic Behavior of Rao–Nakra Sandwich Beam with Nonlinear Localized Damping and Source Terms 具有非线性局域阻尼和源项的Rao-Nakra夹层梁的渐近特性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-16 DOI: 10.1007/s00245-024-10218-2
Marcelo M. Cavalcanti, Baowei Feng, Victor Hugo Gonzalez Martinez, Sabeur Mansouri
{"title":"Asymptotic Behavior of Rao–Nakra Sandwich Beam with Nonlinear Localized Damping and Source Terms","authors":"Marcelo M. Cavalcanti,&nbsp;Baowei Feng,&nbsp;Victor Hugo Gonzalez Martinez,&nbsp;Sabeur Mansouri","doi":"10.1007/s00245-024-10218-2","DOIUrl":"10.1007/s00245-024-10218-2","url":null,"abstract":"<div><p>This paper is concerned with a semilinear Rao–Nakra sandwich beam under the action of three nonlinear localized frictional damping terms in which the core viscoelastic layer is constrained by the pure elasticity or piezoelectric outer layers. The main goal is to prove its asymptotic behavior by applying <i>minimal amount of support to the damping</i>. We firstly prove that the system is global well-posedness by the theory of monotone operators. For asymptotic behavior of solutions, we obtain uniform decay rate results of the system and the energy decay rates are determined by a nonlinear first-order ODE. The existence of a smooth global attractor with finite fractal dimension and generalized exponential attractors are finally obtained.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Strict Efficiency in Vector Optimization Via a Directional Curvature Functional 基于方向曲率泛函的矢量优化的严格效率
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-12 DOI: 10.1007/s00245-025-10220-2
José Cerda-Hernández, Alberto Ramos
{"title":"Strict Efficiency in Vector Optimization Via a Directional Curvature Functional","authors":"José Cerda-Hernández,&nbsp;Alberto Ramos","doi":"10.1007/s00245-025-10220-2","DOIUrl":"10.1007/s00245-025-10220-2","url":null,"abstract":"<div><p>We derive new necessary and sufficient conditions for strict efficiency in vector optimization problems for non-smooth mappings. Unlike other approaches, our conditions are described in terms of a suitable directional curvature functional that allows us to derive no-gap second-order optimality conditions in an abstract setting. Our approach allows us to apply our results even when classical assumptions such as the second-order regularity conditions to the feasible set fail, extending the applicability of our approach. As applications to mathematical programming, we provide new primal and dual Karush-Kuhn-Tucker (KKT) second-order necessary and sufficient conditions. We provide some examples to illustrate our findings.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142963090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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