Many machine learning and optimization algorithms can be cast as instances of stochastic approximation (SA). The convergence rate of these algorithms is known to be slow, with the optimal mean squared error (MSE) of order $O(n^{-1})$. In prior work it was shown that MSE bounds approaching $O(n^{-4})$ can be achieved through the framework of quasi-stochastic approximation (QSA); essentially SA with careful choice of deterministic exploration. These results are extended to two time-scale algorithms, as found in policy gradient methods of reinforcement learning and extremum seeking control. The extensions are made possible in part by a new approach to analysis, allowing for the interpretation of two timescale algorithms as instances of single timescale QSA, made possible by the theory of negative Lyapunov exponents for QSA. The general theory is illustrated with applications to extremum seeking control (ESC).
{"title":"Markovian Foundations for Quasi-Stochastic Approximation in Two Timescales: Extended Version","authors":"Caio Kalil Lauand, Sean Meyn","doi":"arxiv-2409.07842","DOIUrl":"https://doi.org/arxiv-2409.07842","url":null,"abstract":"Many machine learning and optimization algorithms can be cast as instances of\u0000stochastic approximation (SA). The convergence rate of these algorithms is\u0000known to be slow, with the optimal mean squared error (MSE) of order\u0000$O(n^{-1})$. In prior work it was shown that MSE bounds approaching $O(n^{-4})$\u0000can be achieved through the framework of quasi-stochastic approximation (QSA);\u0000essentially SA with careful choice of deterministic exploration. These results\u0000are extended to two time-scale algorithms, as found in policy gradient methods\u0000of reinforcement learning and extremum seeking control. The extensions are made\u0000possible in part by a new approach to analysis, allowing for the interpretation\u0000of two timescale algorithms as instances of single timescale QSA, made possible\u0000by the theory of negative Lyapunov exponents for QSA. The general theory is\u0000illustrated with applications to extremum seeking control (ESC).","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paul Häusner, Aleix Nieto Juscafresa, Jens Sjölund
In this paper, we develop a data-driven approach to generate incomplete LU factorizations of large-scale sparse matrices. The learned approximate factorization is utilized as a preconditioner for the corresponding linear equation system in the GMRES method. Incomplete factorization methods are one of the most commonly applied algebraic preconditioners for sparse linear equation systems and are able to speed up the convergence of Krylov subspace methods. However, they are sensitive to hyper-parameters and might suffer from numerical breakdown or lead to slow convergence when not properly applied. We replace the typically hand-engineered algorithms with a graph neural network based approach that is trained against data to predict an approximate factorization. This allows us to learn preconditioners tailored for a specific problem distribution. We analyze and empirically evaluate different loss functions to train the learned preconditioners and show their effectiveness to decrease the number of GMRES iterations and improve the spectral properties on our synthetic dataset. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.
在本文中，我们开发了一种数据驱动方法，用于生成大规模稀疏矩阵的不完整 LU 因子化。学习到的近似因式分解被用作 GMRES 方法中相应线性方程组系统的预处理。不完全因子化方法是稀疏线性方程组最常用的代数预处理方法之一，能够加快 Krylov 子空间方法的收敛速度。然而，它们对超参数很敏感，如果应用不当，可能会出现数值崩溃或导致收敛速度缓慢。我们用基于图神经网络的方法取代了传统的手工设计算法，这种方法通过数据训练来预测近似因式分解。这样，我们就能学习为特定问题分布量身定制的预处理器。我们分析和实证评估了不同的损失函数，以训练学习到的预处理器，并在我们的合成数据集上展示了它们在减少 GMRES 迭代次数和改善频谱特性方面的有效性。代码可在https://github.com/paulhausner/neural-incomplete-factorization。
{"title":"Learning incomplete factorization preconditioners for GMRES","authors":"Paul Häusner, Aleix Nieto Juscafresa, Jens Sjölund","doi":"arxiv-2409.08262","DOIUrl":"https://doi.org/arxiv-2409.08262","url":null,"abstract":"In this paper, we develop a data-driven approach to generate incomplete LU\u0000factorizations of large-scale sparse matrices. The learned approximate\u0000factorization is utilized as a preconditioner for the corresponding linear\u0000equation system in the GMRES method. Incomplete factorization methods are one\u0000of the most commonly applied algebraic preconditioners for sparse linear\u0000equation systems and are able to speed up the convergence of Krylov subspace\u0000methods. However, they are sensitive to hyper-parameters and might suffer from\u0000numerical breakdown or lead to slow convergence when not properly applied. We\u0000replace the typically hand-engineered algorithms with a graph neural network\u0000based approach that is trained against data to predict an approximate\u0000factorization. This allows us to learn preconditioners tailored for a specific\u0000problem distribution. We analyze and empirically evaluate different loss\u0000functions to train the learned preconditioners and show their effectiveness to\u0000decrease the number of GMRES iterations and improve the spectral properties on\u0000our synthetic dataset. The code is available at\u0000https://github.com/paulhausner/neural-incomplete-factorization.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Obtaining the solution of constrained optimization problems as a function of parameters is very important in a multitude of applications, such as control and planning. Solving such parametric optimization problems in real time can present significant challenges, particularly when it is necessary to obtain highly accurate solutions or batches of solutions. To solve these challenges, we propose a learning-based iterative solver for constrained optimization which can obtain very fast and accurate solutions by customizing the solver to a specific parametric optimization problem. For a given set of parameters of the constrained optimization problem, we propose a first step with a neural network predictor that outputs primal-dual solutions of a reasonable degree of accuracy. This primal-dual solution is then improved to a very high degree of accuracy in a second step by a learned iterative solver in the form of a neural network. A novel loss function based on the Karush-Kuhn-Tucker conditions of optimality is introduced, enabling fully self-supervised training of both neural networks without the necessity of prior sampling of optimizer solutions. The evaluation of a variety of quadratic and nonlinear parametric test problems demonstrates that the predictor alone is already competitive with recent self-supervised schemes for approximating optimal solutions. The second step of our proposed learning-based iterative constrained optimizer achieves solutions with orders of magnitude better accuracy than other learning-based approaches, while being faster to evaluate than state-of-the-art solvers and natively allowing for GPU parallelization.
{"title":"Self-Supervised Learning of Iterative Solvers for Constrained Optimization","authors":"Lukas Lüken, Sergio Lucia","doi":"arxiv-2409.08066","DOIUrl":"https://doi.org/arxiv-2409.08066","url":null,"abstract":"Obtaining the solution of constrained optimization problems as a function of\u0000parameters is very important in a multitude of applications, such as control\u0000and planning. Solving such parametric optimization problems in real time can\u0000present significant challenges, particularly when it is necessary to obtain\u0000highly accurate solutions or batches of solutions. To solve these challenges,\u0000we propose a learning-based iterative solver for constrained optimization which\u0000can obtain very fast and accurate solutions by customizing the solver to a\u0000specific parametric optimization problem. For a given set of parameters of the\u0000constrained optimization problem, we propose a first step with a neural network\u0000predictor that outputs primal-dual solutions of a reasonable degree of\u0000accuracy. This primal-dual solution is then improved to a very high degree of\u0000accuracy in a second step by a learned iterative solver in the form of a neural\u0000network. A novel loss function based on the Karush-Kuhn-Tucker conditions of\u0000optimality is introduced, enabling fully self-supervised training of both\u0000neural networks without the necessity of prior sampling of optimizer solutions.\u0000The evaluation of a variety of quadratic and nonlinear parametric test problems\u0000demonstrates that the predictor alone is already competitive with recent\u0000self-supervised schemes for approximating optimal solutions. The second step of\u0000our proposed learning-based iterative constrained optimizer achieves solutions\u0000with orders of magnitude better accuracy than other learning-based approaches,\u0000while being faster to evaluate than state-of-the-art solvers and natively\u0000allowing for GPU parallelization.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wanru Chen, Rolf N. van Lieshout, Dezhi Zhang, Tom Van Woensel
This paper addresses a multi-period line planning problem in an integrated passenger-freight railway system, aiming to maximize profit while serving passengers and freight using a combination of dedicated passenger trains, dedicated freight trains, and mixed trains. To accommodate demand with different time sensitivities, we develop a period-extended change&go-network that tracks the paths taken by passengers and freight. The problem is formulated as a path-based mixed integer programming model, with the linear relaxation solved using column generation. Paths for passengers and freight are dynamically generated by solving pricing problems defined as elementary shortest-path problems with duration constraints. We propose two heuristic approaches: price-and-branch and a diving heuristic, with acceleration strategies, to find integer feasible solutions efficiently. Computational experiments on the Chinese high-speed railway network demonstrate that the diving heuristic outperforms the price-and-branch heuristic in both computational time and solution quality. Additionally, the experiments highlight the benefits of integrating freight, the advantages of multi-period line planning, and the impact of different demand patterns on line operations.
{"title":"Multi-period railway line planning for integrated passenger-freight transportation","authors":"Wanru Chen, Rolf N. van Lieshout, Dezhi Zhang, Tom Van Woensel","doi":"arxiv-2409.08256","DOIUrl":"https://doi.org/arxiv-2409.08256","url":null,"abstract":"This paper addresses a multi-period line planning problem in an integrated\u0000passenger-freight railway system, aiming to maximize profit while serving\u0000passengers and freight using a combination of dedicated passenger trains,\u0000dedicated freight trains, and mixed trains. To accommodate demand with\u0000different time sensitivities, we develop a period-extended change&go-network\u0000that tracks the paths taken by passengers and freight. The problem is\u0000formulated as a path-based mixed integer programming model, with the linear\u0000relaxation solved using column generation. Paths for passengers and freight are\u0000dynamically generated by solving pricing problems defined as elementary\u0000shortest-path problems with duration constraints. We propose two heuristic\u0000approaches: price-and-branch and a diving heuristic, with acceleration\u0000strategies, to find integer feasible solutions efficiently. Computational\u0000experiments on the Chinese high-speed railway network demonstrate that the\u0000diving heuristic outperforms the price-and-branch heuristic in both\u0000computational time and solution quality. Additionally, the experiments\u0000highlight the benefits of integrating freight, the advantages of multi-period\u0000line planning, and the impact of different demand patterns on line operations.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore intertemporal preferences that are recursive and account for local intertemporal substitution. First, we establish a rigorous foundation for these preferences and analyze their properties. Next, we examine the associated optimal consumption problem, proving the existence and uniqueness of the optimal consumption plan. We present an infinite-dimensional version of the Kuhn-Tucker theorem, which provides the necessary and sufficient conditions for optimality. Additionally, we investigate quantitative properties and the construction of the optimal consumption plan. Finally, we offer a detailed description of the structure of optimal consumption within a geometric Poisson framework.
{"title":"Optimal Consumption for Recursive Preferences with Local Substitution under Risk","authors":"Hanwu Li, Frank Riedel","doi":"arxiv-2409.07799","DOIUrl":"https://doi.org/arxiv-2409.07799","url":null,"abstract":"We explore intertemporal preferences that are recursive and account for local\u0000intertemporal substitution. First, we establish a rigorous foundation for these\u0000preferences and analyze their properties. Next, we examine the associated\u0000optimal consumption problem, proving the existence and uniqueness of the\u0000optimal consumption plan. We present an infinite-dimensional version of the\u0000Kuhn-Tucker theorem, which provides the necessary and sufficient conditions for\u0000optimality. Additionally, we investigate quantitative properties and the\u0000construction of the optimal consumption plan. Finally, we offer a detailed\u0000description of the structure of optimal consumption within a geometric Poisson\u0000framework.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a regular mean field game (MFG), the agents are assumed to be insignificant, they do not realize their effect on the population level and this may result in a phenomenon coined as the Tragedy of the Commons by the economists. However, in real life this phenomenon is often avoided thanks to the underlying altruistic behavior of (all or some of the) agents. Motivated by this observation, we introduce and analyze two different mean field models to include altruism in the decision making of agents. In the first model, mixed individual MFGs, there are infinitely many agents who are partially altruistic (i.e., they behave partially cooperatively) and partially non-cooperative. In the second model, mixed population MFGs, one part of the population behaves cooperatively and the remaining agents behave non-cooperatively. Both models are introduced in a general linear quadratic framework for which we characterize the equilibrium via forward backward stochastic differential equations. Furthermore, we give explicit solutions in terms of ordinary differential equations, and prove the existence and uniqueness results.
{"title":"How can the tragedy of the commons be prevented?: Introducing Linear Quadratic Mixed Mean Field Games","authors":"Gokce Dayanikli, Mathieu Lauriere","doi":"arxiv-2409.08235","DOIUrl":"https://doi.org/arxiv-2409.08235","url":null,"abstract":"In a regular mean field game (MFG), the agents are assumed to be\u0000insignificant, they do not realize their effect on the population level and\u0000this may result in a phenomenon coined as the Tragedy of the Commons by the\u0000economists. However, in real life this phenomenon is often avoided thanks to\u0000the underlying altruistic behavior of (all or some of the) agents. Motivated by\u0000this observation, we introduce and analyze two different mean field models to\u0000include altruism in the decision making of agents. In the first model, mixed\u0000individual MFGs, there are infinitely many agents who are partially altruistic\u0000(i.e., they behave partially cooperatively) and partially non-cooperative. In\u0000the second model, mixed population MFGs, one part of the population behaves\u0000cooperatively and the remaining agents behave non-cooperatively. Both models\u0000are introduced in a general linear quadratic framework for which we\u0000characterize the equilibrium via forward backward stochastic differential\u0000equations. Furthermore, we give explicit solutions in terms of ordinary\u0000differential equations, and prove the existence and uniqueness results.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take ``infinite values''.
{"title":"Duality theory in linear optimization and its extensions -- formally verified","authors":"Martin Dvorak, Vladimir Kolmogorov","doi":"arxiv-2409.08119","DOIUrl":"https://doi.org/arxiv-2409.08119","url":null,"abstract":"Farkas established that a system of linear inequalities has a solution if and\u0000only if we cannot obtain a contradiction by taking a linear combination of the\u0000inequalities. We state and formally prove several Farkas-like theorems over\u0000linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the\u0000case when some coefficients are allowed to take ``infinite values''.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multi-time-scale stochastic approximation is an iterative algorithm for finding the fixed point of a set of $N$ coupled operators given their noisy samples. It has been observed that due to the coupling between the decision variables and noisy samples of the operators, the performance of this method decays as $N$ increases. In this work, we develop a new accelerated variant of multi-time-scale stochastic approximation, which significantly improves the convergence rates of its standard counterpart. Our key idea is to introduce auxiliary variables to dynamically estimate the operators from their samples, which are then used to update the decision variables. These auxiliary variables help not only to control the variance of the operator estimates but also to decouple the sampling noise and the decision variables. This allows us to select more aggressive step sizes to achieve an optimal convergence rate. Specifically, under a strong monotonicity condition, we show that for any value of $N$ the $t^{text{th}}$ iterate of the proposed algorithm converges to the desired solution at a rate $widetilde{O}(1/t)$ when the operator samples are generated from a single from Markov process trajectory. A second contribution of this work is to demonstrate that the objective of a range of problems in reinforcement learning and multi-agent games can be expressed as a system of fixed-point equations. As such, the proposed approach can be used to design new learning algorithms for solving these problems. We illustrate this observation with numerical simulations in a multi-agent game and show the advantage of the proposed method over the standard multi-time-scale stochastic approximation algorithm.
{"title":"Accelerated Multi-Time-Scale Stochastic Approximation: Optimal Complexity and Applications in Reinforcement Learning and Multi-Agent Games","authors":"Sihan Zeng, Thinh T. Doan","doi":"arxiv-2409.07767","DOIUrl":"https://doi.org/arxiv-2409.07767","url":null,"abstract":"Multi-time-scale stochastic approximation is an iterative algorithm for\u0000finding the fixed point of a set of $N$ coupled operators given their noisy\u0000samples. It has been observed that due to the coupling between the decision\u0000variables and noisy samples of the operators, the performance of this method\u0000decays as $N$ increases. In this work, we develop a new accelerated variant of\u0000multi-time-scale stochastic approximation, which significantly improves the\u0000convergence rates of its standard counterpart. Our key idea is to introduce\u0000auxiliary variables to dynamically estimate the operators from their samples,\u0000which are then used to update the decision variables. These auxiliary variables\u0000help not only to control the variance of the operator estimates but also to\u0000decouple the sampling noise and the decision variables. This allows us to\u0000select more aggressive step sizes to achieve an optimal convergence rate.\u0000Specifically, under a strong monotonicity condition, we show that for any value\u0000of $N$ the $t^{text{th}}$ iterate of the proposed algorithm converges to the\u0000desired solution at a rate $widetilde{O}(1/t)$ when the operator samples are\u0000generated from a single from Markov process trajectory. A second contribution of this work is to demonstrate that the objective of a\u0000range of problems in reinforcement learning and multi-agent games can be\u0000expressed as a system of fixed-point equations. As such, the proposed approach\u0000can be used to design new learning algorithms for solving these problems. We\u0000illustrate this observation with numerical simulations in a multi-agent game\u0000and show the advantage of the proposed method over the standard\u0000multi-time-scale stochastic approximation algorithm.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marta Rodríguez Barreiro, María José Ginzo Villamayor, Fernando Pérez Porras, María Luisa Carpente Rodríguez, Silvia María Lorenzo Freire
During a wildfire, the work of the aerial coordinator is crucial for the control of the wildfire and the minimization of the burned area and the damage caused. Since it could be very useful for the coordinator to have decision-making tools at his/her disposal, this framework deals with an optimization model to obtain the optimal planning of firefighting helicopters, deciding the points where the aircraft should load water, the areas of the wildfire where they should work, and the rest bases to which each helicopter should be assigned. It was developed a Mixed Integer Linear Programming model which takes into account the configuration of helicopters, in closed circuits, as well as the flight aerial regulations in Spain. Due to the complexity of the model, two algorithms are developed, based on the Simulated Annealing and Iterated Local Search metaheuristic techniques. Both algorithms are tested with real data instances, obtaining very promising results for future application in the planning of aircraft throughout a wildfire evolution.
{"title":"On an optimization model for firefighting helicopter planning","authors":"Marta Rodríguez Barreiro, María José Ginzo Villamayor, Fernando Pérez Porras, María Luisa Carpente Rodríguez, Silvia María Lorenzo Freire","doi":"arxiv-2409.07937","DOIUrl":"https://doi.org/arxiv-2409.07937","url":null,"abstract":"During a wildfire, the work of the aerial coordinator is crucial for the\u0000control of the wildfire and the minimization of the burned area and the damage\u0000caused. Since it could be very useful for the coordinator to have\u0000decision-making tools at his/her disposal, this framework deals with an\u0000optimization model to obtain the optimal planning of firefighting helicopters,\u0000deciding the points where the aircraft should load water, the areas of the\u0000wildfire where they should work, and the rest bases to which each helicopter\u0000should be assigned. It was developed a Mixed Integer Linear Programming model\u0000which takes into account the configuration of helicopters, in closed circuits,\u0000as well as the flight aerial regulations in Spain. Due to the complexity of the\u0000model, two algorithms are developed, based on the Simulated Annealing and\u0000Iterated Local Search metaheuristic techniques. Both algorithms are tested with\u0000real data instances, obtaining very promising results for future application in\u0000the planning of aircraft throughout a wildfire evolution.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the study of nonsmooth optimal control problems (P) involving a controlled sweeping process with three main characteristics is launched. First, the sweeping sets C(t) are nonsmooth, unbounded, time-dependent, uniformly prox-regular, and satisfy minimal assumptions. Second, the sweeping process is coupled with a controlled differential equation. Third, joint-state endpoints constraint set S, including periodic conditions, is present. The existence and uniqueness of a Lipschitz solution for our dynamic is established, the existence of an optimal solution for our general form of optimal control is obtained, and the full form of the nonsmooth Pontryagin maximum principle for strong local minimizers in (P) is derived under minimal hypotheses. One of the novelties of this paper is the idea to work with a well-constructed problem corresponding to truncated sweeping sets and joint endpoint constraints that shares the same strong local minimizer as (P) and for which the exponential-penalty approximation technique can be developed using only the assumptions on (P).
{"title":"Optimal control for coupled sweeping processes under minimal assumptions","authors":"Samara Chamoun, Vera Zeidan","doi":"arxiv-2409.07722","DOIUrl":"https://doi.org/arxiv-2409.07722","url":null,"abstract":"In this paper, the study of nonsmooth optimal control problems (P) involving\u0000a controlled sweeping process with three main characteristics is launched.\u0000First, the sweeping sets C(t) are nonsmooth, unbounded, time-dependent,\u0000uniformly prox-regular, and satisfy minimal assumptions. Second, the sweeping\u0000process is coupled with a controlled differential equation. Third, joint-state\u0000endpoints constraint set S, including periodic conditions, is present. The\u0000existence and uniqueness of a Lipschitz solution for our dynamic is\u0000established, the existence of an optimal solution for our general form of\u0000optimal control is obtained, and the full form of the nonsmooth Pontryagin\u0000maximum principle for strong local minimizers in (P) is derived under minimal\u0000hypotheses. One of the novelties of this paper is the idea to work with a\u0000well-constructed problem corresponding to truncated sweeping sets and joint\u0000endpoint constraints that shares the same strong local minimizer as (P) and for\u0000which the exponential-penalty approximation technique can be developed using\u0000only the assumptions on (P).","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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