The equilibrium selection problem in the generalized Nash equilibrium problem (GNEP) has recently been studied as an optimization problem, defined over the set of all variational equilibria achievable first through a non-cooperative game among players. However, to make such a selection fairly for all players, we have to rely on an unrealistic assumption, that is, the availability of a reliable center not possible to cause any bias for all players. In this paper, we propose a new equilibrium selection achievable by solving a further GNEP, named the hierarchical Nash equilibrium problem (HNEP), within only the players. The HNEP covers existing optimization-based equilibrium selections as its simplest cases, while the general style of the HNEP can ensure a fair equilibrium selection without assuming any trusted center or randomness. We also propose an iterative algorithm for the HNEP as an application of the hybrid steepest descent method to a variational inequality newly defined over the fixed point set of a quasi-nonexpansive operator. Numerical experiments show the effectiveness of the proposed equilibrium selection via the HNEP.
{"title":"Hierarchical Nash Equilibrium over Variational Equilibria via Fixed-point Set Expression of Quasi-nonexpansive Operator","authors":"Shota Matsuo, Keita Kume, Isao Yamada","doi":"arxiv-2409.11094","DOIUrl":"https://doi.org/arxiv-2409.11094","url":null,"abstract":"The equilibrium selection problem in the generalized Nash equilibrium problem\u0000(GNEP) has recently been studied as an optimization problem, defined over the\u0000set of all variational equilibria achievable first through a non-cooperative\u0000game among players. However, to make such a selection fairly for all players,\u0000we have to rely on an unrealistic assumption, that is, the availability of a\u0000reliable center not possible to cause any bias for all players. In this paper,\u0000we propose a new equilibrium selection achievable by solving a further GNEP,\u0000named the hierarchical Nash equilibrium problem (HNEP), within only the\u0000players. The HNEP covers existing optimization-based equilibrium selections as\u0000its simplest cases, while the general style of the HNEP can ensure a fair\u0000equilibrium selection without assuming any trusted center or randomness. We\u0000also propose an iterative algorithm for the HNEP as an application of the\u0000hybrid steepest descent method to a variational inequality newly defined over\u0000the fixed point set of a quasi-nonexpansive operator. Numerical experiments\u0000show the effectiveness of the proposed equilibrium selection via the HNEP.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254633","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}
Juan Zhou, Kangkang Deng, Hongxia Wang, Zheng Peng
Gradient descent methods are fundamental first-order optimization algorithms in both Euclidean spaces and Riemannian manifolds. However, the exact gradient is not readily available in many scenarios. This paper proposes a novel inexact Riemannian gradient descent algorithm for nonconvex problems, accompanied by a convergence guarantee. In particular, we establish two inexact gradient conditions on Riemannian manifolds for the first time, enabling precise gradient approximations. Our method demonstrates strong convergence results for both gradient sequences and function values. The global convergence with constructive convergence rates for the sequence of iterates is ensured under the Riemannian Kurdyka-L ojasiewicz property. Furthermore, our algorithm encompasses two specific applications: Riemannian sharpness-aware minimization and Riemannian extragradient algorithm, both of which inherit the global convergence properties of the inexact gradient methods. Numerical experiments on low-rank matrix completion and principal component analysis problems validate the efficiency and practical relevance of the proposed approaches.
{"title":"Inexact Riemannian Gradient Descent Method for Nonconvex Optimization","authors":"Juan Zhou, Kangkang Deng, Hongxia Wang, Zheng Peng","doi":"arxiv-2409.11181","DOIUrl":"https://doi.org/arxiv-2409.11181","url":null,"abstract":"Gradient descent methods are fundamental first-order optimization algorithms\u0000in both Euclidean spaces and Riemannian manifolds. However, the exact gradient\u0000is not readily available in many scenarios. This paper proposes a novel inexact\u0000Riemannian gradient descent algorithm for nonconvex problems, accompanied by a\u0000convergence guarantee. In particular, we establish two inexact gradient\u0000conditions on Riemannian manifolds for the first time, enabling precise\u0000gradient approximations. Our method demonstrates strong convergence results for\u0000both gradient sequences and function values. The global convergence with\u0000constructive convergence rates for the sequence of iterates is ensured under\u0000the Riemannian Kurdyka-L ojasiewicz property. Furthermore, our algorithm\u0000encompasses two specific applications: Riemannian sharpness-aware minimization\u0000and Riemannian extragradient algorithm, both of which inherit the global\u0000convergence properties of the inexact gradient methods. Numerical experiments\u0000on low-rank matrix completion and principal component analysis problems\u0000validate the efficiency and practical relevance of the proposed approaches.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254612","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}
Present bias, the tendency to overvalue immediate rewards while undervaluing future ones, is a well-known barrier to achieving long-term goals. As artificial intelligence and behavioral economics increasingly focus on this phenomenon, the need for robust mathematical models to predict behavior and guide effective interventions has become crucial. However, existing models are constrained by their reliance on the discreteness of time and limited discount functions. This study introduces a novel continuous-time mathematical model for agents influenced by present bias. Using the variational principle, we model human behavior, where individuals repeatedly act according to a sequence of states that minimize their perceived cost. Our model not only retains analytical tractability but also accommodates various discount functions. Using this model, we consider intervention optimization problems under exponential and hyperbolic discounting and theoretically derive optimal intervention strategies, offering new insights into managing present-biased behavior.
{"title":"A Continuous-time Tractable Model for Present-biased Agents","authors":"Yasunori Akagi, Hideaki Kim, Takeshi Kurashima","doi":"arxiv-2409.11225","DOIUrl":"https://doi.org/arxiv-2409.11225","url":null,"abstract":"Present bias, the tendency to overvalue immediate rewards while undervaluing\u0000future ones, is a well-known barrier to achieving long-term goals. As\u0000artificial intelligence and behavioral economics increasingly focus on this\u0000phenomenon, the need for robust mathematical models to predict behavior and\u0000guide effective interventions has become crucial. However, existing models are\u0000constrained by their reliance on the discreteness of time and limited discount\u0000functions. This study introduces a novel continuous-time mathematical model for\u0000agents influenced by present bias. Using the variational principle, we model\u0000human behavior, where individuals repeatedly act according to a sequence of\u0000states that minimize their perceived cost. Our model not only retains\u0000analytical tractability but also accommodates various discount functions. Using\u0000this model, we consider intervention optimization problems under exponential\u0000and hyperbolic discounting and theoretically derive optimal intervention\u0000strategies, offering new insights into managing present-biased behavior.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254486","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 work, we propose novel LMI-based controller synthesis frameworks for periodically time-varying Markov-jump linear systems. We first discuss the necessary conditions for mean square stability and derive Lyapunov-like conditions for stability assurance. To relax strict stability requirements, we introduce a new criterion that doesn't require the Lyapunov function to decrease at each time step. Further, we incorporate these stability theorems in LMI-based controller synthesis frameworks while considering two separate problems: minimizing a quadratic cost, and maximizing the region of attraction. Numerical simulations verify the controllers' stability and showcase its applicability to fault-tolerant control.
{"title":"Robust Controller Synthesis under Markovian Mode Switching with Periodic LTV Dynamics","authors":"Shaurya Shrivastava, Kenshiro Oguri","doi":"arxiv-2409.11537","DOIUrl":"https://doi.org/arxiv-2409.11537","url":null,"abstract":"In this work, we propose novel LMI-based controller synthesis frameworks for\u0000periodically time-varying Markov-jump linear systems. We first discuss the\u0000necessary conditions for mean square stability and derive Lyapunov-like\u0000conditions for stability assurance. To relax strict stability requirements, we\u0000introduce a new criterion that doesn't require the Lyapunov function to\u0000decrease at each time step. Further, we incorporate these stability theorems in\u0000LMI-based controller synthesis frameworks while considering two separate\u0000problems: minimizing a quadratic cost, and maximizing the region of attraction.\u0000Numerical simulations verify the controllers' stability and showcase its\u0000applicability to fault-tolerant control.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254605","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 introduce the nested $p$-center problem, which is a multi-period variant of the well-known $p$-center problem. The use of the nesting concept allows to obtain solutions, which are consistent over the considered time horizon, i.e., facilities which are opened in a given time period stay open for subsequent time periods. This is important in real-life applications, as closing (and potential later re-opening) of facilities between time periods can be undesirable. We consider two different versions of our problem, with the difference being the objective function. The first version considers the sum of the absolute regrets (of nesting) over all time periods, and the second version considers minimizing the maximum relative regret over the time periods. We present three mixed-integer programming formulations for the version with absolute regret objective and two formulations for the version with relative regret objective. For all the formulations, we present valid inequalities. Based on the formulations and the valid inequalities, we develop branch-and-bound/branch-and-cut solution algorithms. These algorithms include a preprocessing procedure that exploits the nesting property and also begins heuristics and primal heuristics. We conducted a computational study on instances from the literature for the $p$-center problem, which we adapted to our problems. We also analyse the effect of nesting on the solution cost and the number of open facilities.
{"title":"Mixed-integer linear programming approaches for nested $p$-center problems with absolute and relative regret objectives","authors":"Christof Brandstetter, Markus Sinnl","doi":"arxiv-2409.11346","DOIUrl":"https://doi.org/arxiv-2409.11346","url":null,"abstract":"We introduce the nested $p$-center problem, which is a multi-period variant\u0000of the well-known $p$-center problem. The use of the nesting concept allows to\u0000obtain solutions, which are consistent over the considered time horizon, i.e.,\u0000facilities which are opened in a given time period stay open for subsequent\u0000time periods. This is important in real-life applications, as closing (and\u0000potential later re-opening) of facilities between time periods can be\u0000undesirable. We consider two different versions of our problem, with the difference being\u0000the objective function. The first version considers the sum of the absolute\u0000regrets (of nesting) over all time periods, and the second version considers\u0000minimizing the maximum relative regret over the time periods. We present three mixed-integer programming formulations for the version with\u0000absolute regret objective and two formulations for the version with relative\u0000regret objective. For all the formulations, we present valid inequalities.\u0000Based on the formulations and the valid inequalities, we develop\u0000branch-and-bound/branch-and-cut solution algorithms. These algorithms include a\u0000preprocessing procedure that exploits the nesting property and also begins\u0000heuristics and primal heuristics. We conducted a computational study on instances from the literature for the\u0000$p$-center problem, which we adapted to our problems. We also analyse the\u0000effect of nesting on the solution cost and the number of open facilities.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254608","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}
Erik Troedsson, Daniel Falkowski, Carl-Fredrik Lidgren, Herwig Wendt, Marcus Carlsson
The problem of approximate joint diagonalization of a collection of matrices arises in a number of diverse engineering and signal processing problems. This problem is usually cast as an optimization problem, and it is the main goal of this publication to provide a theoretical study of the corresponding cost-functional. As our main result, we prove that this functional tends to infinity in the vicinity of rank-deficient matrices with probability one, thereby proving that the optimization problem is well posed. Secondly, we provide unified expressions for its higher-order derivatives in multilinear form, and explicit expressions for the gradient and the Hessian of the functional in standard form, thereby opening for new improved numerical schemes for the solution of the joint diagonalization problem. A special section is devoted to the important case of self-adjoint matrices.
{"title":"On joint eigen-decomposition of matrices","authors":"Erik Troedsson, Daniel Falkowski, Carl-Fredrik Lidgren, Herwig Wendt, Marcus Carlsson","doi":"arxiv-2409.10292","DOIUrl":"https://doi.org/arxiv-2409.10292","url":null,"abstract":"The problem of approximate joint diagonalization of a collection of matrices\u0000arises in a number of diverse engineering and signal processing problems. This\u0000problem is usually cast as an optimization problem, and it is the main goal of\u0000this publication to provide a theoretical study of the corresponding\u0000cost-functional. As our main result, we prove that this functional tends to\u0000infinity in the vicinity of rank-deficient matrices with probability one,\u0000thereby proving that the optimization problem is well posed. Secondly, we\u0000provide unified expressions for its higher-order derivatives in multilinear\u0000form, and explicit expressions for the gradient and the Hessian of the\u0000functional in standard form, thereby opening for new improved numerical schemes\u0000for the solution of the joint diagonalization problem. A special section is\u0000devoted to the important case of self-adjoint matrices.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254642","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}
Deepak Prakash Kumar, Swaroop Darbha, Satyanarayana Gupta Manyam, David Casbeer
In this paper, motion planning for a Dubins vehicle on a unit sphere to attain a desired final location is considered. The radius of the Dubins path on the sphere is lower bounded by $r$. In a previous study, this problem was addressed, wherein it was shown that the optimal path is of type $CG, CC,$ or a degenerate path of the same for $r leq frac{1}{2}.$ Here, $C = L, R$ denotes an arc of a tight left or right turn of minimum turning radius $r,$ and $G$ denotes an arc of a great circle. In this study, the candidate paths for the same problem are generalized to model vehicles with a larger turning radius. In particular, it is shown that the candidate optimal paths are of type $CG, CC,$ or a degenerate path of the same for $r leq frac{sqrt{3}}{2}.$ Noting that at most two $LG$ paths and two $RG$ paths can exist for a given final location, this article further reduces the candidate optimal paths by showing that only one $LG$ and one $RG$ path can be optimal, yielding a total of seven candidate paths for $r leq frac{sqrt{3}}{2}.$ Additional conditions for the optimality of $CC$ paths are also derived in this study.
{"title":"Generalization of Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation","authors":"Deepak Prakash Kumar, Swaroop Darbha, Satyanarayana Gupta Manyam, David Casbeer","doi":"arxiv-2409.09954","DOIUrl":"https://doi.org/arxiv-2409.09954","url":null,"abstract":"In this paper, motion planning for a Dubins vehicle on a unit sphere to\u0000attain a desired final location is considered. The radius of the Dubins path on\u0000the sphere is lower bounded by $r$. In a previous study, this problem was\u0000addressed, wherein it was shown that the optimal path is of type $CG, CC,$ or a\u0000degenerate path of the same for $r leq frac{1}{2}.$ Here, $C = L, R$ denotes\u0000an arc of a tight left or right turn of minimum turning radius $r,$ and $G$\u0000denotes an arc of a great circle. In this study, the candidate paths for the\u0000same problem are generalized to model vehicles with a larger turning radius. In\u0000particular, it is shown that the candidate optimal paths are of type $CG, CC,$\u0000or a degenerate path of the same for $r leq frac{sqrt{3}}{2}.$ Noting that\u0000at most two $LG$ paths and two $RG$ paths can exist for a given final location,\u0000this article further reduces the candidate optimal paths by showing that only\u0000one $LG$ and one $RG$ path can be optimal, yielding a total of seven candidate\u0000paths for $r leq frac{sqrt{3}}{2}.$ Additional conditions for the optimality\u0000of $CC$ paths are also derived in this study.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254538","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}
Simon Thomä, Maximilian Schiffer, Wolfram Wiesemann
Multi-stage decision-making under uncertainty, where decisions are taken under sequentially revealing uncertain problem parameters, is often essential to faithfully model managerial problems. Given the significant computational challenges involved, these problems are typically solved approximately. This short note introduces an algorithmic framework that revisits a popular approximation scheme for multi-stage stochastic programs by Georghiou et al. (2015) and improves upon it to deliver superior policies in the stochastic setting, as well as extend its applicability to robust optimization and a contemporary Wasserstein-based data-driven setting. We demonstrate how the policies of our framework can be computed efficiently, and we present numerical experiments that highlight the benefits of our method.
{"title":"A Note on Piecewise Affine Decision Rules for Robust, Stochastic, and Data-Driven Optimization","authors":"Simon Thomä, Maximilian Schiffer, Wolfram Wiesemann","doi":"arxiv-2409.10295","DOIUrl":"https://doi.org/arxiv-2409.10295","url":null,"abstract":"Multi-stage decision-making under uncertainty, where decisions are taken\u0000under sequentially revealing uncertain problem parameters, is often essential\u0000to faithfully model managerial problems. Given the significant computational\u0000challenges involved, these problems are typically solved approximately. This\u0000short note introduces an algorithmic framework that revisits a popular\u0000approximation scheme for multi-stage stochastic programs by Georghiou et al.\u0000(2015) and improves upon it to deliver superior policies in the stochastic\u0000setting, as well as extend its applicability to robust optimization and a\u0000contemporary Wasserstein-based data-driven setting. We demonstrate how the\u0000policies of our framework can be computed efficiently, and we present numerical\u0000experiments that highlight the benefits of our method.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254495","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 propose a new relative-error inexact version of the alternating direction method of multipliers (ADMM) for convex optimization. We prove the asymptotic convergence of our main algorithm as well as pointwise and ergodic iteration-complexities for residuals. We also justify the effectiveness of the proposed algorithm through some preliminary numerical experiments on regression problems.
{"title":"A relative-error inexact ADMM splitting algorithm for convex optimization with inertial effects","authors":"M. Marques Alves, M. Geremia","doi":"arxiv-2409.10311","DOIUrl":"https://doi.org/arxiv-2409.10311","url":null,"abstract":"We propose a new relative-error inexact version of the alternating direction\u0000method of multipliers (ADMM) for convex optimization. We prove the asymptotic\u0000convergence of our main algorithm as well as pointwise and ergodic\u0000iteration-complexities for residuals. We also justify the effectiveness of the\u0000proposed algorithm through some preliminary numerical experiments on regression\u0000problems.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254496","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}
Deepak Prakash Kumar, Swaroop Darbha, Satyanarayana Gupta Manyam, Dzung Tran, David W. Casbeer
In this paper, motion planning for a vehicle moving on a unit sphere with unit speed is considered, wherein the desired terminal location is fixed, but the terminal orientation is free. The motion of the vehicle is modeled to be constrained by a maximum geodesic curvature $U_{max},$ which controls the rate of change of heading of the vehicle such that the maximum heading change occurs when the vehicle travels on a tight circular arc of radius $r = frac{1}{sqrt{1 + U_{max}^2}}$. Using Pontryagin's Minimum Principle, the main result of this paper shows that for $r leq frac{1}{2}$, the optimal path connecting a given initial configuration and a final location on the sphere belongs to a set of at most seven paths. The candidate paths are of type $CG, CC,$ and degenerate paths of the same, where $C in {L, R}$ denotes a tight left or right turn, respectively, and $G$ denotes a great circular arc.
{"title":"Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation","authors":"Deepak Prakash Kumar, Swaroop Darbha, Satyanarayana Gupta Manyam, Dzung Tran, David W. Casbeer","doi":"arxiv-2409.10363","DOIUrl":"https://doi.org/arxiv-2409.10363","url":null,"abstract":"In this paper, motion planning for a vehicle moving on a unit sphere with\u0000unit speed is considered, wherein the desired terminal location is fixed, but\u0000the terminal orientation is free. The motion of the vehicle is modeled to be\u0000constrained by a maximum geodesic curvature $U_{max},$ which controls the rate\u0000of change of heading of the vehicle such that the maximum heading change occurs\u0000when the vehicle travels on a tight circular arc of radius $r =\u0000frac{1}{sqrt{1 + U_{max}^2}}$. Using Pontryagin's Minimum Principle, the main\u0000result of this paper shows that for $r leq frac{1}{2}$, the optimal path\u0000connecting a given initial configuration and a final location on the sphere\u0000belongs to a set of at most seven paths. The candidate paths are of type $CG,\u0000CC,$ and degenerate paths of the same, where $C in {L, R}$ denotes a tight\u0000left or right turn, respectively, and $G$ denotes a great circular arc.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254493","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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