Andrey Veprikov, Alexander Bogdanov, Vladislav Minashkin, Alexander Beznosikov
This paper deals with the black-box optimization problem. In this setup, we do not have access to the gradient of the objective function, therefore, we need to estimate it somehow. We propose a new type of approximation JAGUAR, that memorizes information from previous iterations and requires $mathcal{O}(1)$ oracle calls. We implement this approximation in the Frank-Wolfe and Gradient Descent algorithms and prove the convergence of these methods with different types of zero-order oracle. Our theoretical analysis covers scenarios of non-convex, convex and PL-condition cases. Also in this paper, we consider the stochastic minimization problem on the set $Q$ with noise in the zero-order oracle; this setup is quite unpopular in the literature, but we prove that the JAGUAR approximation is robust not only in deterministic minimization problems, but also in the stochastic case. We perform experiments to compare our gradient estimator with those already known in the literature and confirm the dominance of our methods.
{"title":"New Aspects of Black Box Conditional Gradient: Variance Reduction and One Point Feedback","authors":"Andrey Veprikov, Alexander Bogdanov, Vladislav Minashkin, Alexander Beznosikov","doi":"arxiv-2409.10442","DOIUrl":"https://doi.org/arxiv-2409.10442","url":null,"abstract":"This paper deals with the black-box optimization problem. In this setup, we\u0000do not have access to the gradient of the objective function, therefore, we\u0000need to estimate it somehow. We propose a new type of approximation JAGUAR,\u0000that memorizes information from previous iterations and requires\u0000$mathcal{O}(1)$ oracle calls. We implement this approximation in the\u0000Frank-Wolfe and Gradient Descent algorithms and prove the convergence of these\u0000methods with different types of zero-order oracle. Our theoretical analysis\u0000covers scenarios of non-convex, convex and PL-condition cases. Also in this\u0000paper, we consider the stochastic minimization problem on the set $Q$ with\u0000noise in the zero-order oracle; this setup is quite unpopular in the\u0000literature, but we prove that the JAGUAR approximation is robust not only in\u0000deterministic minimization problems, but also in the stochastic case. We\u0000perform experiments to compare our gradient estimator with those already known\u0000in the literature and confirm the dominance of our methods.","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":"142254492","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 study the oracle complexity of nonsmooth nonconvex optimization, with the algorithm assumed to have access only to local function information. It has been shown by Davis, Drusvyatskiy, and Jiang (2023) that for nonsmooth Lipschitz functions satisfying certain regularity and strictness conditions, perturbed gradient descent converges to local minimizers asymptotically. Motivated by this result and by other recent algorithmic advances in nonconvex nonsmooth optimization concerning Goldstein stationarity, we consider the question of obtaining a non-asymptotic rate of convergence to local minima for this problem class. We provide the following negative answer to this question: Local algorithms acting on regular Lipschitz functions cannot, in the worst case, provide meaningful local guarantees in terms of function value in sub-exponential time, even when all near-stationary points are global minima. This sharply contrasts with the smooth setting, for which it is well-known that standard gradient methods can do so in a dimension-independent rate. Our result complements the rich body of work in the theoretical computer science literature that provide hardness results conditional on conjectures such as $mathsf{P}neqmathsf{NP}$ or cryptographic assumptions, in that ours holds unconditional of any such assumptions.
{"title":"On the Hardness of Meaningful Local Guarantees in Nonsmooth Nonconvex Optimization","authors":"Guy Kornowski, Swati Padmanabhan, Ohad Shamir","doi":"arxiv-2409.10323","DOIUrl":"https://doi.org/arxiv-2409.10323","url":null,"abstract":"We study the oracle complexity of nonsmooth nonconvex optimization, with the\u0000algorithm assumed to have access only to local function information. It has\u0000been shown by Davis, Drusvyatskiy, and Jiang (2023) that for nonsmooth\u0000Lipschitz functions satisfying certain regularity and strictness conditions,\u0000perturbed gradient descent converges to local minimizers asymptotically.\u0000Motivated by this result and by other recent algorithmic advances in nonconvex\u0000nonsmooth optimization concerning Goldstein stationarity, we consider the\u0000question of obtaining a non-asymptotic rate of convergence to local minima for\u0000this problem class. We provide the following negative answer to this question: Local algorithms\u0000acting on regular Lipschitz functions cannot, in the worst case, provide\u0000meaningful local guarantees in terms of function value in sub-exponential time,\u0000even when all near-stationary points are global minima. This sharply contrasts\u0000with the smooth setting, for which it is well-known that standard gradient\u0000methods can do so in a dimension-independent rate. Our result complements the\u0000rich body of work in the theoretical computer science literature that provide\u0000hardness results conditional on conjectures such as $mathsf{P}neqmathsf{NP}$\u0000or cryptographic assumptions, in that ours holds unconditional of any such\u0000assumptions.","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":"142254543","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}
This paper is concerned with finite element error estimates for Neumann boundary control problems posed on convex and polyhedral domains. Different discretization concepts are considered and for each optimal discretization error estimates are established. In particular, for a full discretization with piecewise linear and globally continuous functions for the control and standard linear finite elements for the state optimal convergence rates for the controls are proven which solely depend on the largest interior edge angle. To be more precise, below the critical edge angle of $2pi/3$, a convergence rate of two (times a log-factor) can be achieved for the discrete controls in the $L^2$-norm on the boundary. For larger interior edge angles the convergence rates are reduced depending on their size, which is due the impact of singular (domain dependent) terms in the solution. The results are comparable to those for the two dimensional case. However, new techniques in this context are used to prove the estimates on the boundary which also extend to the two dimensional case. Moreover, it is shown that the discrete states converge with a rate of two in the $L^2$-norm in the domain independent of the interior edge angles, i.e. for any convex and polyhedral domain. It is remarkable that this not only holds for a full discretization using piecewise linear and globally continuous functions for the control, but also for a full discretization using piecewise constant functions for the control, where the discrete controls only converge with a rate of one in the $L^2$-norm on the boundary at best. At the end, the theoretical results are confirmed by several numerical experiments.
{"title":"Numerical Analysis for Neumann Optimal Control Problems on Convex Polyhedral Domains","authors":"Johannes Pfefferer, Boris Vexler","doi":"arxiv-2409.10736","DOIUrl":"https://doi.org/arxiv-2409.10736","url":null,"abstract":"This paper is concerned with finite element error estimates for Neumann\u0000boundary control problems posed on convex and polyhedral domains. Different\u0000discretization concepts are considered and for each optimal discretization\u0000error estimates are established. In particular, for a full discretization with\u0000piecewise linear and globally continuous functions for the control and standard\u0000linear finite elements for the state optimal convergence rates for the controls\u0000are proven which solely depend on the largest interior edge angle. To be more\u0000precise, below the critical edge angle of $2pi/3$, a convergence rate of two\u0000(times a log-factor) can be achieved for the discrete controls in the\u0000$L^2$-norm on the boundary. For larger interior edge angles the convergence\u0000rates are reduced depending on their size, which is due the impact of singular\u0000(domain dependent) terms in the solution. The results are comparable to those\u0000for the two dimensional case. However, new techniques in this context are used\u0000to prove the estimates on the boundary which also extend to the two dimensional\u0000case. Moreover, it is shown that the discrete states converge with a rate of\u0000two in the $L^2$-norm in the domain independent of the interior edge angles,\u0000i.e. for any convex and polyhedral domain. It is remarkable that this not only\u0000holds for a full discretization using piecewise linear and globally continuous\u0000functions for the control, but also for a full discretization using piecewise\u0000constant functions for the control, where the discrete controls only converge\u0000with a rate of one in the $L^2$-norm on the boundary at best. At the end, the\u0000theoretical results are confirmed by several numerical experiments.","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":"142254494","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}
Andrey Sadchikov, Savelii Chezhegov, Aleksandr Beznosikov, Alexander Gasnikov
Distributed optimization plays an important role in modern large-scale machine learning and data processing systems by optimizing the utilization of computational resources. One of the classical and popular approaches is Local Stochastic Gradient Descent (Local SGD), characterized by multiple local updates before averaging, which is particularly useful in distributed environments to reduce communication bottlenecks and improve scalability. A typical feature of this method is the dependence on the frequency of communications. But in the case of a quadratic target function with homogeneous data distribution over all devices, the influence of frequency of communications vanishes. As a natural consequence, subsequent studies include the assumption of a Lipschitz Hessian, as this indicates the similarity of the optimized function to a quadratic one to some extent. However, in order to extend the completeness of the Local SGD theory and unlock its potential, in this paper we abandon the Lipschitz Hessian assumption by introducing a new concept of $textit{approximate quadraticity}$. This assumption gives a new perspective on problems that have near quadratic properties. In addition, existing theoretical analyses of Local SGD often assume bounded variance. We, in turn, consider the unbounded noise condition, which allows us to broaden the class of studied problems.
{"title":"Local SGD for Near-Quadratic Problems: Improving Convergence under Unconstrained Noise Conditions","authors":"Andrey Sadchikov, Savelii Chezhegov, Aleksandr Beznosikov, Alexander Gasnikov","doi":"arxiv-2409.10478","DOIUrl":"https://doi.org/arxiv-2409.10478","url":null,"abstract":"Distributed optimization plays an important role in modern large-scale\u0000machine learning and data processing systems by optimizing the utilization of\u0000computational resources. One of the classical and popular approaches is Local\u0000Stochastic Gradient Descent (Local SGD), characterized by multiple local\u0000updates before averaging, which is particularly useful in distributed\u0000environments to reduce communication bottlenecks and improve scalability. A\u0000typical feature of this method is the dependence on the frequency of\u0000communications. But in the case of a quadratic target function with homogeneous\u0000data distribution over all devices, the influence of frequency of\u0000communications vanishes. As a natural consequence, subsequent studies include\u0000the assumption of a Lipschitz Hessian, as this indicates the similarity of the\u0000optimized function to a quadratic one to some extent. However, in order to\u0000extend the completeness of the Local SGD theory and unlock its potential, in\u0000this paper we abandon the Lipschitz Hessian assumption by introducing a new\u0000concept of $textit{approximate quadraticity}$. This assumption gives a new\u0000perspective on problems that have near quadratic properties. In addition,\u0000existing theoretical analyses of Local SGD often assume bounded variance. We,\u0000in turn, consider the unbounded noise condition, which allows us to broaden the\u0000class of studied problems.","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":"142254535","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}
This review surveys previous and recent results on null controllability and inverse problems for parabolic systems with dynamic boundary conditions. We aim to demonstrate how classical methods such as Carleman estimates can be extended to prove null controllability for parabolic systems and Lipschitz stability estimates for inverse problems with dynamic boundary conditions of surface diffusion type. We mainly focus on the substantial difficulties compared to static boundary conditions. Finally, some conclusions and open problems will be mentioned.
{"title":"Controllability and Inverse Problems for Parabolic Systems with Dynamic Boundary Conditions","authors":"S. E. Chorfi, L. Maniar","doi":"arxiv-2409.10302","DOIUrl":"https://doi.org/arxiv-2409.10302","url":null,"abstract":"This review surveys previous and recent results on null controllability and\u0000inverse problems for parabolic systems with dynamic boundary conditions. We aim\u0000to demonstrate how classical methods such as Carleman estimates can be extended\u0000to prove null controllability for parabolic systems and Lipschitz stability\u0000estimates for inverse problems with dynamic boundary conditions of surface\u0000diffusion type. We mainly focus on the substantial difficulties compared to\u0000static boundary conditions. Finally, some conclusions and open problems will be\u0000mentioned.","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":"142254497","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}
The data-driven techniques have been developed to deal with the output regulation problem of unknown linear systems by various approaches. In this paper, we first extend an existing algorithm from single-input single-output linear systems to multi-input multi-output linear systems. Then, by separating the dynamics used in the learning phase and the control phase, we further propose an improved algorithm that significantly reduces the computational cost and weakens the solvability conditions over the first algorithm.
{"title":"Data-Driven Output Regulation via Internal Model Principle","authors":"Liquan Lin, Jie Huang","doi":"arxiv-2409.09571","DOIUrl":"https://doi.org/arxiv-2409.09571","url":null,"abstract":"The data-driven techniques have been developed to deal with the output\u0000regulation problem of unknown linear systems by various approaches. In this\u0000paper, we first extend an existing algorithm from single-input single-output\u0000linear systems to multi-input multi-output linear systems. Then, by separating\u0000the dynamics used in the learning phase and the control phase, we further\u0000propose an improved algorithm that significantly reduces the computational cost\u0000and weakens the solvability conditions over the first algorithm.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254547","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}
The stable marriage problem with incomplete lists and ties (SMTI) and the hospitals/residents problem with ties (HRT) are important in matching theory with broad practical applications. In this paper, we introduce a tie-breaking based local search algorithm (TBLS) designed to achieve a weakly stable matching of maximum size for both the SMTI and HRT problems. TBLS begins by arbitrarily resolving all ties and iteratively refines the tie-breaking strategy by adjusting the relative order within ties based on preference ranks and the current stable matching. Additionally, we introduce TBLS-E, an equity-focused variant of TBLS, specifically designed for the SMTI problem. This variant maintains the objective of maximizing matching size, while enhancing equity through two simple modifications. In comparison with ten other approximation and local search algorithms, TBLS achieves the highest matching size, while TBLS-E exhibits the lowest sex equality cost. Significantly, TBLS-E preserves a matching size comparable to that of TBLS. Both our algorithms demonstrate faster computational speed than other local search algorithms in solving large-sized instances.
{"title":"A Tie-breaking based Local Search Algorithm for Stable Matching Problems","authors":"Junyuan Qiu","doi":"arxiv-2409.10575","DOIUrl":"https://doi.org/arxiv-2409.10575","url":null,"abstract":"The stable marriage problem with incomplete lists and ties (SMTI) and the\u0000hospitals/residents problem with ties (HRT) are important in matching theory\u0000with broad practical applications. In this paper, we introduce a tie-breaking\u0000based local search algorithm (TBLS) designed to achieve a weakly stable\u0000matching of maximum size for both the SMTI and HRT problems. TBLS begins by\u0000arbitrarily resolving all ties and iteratively refines the tie-breaking\u0000strategy by adjusting the relative order within ties based on preference ranks\u0000and the current stable matching. Additionally, we introduce TBLS-E, an\u0000equity-focused variant of TBLS, specifically designed for the SMTI problem.\u0000This variant maintains the objective of maximizing matching size, while\u0000enhancing equity through two simple modifications. In comparison with ten other\u0000approximation and local search algorithms, TBLS achieves the highest matching\u0000size, while TBLS-E exhibits the lowest sex equality cost. Significantly, TBLS-E\u0000preserves a matching size comparable to that of TBLS. Both our algorithms\u0000demonstrate faster computational speed than other local search algorithms in\u0000solving large-sized instances.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254489","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}
Wei Lv, Cheng-Yang Yu, Jie Liang, Wei-Kun Chen, Yu-Hong Dai
This paper considers the generalized maximal covering location problem (GMCLP) which establishes a fixed number of facilities to maximize the weighted sum of the covered customers, allowing customers' weights to be positive or negative. The GMCLP can be modeled as a mixed integer programming (MIP) formulation and solved by off-the-shelf MIP solvers. However, due to the large problem size and particularly, poor linear programming (LP) relaxation, the GMCLP is extremely difficult to solve by state-of-the-art MIP solvers. To improve the computational performance of MIP-based approaches for solving GMCLPs, we propose customized presolving and cutting plane techniques, which are the isomorphic aggregation, dominance reduction, and two-customer inequalities. The isomorphic aggregation and dominance reduction can not only reduce the problem size but also strengthen the LP relaxation of the MIP formulation of the GMCLP. The two-customer inequalities can be embedded into a branch-and-cut framework to further strengthen the LP relaxation of the MIP formulation on the fly. By extensive computational experiments, we show that all three proposed techniques can substantially improve the capability of MIP solvers in solving GMCLPs. In particular, for a testbed of 40 instances with identical numbers of customers and facilities in the literature, the proposed techniques enable to provide optimal solutions for 13 previously unsolved benchmark instances; for a testbed of 56 instances where the number of customers is much larger than the number of facilities, the proposed techniques can turn most of them from intractable to easily solvable.
{"title":"Presolving and cutting planes for the generalized maximal covering location problem","authors":"Wei Lv, Cheng-Yang Yu, Jie Liang, Wei-Kun Chen, Yu-Hong Dai","doi":"arxiv-2409.09834","DOIUrl":"https://doi.org/arxiv-2409.09834","url":null,"abstract":"This paper considers the generalized maximal covering location problem\u0000(GMCLP) which establishes a fixed number of facilities to maximize the weighted\u0000sum of the covered customers, allowing customers' weights to be positive or\u0000negative. The GMCLP can be modeled as a mixed integer programming (MIP)\u0000formulation and solved by off-the-shelf MIP solvers. However, due to the large\u0000problem size and particularly, poor linear programming (LP) relaxation, the\u0000GMCLP is extremely difficult to solve by state-of-the-art MIP solvers. To\u0000improve the computational performance of MIP-based approaches for solving\u0000GMCLPs, we propose customized presolving and cutting plane techniques, which\u0000are the isomorphic aggregation, dominance reduction, and two-customer\u0000inequalities. The isomorphic aggregation and dominance reduction can not only\u0000reduce the problem size but also strengthen the LP relaxation of the MIP\u0000formulation of the GMCLP. The two-customer inequalities can be embedded into a\u0000branch-and-cut framework to further strengthen the LP relaxation of the MIP\u0000formulation on the fly. By extensive computational experiments, we show that\u0000all three proposed techniques can substantially improve the capability of MIP\u0000solvers in solving GMCLPs. In particular, for a testbed of 40 instances with\u0000identical numbers of customers and facilities in the literature, the proposed\u0000techniques enable to provide optimal solutions for 13 previously unsolved\u0000benchmark instances; for a testbed of 56 instances where the number of\u0000customers is much larger than the number of facilities, the proposed techniques\u0000can turn most of them from intractable to easily solvable.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254542","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}
Jolan Wauters, Tom Lefebvre, Joris Degroote, Ivo Couckuyt, Guillaume Crevecoeur
In recent years, there has been a notable evolution in various multidisciplinary design methodologies for dynamic systems. Among these approaches, a noteworthy concept is that of concurrent conceptual and control design or co-design. This approach involves the tuning of feedforward and/or feedback control strategies in conjunction with the conceptual design of the dynamic system. The primary aim is to discover integrated solutions that surpass those attainable through a disjointed or decoupled approach. This concurrent design paradigm exhibits particular promise in the context of hybrid unmanned aerial systems (UASs), such as tail-sitters, where the objectives of versatility (driven by control considerations) and efficiency (influenced by conceptual design) often present conflicting demands. Nevertheless, a persistent challenge lies in the potential disparity between the theoretical models that underpin the design process and the real-world operational environment, the so-called reality gap. Such disparities can lead to suboptimal performance when the designed system is deployed in reality. To address this issue, this paper introduces DAIMYO, a novel design architecture that incorporates a high-fidelity environment, which emulates real-world conditions, into the procedure in pursuit of a `first-time-right' design. The outcome of this innovative approach is a design procedure that yields versatile and efficient UAS designs capable of withstanding the challenges posed by the reality gap.
{"title":"Introducing DAIMYO: a first-time-right dynamic design architecture and its application to tail-sitter UAS development","authors":"Jolan Wauters, Tom Lefebvre, Joris Degroote, Ivo Couckuyt, Guillaume Crevecoeur","doi":"arxiv-2409.09820","DOIUrl":"https://doi.org/arxiv-2409.09820","url":null,"abstract":"In recent years, there has been a notable evolution in various\u0000multidisciplinary design methodologies for dynamic systems. Among these\u0000approaches, a noteworthy concept is that of concurrent conceptual and control\u0000design or co-design. This approach involves the tuning of feedforward and/or\u0000feedback control strategies in conjunction with the conceptual design of the\u0000dynamic system. The primary aim is to discover integrated solutions that\u0000surpass those attainable through a disjointed or decoupled approach. This\u0000concurrent design paradigm exhibits particular promise in the context of hybrid\u0000unmanned aerial systems (UASs), such as tail-sitters, where the objectives of\u0000versatility (driven by control considerations) and efficiency (influenced by\u0000conceptual design) often present conflicting demands. Nevertheless, a\u0000persistent challenge lies in the potential disparity between the theoretical\u0000models that underpin the design process and the real-world operational\u0000environment, the so-called reality gap. Such disparities can lead to suboptimal\u0000performance when the designed system is deployed in reality. To address this\u0000issue, this paper introduces DAIMYO, a novel design architecture that\u0000incorporates a high-fidelity environment, which emulates real-world conditions,\u0000into the procedure in pursuit of a `first-time-right' design. The outcome of\u0000this innovative approach is a design procedure that yields versatile and\u0000efficient UAS designs capable of withstanding the challenges posed by the\u0000reality gap.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254544","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}
Motion camouflage can be a useful tactic for a pursuer attempting to conceal their true trajectory from their target. Many previous studies determine optimal trajectories subject to motion camouflage constraints, but these analyses do not address when it is optimal to use, nor do they account for the pursuer's inability to predict if and when the target will try to escape. We present an optimal control framework to determine when the pursuer should use motion camouflage amidst uncertainty in the target's escape attempt time. Focusing on the illustrative problem of a male hover fly pursuing a female hover fly for mating, we model the female fly's escape response as the result of a non-homogeneous Poisson point process with a biologically informed rate function, and we obtain and numerically solve two Hamilton-Jacobi-Bellman (HJB) PDEs which encode the pursuer's optimal trajectories. Our numerical experiments and statistics illustrate when it is optimal to use motion camouflage pursuit tactics under varying degrees of the target's visual acuity and tolerance to the pursuer's presence.
{"title":"Optimality of Motion Camouflage Under Escape Uncertainty","authors":"Mallory Gaspard","doi":"arxiv-2409.09890","DOIUrl":"https://doi.org/arxiv-2409.09890","url":null,"abstract":"Motion camouflage can be a useful tactic for a pursuer attempting to conceal\u0000their true trajectory from their target. Many previous studies determine\u0000optimal trajectories subject to motion camouflage constraints, but these\u0000analyses do not address when it is optimal to use, nor do they account for the\u0000pursuer's inability to predict if and when the target will try to escape. We\u0000present an optimal control framework to determine when the pursuer should use\u0000motion camouflage amidst uncertainty in the target's escape attempt time.\u0000Focusing on the illustrative problem of a male hover fly pursuing a female\u0000hover fly for mating, we model the female fly's escape response as the result\u0000of a non-homogeneous Poisson point process with a biologically informed rate\u0000function, and we obtain and numerically solve two Hamilton-Jacobi-Bellman (HJB)\u0000PDEs which encode the pursuer's optimal trajectories. Our numerical experiments\u0000and statistics illustrate when it is optimal to use motion camouflage pursuit\u0000tactics under varying degrees of the target's visual acuity and tolerance to\u0000the pursuer's presence.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254541","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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