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This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether's theorem and other theorem characterizing the Lie subalgebra of symmetries of a forced discrete Lagrangian system. Moreover, we develop a Hamilton-Jacobi theory for forced discrete Hamiltonian systems. These results are useful for the construction of so-called variational integrators, which, as we illustrate with some examples, are remarkably superior to the usual numerical integrators such as the Runge-Kutta method.

Many descent methods for multiobjective optimization problems have been developed in recent years. In 2000, the steepest descent method was proposed for differentiable multiobjective optimization problems. Afterward, the proximal gradient method, which can solve composite problems, was also considered. However, the accelerated versions are not sufficiently studied. In this paper, we propose a multiobjective accelerated proximal gradient algorithm, in which we solve subproblems with terms that only appear in the multiobjective case. We also show the proposed method's global convergence rate ($O(1/k^2)$) under reasonable assumptions, using a merit function to measure the complexity. Moreover, we present an efficient way to solve the subproblem via its dual, and we confirm the validity of the proposed method through preliminary numerical experiments.

When a mobile manipulator's wheel loses contact with the ground, tipping-over may occur, causing material damage, and in the worst case, it can put human lives in danger. The tip-over stability of wheeled mobile manipulators must not be overlooked at any stage of a mobile manipulator's life, starting from the design phase, continuing through the commissioning period and extending to the operational phase. Many tip-over stability criteria formulated throughout the years do not explicitly consider the normal wheel loads, with most of them relying on prescribed stability margins in terms of overturning moments. In these formulations, it is commonly argued that overturning will occur about one of the axes connecting adjacent manipulator's contact points with the ground. This claim may not always be valid and is certainly restrictive. Explicit expressions for the manipulator supporting forces provide the best insight into relevant affecting terms which contribute to the tip-over (in)stability. They also remove the necessity for thinking about which axis the manipulator could tip over and simultaneously enable the formulation of more intuitive stability margins and on-line tip-over prevention techniques. The present study presents a general dynamics modelling approach in the Newton--Euler framework using 6D vectors and gives normal wheel load equations in a typical 4-wheeled mobile manipulator negotiating a slope. The given expressions are expected to become standard in wheeled mobile manipulators and to provide a basis for effective tip-over stability criteria and tip-over avoidance techniques. Based on the presented results, specific improvements of the state-of-the-art criteria are discussed.

Owing to the unique advantages of low cost and controllability, reconfigurable intelligent surface (RIS) is a promising candidate to address the blockage issue in millimeter wave (mmWave) communication systems, consequently has captured widespread attention in recent years. However, the joint active beamforming and passive beamforming design is an arduous task due to the high computational complexity and the dynamic changes of wireless environment. In this paper, we consider a RIS-assisted multi-user multiple-input single-output (MU-MISO) mmWave system and aim to develop a deep reinforcement learning (DRL) based algorithm to jointly design active hybrid beamformer at the base station (BS) side and passive beamformer at the RIS side. By employing an advanced soft actor-critic (SAC) algorithm, we propose a maximum entropy based DRL algorithm, which can explore more stochastic policies than deterministic policy, to design active analog precoder and passive beamformer simultaneously. Then, the digital precoder is determined by minimum mean square error (MMSE) method. The experimental results demonstrate that our proposed SAC algorithm can achieve better performance compared with conventional optimization algorithm and DRL algorithm.

In the Multi-Agent Path Finding (MAPF) problem, the goal is to find non-colliding paths for agents in an environment, such that each agent reaches its goal from its initial location. In safety-critical applications, a human supervisor may want to verify that the plan is indeed collision-free. To this end, a recent work introduces a notion of explainability for MAPF based on a visualization of the plan as a short sequence of images representing time segments, where in each time segment the trajectories of the agents are disjoint. Then, the explainable MAPF problem asks for a set of non-colliding paths that admits a short-enough explanation. Explainable MAPF adds a new difficulty to MAPF, in that it is NP-hard with respect to the size of the environment, and not just the number of agents. Thus, traditional MAPF algorithms are not equipped to directly handle explainable-MAPF. In this work, we adapt Conflict Based Search (CBS), a well-studied algorithm for MAPF, to handle explainable MAPF. We show how to add explainability constraints on top of the standard CBS tree and its underlying A* search. We examine the usefulness of this approach and, in particular, the tradeoff between planning time and explainability.

In this work, we consider a remote monitoring scenario in which multiple sensors share a wireless channel to deliver their status updates to a process monitor via an access point (AP). Moreover, we consider that the sensors randomly arrive and depart from the network as they become active and inactive. The goal of the sensors is to devise a medium access strategy to collectively minimize the long-term mean network \ac{AoI} of their respective processes at the remote monitor. For this purpose, we propose specific modifications to ALOHA-QT algorithm, a distributed medium access algorithm that employs a policy tree (PT) and reinforcement learning (RL) to achieve high throughput. We provide the upper bound on the mean network Age of Information (AoI) for the proposed algorithm along with pointers for selecting its key parameter. The results reveal that the proposed algorithm reduces mean network \ac{AoI} by more than 50 percent for state of the art stationary randomized policies while successfully adjusting to a changing number of active users in the network. The algorithm needs less memory and computation than ALOHA-QT while performing better in terms of AoI.

The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters $k$, $h$ respectively in terms of the parameter $\epsilon$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ \epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of $\epsilon$. These estimates and a suitable localization technique, via the second order condition (see \cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

Image foreground extraction is a classical problem in image processing and vision, with a large range of applications. In this dissertation, we focus on the extraction of text and graphics in mixed-content images, and design novel approaches for various aspects of this problem. We first propose a sparse decomposition framework, which models the background by a subspace containing smooth basis vectors, and foreground as a sparse and connected component. We then formulate an optimization framework to solve this problem, by adding suitable regularizations to the cost function to promote the desired characteristics of each component. We present two techniques to solve the proposed optimization problem, one based on alternating direction method of multipliers (ADMM), and the other one based on robust regression. Promising results are obtained for screen content image segmentation using the proposed algorithm. We then propose a robust subspace learning algorithm for the representation of the background component using training images that could contain both background and foreground components, as well as noise. With the learnt subspace for the background, we can further improve the segmentation results, compared to using a fixed subspace. Lastly, we investigate a different class of signal/image decomposition problem, where only one signal component is active at each signal element. In this case, besides estimating each component, we need to find their supports, which can be specified by a binary mask. We propose a mixed-integer programming problem, that jointly estimates the two components and their supports through an alternating optimization scheme. We show the application of this algorithm on various problems, including image segmentation, video motion segmentation, and also separation of text from textured images.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.