Search

Control design for linear, time-invariant mechanical systems typically requires an accurate low-order approximation in the low frequency range. For example a series expansion of the transfer function around zero consisting of a mass, velocity, and compliance term. Because computing such a series expansion of the transfer function can be cumbersome, a new method to compute low-order approximations of mechanical systems is developed in this paper. The method does not require an explicit expression for the transfer function, which is not always available for infinite-dimensional systems. The advantages of the proposed method is demonstrated in three examples., Comment: 32 pages, 7 figures

The search for global minima is a critical challenge across multiple fields including engineering, finance, and artificial intelligence, particularly with non-convex functions that feature multiple local optima, complicating optimization efforts. We introduce the Quantum Global Minimum Finder (QGMF), an innovative quantum computing approach that efficiently identifies global minima. QGMF combines binary search techniques to shift the objective function to a suitable position and then employs Variational Quantum Search to precisely locate the global minimum within this targeted subspace. Designed with a low-depth circuit architecture, QGMF is optimized for Noisy Intermediate-Scale Quantum (NISQ) devices, utilizing the logarithmic benefits of binary search to enhance scalability and efficiency. This work demonstrates the impact of QGMF in advancing the capabilities of quantum computing to overcome complex non-convex optimization challenges effectively., Comment: 12 pages, 3 figures

This paper presents a convex optimization framework for eco-driving and vehicle energy management problems. We will first show that several types of eco-driving and vehicle energy management problems can be modelled using the same notions of energy storage buffers and energy storage converters that are connected to a power network. It will be shown that these problems can be formulated as optimization problems with linear cost functions and linear dynamics, and nonlinear constraints representing the power converters. We will show that under some mild conditions, the (non-convex) optimization problem has the same (globally) optimal solution as a convex relaxation. This means that the problems can be solved efficiently and that the solution is guaranteed to be globally optimal. Finally, a numerical example of the eco-driving problem is used to illustrate this claim.

Buildings account for 40 % of global energy consumption. A considerable portion of building energy consumption stems from heating, ventilation, and air conditioning (HVAC), and thus implementing smart, energy-efficient HVAC systems has the potential to significantly impact the course of climate change. In recent years, model-free reinforcement learning algorithms have been increasingly assessed for this purpose due to their ability to learn and adapt purely from experience. They have been shown to outperform classical controllers in terms of energy cost and consumption, as well as thermal comfort. However, their weakness lies in their relatively poor data efficiency, requiring long periods of training to reach acceptable policies, making them inapplicable to real-world controllers directly. Hence, common research goals are to improve the learning speed, as well as to improve their ability to generalize, in order to facilitate transfer learning to unseen building environments. In this paper, we take a federated learning approach to training the reinforcement learning controller of an HVAC system. A global control policy is learned by aggregating local policies trained on multiple data centers located in different climate zones. The goal of the policy is to simultaneously minimize energy consumption and maximize thermal comfort. The federated optimization strategy indirectly increases both the rate at which experience data is collected and the variation in the data. We demonstrate through experimental evaluation that these effects lead to a faster learning speed, as well as greater generalization capabilities in the federated policy compared to any individually trained policy.

The paper is concerned with a zero-sum differential game in the case where a payoff is determined by the exit time, that is, the first time when the system leaves the game domain. Additionally, we assume that a part of domain's boundary is a lifeline where the payoff is infinite. Hereby, the examined problem generalizes the well-known time-optimal problem as well as time-optimal problem with lifeline. The main result of the paper relies on the solution to the Direchlet problem for the Hamilton-Jacobi equation associated with the game with exit time. We prove the existence of the value function for examined problem and construct suboptimal feedback strategies under assumption that the associated Dirichlet problem for the Hamilton-Jacobi equation admits a viscosity/minimax solution. Additionally, we derive a sufficient condition of existence result to this Dirichlet problem., Comment: 22 pages

Achieving control objectives (e.g., stabilization or convergence of tracking error to zero, input-to-state stabilization) in "prescribed time" has attracted significant research interest in recent years. The key property of prescribed-time results unlike traditional "asymptotic" results is that the convergence or other control objectives are achieved within an arbitrary designer-specified time interval instead of asymptotically as time goes to infinity. In this paper, we consider cascade and feedback interconnections of prescribed-time input-to-state stable (ISS) systems and study conditions under which the overall states of such interconnected systems also converge to the origin in the prescribed time interval. We show that these conditions are intrinsically related to properties of the time-varying "blow-up" functions that are central to prescribed-time control designs. We also generalize the results to interconnections of an arbitrary number of systems. As an illustrative example, we consider an interconnection of two uncertain systems that are prescribed-time stabilized using two different control design methods and show that the two separate controllers can be put together to achieve prescribed-time stability of the interconnected system., Comment: 2 figures

Stochastic optimization problems are generally known to be ill-conditioned to the form of the underlying uncertainty. A framework is introduced for optimal control problems with partial differential equations as constraints that is robust to inaccuracies in the precise form of the problem uncertainty. The framework is based on problem relaxation and involves optimizing a bivariate, "Rockafellian" objective functional that features both a standard control variable and an additional perturbation variable that handles the distributional ambiguity. In the presence of distributional corruption, the Rockafellian objective functionals are shown in the appropriate settings to $\Gamma$-converge to uncorrupted objective functionals in the limit of vanishing corruption. Numerical examples illustrate the framework's utility for outlier detection and removal and for variance reduction.

Real-stable, Lorentzian, and log-concave polynomials are well-studied classes of polynomials, and have been powerful tools in resolving several conjectures. We show that the problems of deciding whether a polynomial of fixed degree is real stable or log concave are coNP-complete. On the other hand, while all homogeneous real-stable polynomials are Lorentzian and all Lorentzian polynomials are log concave on the positive orthant, the problem of deciding whether a polynomial of fixed degree is Lorentzian can be solved in polynomial time., Comment: 21 pages, 1 figure

Designing policies for a network of agents is typically done by formulating an optimization problem where each agent has access to state measurements of all the other agents in the network. Such policy designs with centralized information exchange result in optimization problems that are typically hard to solve, require establishing substantial communication links, and do not promote privacy since all information is shared among the agents. Designing policies based on arbitrary communication structures can lead to non-convex optimization problems which are typically NP-hard. In this work, we propose an optimization framework for decentralized policy designs. In contrast to the centralized information exchange, our approach requires only local communication exchange among the neighboring agents matching the physical coupling of the network. Thus, each agent only requires information from its direct neighbors, minimizing the need for excessive communication and promoting privacy amongst the agents. Using robust optimization techniques, we formulate a convex optimization problem with a loosely coupled structure that can be solved efficiently. We numerically demonstrate the efficacy of the proposed approach in energy management and supply chain applications. We show that the proposed approach leads to solutions that closely approximate those obtained by the centralized formulation only at a fraction of the computational effort., Comment: arXiv admin note: text overlap with arXiv:1803.07660

We revisit the work of Mitter and Newton on an information-theoretic interpretation of Bayes' formula through the Gibbs variational principle. This formulation allowed them to pose nonlinear estimation for diffusion processes as a problem in stochastic optimal control, so that the posterior density of the signal given the observation path could be sampled by adding a drift to the signal process. We show that this control-theoretic approach to sampling provides a common mechanism underlying several distinct problems involving diffusion processes, specifically importance sampling using Feynman-Kac averages, time reversal, and Schr\"odinger bridges., Comment: 22 pages; dedicated to the memory of Sanjoy K. Mitter (1933-2023)

State transition algorithm (STA) is a metaheuristic method for global optimization. Recently, a modified STA named parameter optimal state transition algorithm (POSTA) is proposed. In POSTA, the performance of expansion operator, rotation operator and axesion operator is optimized through a parameter selection mechanism. But due to the insufficient utilization of historical information, POSTA still suffers from slow convergence speed and low solution accuracy on specific problems. To make better use of the historical information, Nelder-Mead (NM) simplex search and quadratic interpolation (QI) are integrated into POSTA. The enhanced POSTA is tested against 14 benchmark functions with 20-D, 30-D and 50-D space. An experimental comparison with several competitive metaheuristic methods demonstrates the effectiveness of the proposed method.

This paper develops an adaptive traffic control policy inspired by Maximum Pressure (MP) while imposing coordination across intersections. The proposed Coordinated Maximum Pressure-plus-Penalty (CMPP) control policy features a local objective for each intersection that consists of the total pressure within the neighborhood and a penalty accounting for the queue capacities and continuous green time for certain movements. The corresponding control task is reformulated as a distributed optimization problem and solved via two customized algorithms: one based on the alternating direction method of multipliers (ADMM) and the other follows a greedy heuristic augmented with a majority vote. CMPP not only provides a theoretical guarantee of queuing network stability but also outperforms several benchmark controllers in simulations on a large-scale real traffic network with lower average travel and waiting time per vehicle, as well as less network congestion. Furthermore, CPMM with the greedy algorithm enjoys comparable computational efficiency as fully decentralized controllers without significantly compromising the control performance, which highlights its great potential for real-world deployment.

Motivated by the widespread use of approximate derivatives in machine learning and optimization, we study inexact subgradient methods with non-vanishing additive errors and step sizes. In the nonconvex semialgebraic setting, under boundedness assumptions, we prove that the method provides points that eventually fluctuate close to the critical set at a distance proportional to $\epsilon^\rho$ where $\epsilon$ is the error in subgradient evaluation and $\rho$ relates to the geometry of the problem. In the convex setting, we provide complexity results for the averaged values. We also obtain byproducts of independent interest, such as descent-like lemmas for nonsmooth nonconvex problems and some results on the limit of affine interpolants of differential inclusions.

In this paper, we propose a two-stage robust planning model for an Integrated Energy System (IES) that serves an industrial park. The term 'Park-level IES' is used to refers to IES of a smaller scale but have high demands for various forms of energy. The proposed planning model considers uncertainties like load demand fluctuations and equipment contingencies, and provides a reliable scheme of equipment selection and sizing for IES investors. Inspired by the unit commitment problem, we formulate an equipment contingency uncertainty set to accurately describe the potential equipment contingencies which happen and can be repaired within a day. Then, a novel and modified nested column-and-constraint generation algorithm is applied to solve this two-stage robust planning model with integer recourse efficiently. In the case study, the role of energy storage system for IES reliability enhancement is analyzed in detail. Computational results demonstrate the advantage of the proposed models over the deterministic planning model in terms of improving reliability.

In this paper we compare two numerical methods to integrate Riemannian cubic polynomials on the Stiefel manifold $\textbf{St}_{n,k}$. The first one is the adjusted de Casteljau algorithm, and the second one is a symplectic integrator constructed through discretization maps. In particular, we choose the cases of $n=3$ together with $k=1$ and $k=2$. The first case is diffeomorphic to the sphere and the quasi-geodesics appearing in the adjusted de Casteljau algorithm are actually geodesics. The second case is an example where we have a pure quasi-geodesic different from a geodesic. We provide a numerical comparison of both methods and discuss the obtained results to highlight the benefits of each method., Comment: 12 pages, 3 figures

In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several scenarios., Comment: 45 pages, 5 figures, 4 tables

Given a determinate (multivariate) probability measure $\mu$, we characterize Gaussian mixtures $\nu\_\phi$ which minimize the Wasserstein distance $W\_2(\mu,\nu\_\phi)$ to $\mu$ when the mixing probability measure $\phi$ on the parameters $(m,\Sigma)$ of the Gaussians is supported on a compact set $S$.(i) We first show that such mixtures are optimal solutions of a particular optimal transport (OT) problem where the marginal $\nu\_{\phi}$ of the OT problem is also unknown via the mixing measure variable $\phi$. Next (ii) by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) and if the set $S$ of mixture parameters $(m,\Sigma)$ is a basic compact semi-algebraic set, we provide a "mesh-free" numerical scheme to approximate as closely as desired the optimal distance by solving a hierarchy of semidefinite relaxations of increasing size. In particular, we neither assume that the mixing measure is finitely supported nor that the variance is the same for all components. If the original measure $\mu$ is not a Gaussian mixture with parameters $(m,\Sigma)\in S$, then a strictly positive distance is detected at a finite step of the hierarchy. If the original measure $\mu$ is a Gaussian mixture with parameters $(m,\Sigma)\in S$, then all semidefinite relaxations of the hierarchy have same zero optimal value. Moreover if the mixing measure is atomic with finite support, its components can sometimes be extracted from an optimal solution at some semidefinite relaxation of the hierarchy when Curto & Fialkow's flatness condition holds for some moment matrix.

Although quantum entanglement is an important resource, its characterization is quite challenging. The partial transposition is a common method to detect bipartite entanglement. In this paper, the authors study the partial-transpose(PT)-moments of two-qubit states,and completely describe the whole region, composed of the second and third PT-moments, for all two-qubit states. Furthermore, they determine the accurate region corresponding to all entangled two-qubit states. The states corresponding to those boundary points of the whole region, and to the border lines between separable and entangled states are analyzed. As an application, they characterize the entangled region of PT-moments for the two families of Werner states and Bell-diagonal states. The relations between entanglement and the pairs of PT-moments are revealed from these typical examples. They also numerically plot the whole region of possible PT-moments for all two-qubit X-states, and find that this region is almost the same as the whole region of PT-moments for all two-qubit states. Moreover, they extend their results to detect the entanglement of multiqubit states. By utilizing the PT-moment-based method to characterize the entanglement of the multiqubit states mixed by the GHZ and W states, they propose an operational way of verifying the genuine entanglement in such states., Comment: 31 pages, LaTeX, 9 figures

Recent empirical and theoretical work has shown that the dynamics of the large eigenvalues of the training loss Hessian have some remarkably robust features across models and datasets in the full batch regime. There is often an early period of progressive sharpening where the large eigenvalues increase, followed by stabilization at a predictable value known as the edge of stability. Previous work showed that in the stochastic setting, the eigenvalues increase more slowly - a phenomenon we call conservative sharpening. We provide a theoretical analysis of a simple high-dimensional model which shows the origin of this slowdown. We also show that there is an alternative stochastic edge of stability which arises at small batch size that is sensitive to the trace of the Neural Tangent Kernel rather than the large Hessian eigenvalues. We conduct an experimental study which highlights the qualitative differences from the full batch phenomenology, and suggests that controlling the stochastic edge of stability can help optimization.

This paper studies the convergence properties of a family of Relaxed $\ell$-Minimal Gradient Descent methods for quadratic optimization; the family includes the omnipresent Steepest Descent method, as well as the Minimal Gradient method. Simple proofs are provided that show, in an appropriately chosen norm, the gradient and the distance of the iterates from optimality converge linearly, for all members of the family. Moreover, the function values decrease linearly, and iteration complexity results are provided. All theoretical results hold when (fixed) relaxation is employed. It is also shown that, given a fixed overhead and storage budget, every Relaxed $\ell$-Minimal Gradient Descent method can be implemented using exactly one matrix vector product. Numerical experiments are presented that illustrate the benefits of relaxation across the family., Comment: 23 pages, 6 figures, 2 tables

Reinforcement learning for multi-agent games has attracted lots of attention recently. However, given the challenge of solving Nash equilibria for large population games, existing works with guaranteed polynomial complexities either focus on variants of zero-sum and potential games, or aim at solving (coarse) correlated equilibria, or require access to simulators, or rely on certain assumptions that are hard to verify. This work proposes MF-OML (Mean-Field Occupation-Measure Learning), an online mean-field reinforcement learning algorithm for computing approximate Nash equilibria of large population sequential symmetric games. MF-OML is the first fully polynomial multi-agent reinforcement learning algorithm for provably solving Nash equilibria (up to mean-field approximation gaps that vanish as the number of players $N$ goes to infinity) beyond variants of zero-sum and potential games. When evaluated by the cumulative deviation from Nash equilibria, the algorithm is shown to achieve a high probability regret bound of $\tilde{O}(M^{3/4}+N^{-1/2}M)$ for games with the strong Lasry-Lions monotonicity condition, and a regret bound of $\tilde{O}(M^{11/12}+N^{- 1/6}M)$ for games with only the Lasry-Lions monotonicity condition, where $M$ is the total number of episodes and $N$ is the number of agents of the game. As a byproduct, we also obtain the first tractable globally convergent computational algorithm for computing approximate Nash equilibria of monotone mean-field games.

The paper explores the Biased Random-Key Genetic Algorithm (BRKGA) in the domain of logistics and vehicle routing. Specifically, the application of the algorithm is contextualized within the framework of the Vehicle Routing Problem with Occasional Drivers and Time Window (VRPODTW) that represents a critical challenge in contemporary delivery systems. Within this context, BRKGA emerges as an innovative solution approach to optimize routing plans, balancing cost-efficiency with operational constraints. This research introduces a new BRKGA, characterized by a variable mutant population which can vary from generation to generation, named BRKGA-VM. This novel variant was tested to solve a VRPODTW. For this purpose, an innovative specific decoder procedure was proposed and implemented. Furthermore, a hybridization of the algorithm with a Variable Neighborhood Descent (VND) algorithm has also been considered, showing an improvement of problem-solving capabilities. Computational results show a better performances in term of effectiveness over a previous version of BRKGA, denoted as MP. The improved performance of BRKGA-VM is evident from its ability to optimize solutions across a wide range of scenarios, with significant improvements observed for each type of instance considered. The analysis also reveals that VM achieves preset goals more quickly compared to MP, thanks to the increased variability induced in the mutant population which facilitates the exploration of new regions of the solution space. Furthermore, the integration of VND has shown an additional positive impact on the quality of the solutions found., Comment: 25 pages, 9 figures

With the advent of self-driving cars, experts envision autonomous mobility-on-demand services in the near future to cope with overloaded transportation systems in cities worldwide. Efficient operations are imperative to unlock such a system's maximum improvement potential. Existing approaches either consider a narrow planning horizon or ignore essential characteristics of the underlying problem. In this paper, we develop an algorithmic framework that allows the study of very large-scale pickup and delivery routing problems with more than 20 thousand requests, which arise in the context of integrated request pooling and vehicle-to-request dispatching. We conduct a computational study and present comparative results showing the characteristics of the developed approaches. Furthermore, we apply our algorithm to related benchmark instances from the literature to show the efficacy. Finally, we solve very large-scale instances and derive insights on upper-bound improvements regarding fleet sizing and customer delay acceptance from a practical perspective.

First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds has been shown to increase both the theoretical guarantees and the practical performance. In this work, we define a global lower bound for smooth differentiable objectives that is optimal with respect to the collected oracle information. The bound can be readily employed by the Gradient Method with Memory to improve its performance. Further using the machinery underlying the optimal bounds, we introduce a modified version of the estimate sequence that we use to construct an Optimized Gradient Method with Memory possessing the best known convergence guarantees for its class of algorithms, even in terms of the proportionality constant. We additionally equip the method with an adaptive convergence guarantee adjustment procedure that is an effective replacement for line-search. Simulation results on synthetic but otherwise difficult smooth problems validate the theoretical properties of the bound and proposed methods., Comment: 29 pages, 1 figure

We examine the modeling, simulation, and optimization of district heating systems, which are widely used for thermal transport using steam or hot water as a carrier. We propose a generalizable framework to specify network models and scenario parameters, and develop an optimization method for evaluating system states including pressures, fluid flow rates, and temperatures throughout the network. The network modeling includes pipes, thermal plants, pumps, and passive or controllable loads as system components. We propose basic models for thermodynamic fluid transport and enforce the balance of physical quantities in steady-state flow over co-located outgoing and return networks. We formulate an optimization problem with steam and hot water as the outgoing and return carriers, as in legacy 20th century systems. The physical laws and engineering limitations are specified for each component type, and the thermal network flow optimization (TNFO) problem is formulated and solved for a realistic test network under several scenarios.

Model predictive control (MPC) is a powerful, optimization-based approach for controlling dynamical systems. However, the computational complexity of online optimization can be problematic on embedded devices. Especially, when we need to guarantee fixed control frequencies. Thus, previous work proposed to reduce the computational burden using imitation learning (IL) approximating the MPC policy by a neural network. In this work, we instead learn the whole planned trajectory of the MPC. We introduce a combination of a novel neural network architecture PlanNetX and a simple loss function based on the state trajectory that leverages the parameterized optimal control structure of the MPC. We validate our approach in the context of autonomous driving by learning a longitudinal planner and benchmarking it extensively in the CommonRoad simulator using synthetic scenarios and scenarios derived from real data. Our experimental results show that we can learn the open-loop MPC trajectory with high accuracy while improving the closed-loop performance of the learned control policy over other baselines like behavior cloning., Comment: under revision for the 6th Annual Learning for Dynamics & Control Conference (L4DC 2024)

Regularized optimal transport (OT) has received much attention in recent years starting from Cuturi's paper with Kullback-Leibler (KL) divergence regularized OT. In this paper, we propose to regularize the OT problem using the family of $\alpha$-R\'enyi divergences for $\alpha \in (0, 1)$. R\'enyi divergences are neither $f$-divergences nor Bregman distances, but they recover the KL divergence in the limit $\alpha \nearrow 1$. The advantage of introducing the additional parameter $\alpha$ is that for $\alpha \searrow 0$ we obtain convergence to the unregularized OT problem. For the KL regularized OT problem, this was achieved by letting the regularization parameter tend to zero, which causes numerical instabilities. We present two different ways to obtain premetrics on probability measures, namely by R\'enyi divergence constraints and penalization. The latter premetric interpolates between the unregularized and KL regularized OT problem with weak convergence of the minimizer, generalizing the interpolating property of KL regularized OT. We use a nested mirror descent algorithm for solving the primal formulation. Both on real and synthetic data sets R\'enyi regularized OT plans outperform their KL and Tsallis counterparts in terms of being closer to the unregularized transport plans and recovering the ground truth in inference tasks better., Comment: 40 pages, 9 figures, 3 tables, comments welcome

We prove that the $D_4$ root system (the set of vertices of the regular $24$-cell) is the unique optimal kissing configuration in $\mathbb R^4$, and is an optimal spherical code. For this, we use semidefinite programming to compute an exact optimal solution to the second level of the Lasserre hierarchy. We also improve the upper bound for the kissing number problem in $\mathbb R^6$ to $77$.

Recent growing complexity in space missions has led to an active research field of space logistics and mission design. This research field leverages the key ideas and methods used to handle complex terrestrial logistics to tackle space logistics design problems. A typical goal in space logistics is to optimize the commodity flow to satisfy some mission objectives with the lowest cost. One of the successful space logistics approaches is network flow modeling and optimization using mixed-integer linear programming (MILP). A caveat of the conventional MILP-based network approach for space logistics is its incapability of handling nonlinearity. For example, in the MILP formulation, the spacecraft structure mass and fuel/payload capacity are approximated by a linear relationship. However, this oversimplified relationship cannot characterize a realistic spacecraft design. Other types of nonlinearity can appear when a nonlinear time-dependent trajectory model is considered in an event-driven network, where the time step of each event itself is a variable. In response to this challenge, this Note develops a new systematic general framework to handle nonlinearity in the MILP-based space logistics formulation using machine learning (ML). Specifically, we replace the nonlinear constraints in the space logistics formulation with trained ML models that are compatible with MILP. The MILP-compatible ML model includes linear regression, PWL approximations, neural networks (NN) with Rectified Linear Unit (ReLU) activations, decision tree regression, and random forest regression, among others; these models can be translated into MILP formulations with a definition of additional variables and constraints while maintaining the linearity. This Note provides the first demonstration of using such trained ML models directly in a MILP-based space logistics optimization formulation., Comment: 12 pages, 2 figures, under review by AIAA Journal of Spacecraft and Rockets

In this article, we study the existence of $\eta\in W_0^{1,\infty}(\Omega;\mathbb R^n)$ satisfying $$\textrm{curl} \ \eta\in E \textrm{ a.e. in }\Omega,$$ where $n\in \mathbb N, \Omega\subseteq \mathbb R^n$ is open, bounded and $E\subseteq \Lambda^2.$, Comment: 15 pages

In this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively., Comment: arXiv admin note: text overlap with arXiv:2111.00100

Dependency length minimization is a universally observed quantitative property of natural languages. However, the extent of dependency length minimization, and the cognitive mechanisms through which the language processor achieves this minimization remain unclear. This research offers mechanistic insights by postulating that moving a short preverbal constituent next to the main verb explains preverbal constituent ordering decisions better than global minimization of dependency length in SOV languages. This approach constitutes a least-effort strategy because it's just one operation but simultaneously reduces the length of all preverbal dependencies linked to the main verb. We corroborate this strategy using large-scale corpus evidence across all seven SOV languages that are prominently represented in the Universal Dependency Treebank. These findings align with the concept of bounded rationality, where decision-making is influenced by 'quick-yet-economical' heuristics rather than exhaustive searches for optimal solutions. Overall, this work sheds light on the role of bounded rationality in linguistic decision-making and language evolution., Comment: Accepted at CogSci-2024 as talk with full paper publication

The uncertainty relation is a fundamental concept in quantum theory, plays a pivotal role in various quantum information processing tasks. In this study, we explore the additive uncertainty relation pertaining to two or more observables, in terms of their variance,by utilizing the generalized Gell-Mann representation in qudit systems. We find that the tight state-independent lower bound of the variance sum can be characterized as a quadratic programming problem with nonlinear constraints in optimization theory. As illustrative examples, we derive analytical solutions for these quadratic programming problems in lower-dimensional systems, which align with the state-independent lower bounds. Additionally, we introduce a numerical algorithm tailored for solving these quadratic programming instances, highlighting its efficiency and accuracy. The advantage of our approach lies in its potential ability to simultaneously achieve the optimal value of the quadratic programming problem with nonlinear constraints but also precisely identify the extremal state where this optimal value is attained. This enables us to establish a tight state-independent lower bound for the sum of variances, and further identify the extremal state at which this lower bound is realized., Comment: 35 pages, LaTeX

This scoping review assesses the current use of simulation-based design optimization (SBDO) in marine engineering, focusing on identifying research trends, methodologies, and application areas. Analyzing 277 studies from Scopus and Web of Science, the review finds that SBDO is predominantly applied to optimizing marine vessel hulls, including both surface and underwater types, and extends to key components like bows, sterns, propellers, and fins. It also covers marine structures and renewable energy systems. A notable trend is the preference for deterministic single-objective optimization methods, indicating potential growth areas in multi-objective and stochastic approaches. The review points out the necessity of integrating more comprehensive multidisciplinary optimization methods to address the complex challenges in marine environments. Despite the extensive application of SBDO in marine engineering, there remains a need for enhancing the methodologies' efficiency and robustness. This review offers a critical overview of SBDO's role in marine engineering and highlights opportunities for future research to advance the field.

The energy efficiency optimization of the power generation system and the energy efficiency optimization of the energy consumption system are unified into the same optimization problem, and a simple method to achieve energy efficiency optimization without establishing an accurate mathematical model of the system is proposed. For systems with similar energy efficiency, it is proved that the best load distribution method between equipment is to keep the operating energy efficiency of each operating device equal, Yao's theorem 1. It is proved that the optimal switching method for the number of operating units between equipment with different energy efficiency is to keep the energy efficiency of the switching point equal, or at the maximum load point of the equipment, Yao's Theorem 2. This article gives two cases, a system composed of equipment with similar efficiency and a system composed of equipment with different efficiency.

Hybrid-excited electrical machines aim to combine the advantages of permanent magnet machines (high efficiency and torque density) with those of separately excited machines (ease of flux-weakening at high speed). These machines are of interest to electric vehicles, and only parametric approaches are available in the literature for their optimization. This work proposes a more general topology optimization methodology by extending the formalism of density methods. The difficulty lies in integrating the numerous natures of materials (conductors, permanent magnets, ferromagnetic material, air...) without strongly deconvexifying the optimization problem, which leads to non-physical results with unsatisfactory performance. To address this issue, a recursive material interpolation is introduced. The hybrid-excited rotors optimized by this approach are compared with those of existing techniques, demonstrating a clear superiority of the recursive interpolation.

A linear quadratic (LQ) stochastic optimization system involving large population, which is driven by forward-backward stochastic differential equation (FBSDE), is investigated in this paper. Agents cooperate with each other to minimize the so-called social objective, which is rather different from mean field (MF) game. Employing forward-backward person-by-person optimality principle, we derive an auxiliary LQ control problem by decentralized information. A decentralized strategy is obtained by virtue of an MF-type forward-backward stochastic differential equation consistency condition. Applying Riccati equation decoupling method, we solve the consistency condition system. We also verify the asymptotic social optimality in this framework., Comment: 30 pages

Pose graph optimization (PGO) is a well-known technique for solving the pose-based simultaneous localization and mapping (SLAM) problem. In this paper, we represent the rotation and translation by a unit quaternion and a three-dimensional vector, and propose a new PGO model based on the von Mises-Fisher distribution. The constraints derived from the unit quaternions are spherical manifolds, and the projection onto the constraints can be calculated by normalization. Then a proximal linearized Riemannian alternating direction method of multipliers (PieADMM) is developed to solve the proposed model, which not only has low memory requirements, but also can update the poses in parallel. Furthermore, we establish the iteration complexity of $O(1/\epsilon^{2})$ of PieADMM for finding an $\epsilon$-stationary solution of our model. The efficiency of our proposed algorithm is demonstrated by numerical experiments on two synthetic and four 3D SLAM benchmark datasets.

Multi-purpose reservoirs support socioeconomic development by providing irrigation, domestic water supply, hydropower, and other services. However, impoundment of water impacts instream aquatic ecosystems. Thus, the concept of minimum environmental flows (MEFs) was established to restore the benefits of naturally flowing rivers by specifying minimum flow rates to be maintained downstream of dams.But varying legislative contexts under which multi-purpose reservoirs operate may not always necessitate MEF releases. To what extent the release of MEF affects other sectoral benefits remains an open-ended and possibly a site-specific inquiry. A related issue is - how does the order in which releases are prioritized influences sectoral performances? We analyse these issues for the Nagarjuna Sagar reservoir, one of the largest multipurpose reservoirs in southern India. We formulate two versions of a multi-objective decision problem. PF_MEF formulation prioritizes MEF releases over releases for water demand satisfaction, followed by hydropower releases. PF_nMEF formulation follows the regional legislative rule releasing first for demand satisfaction, followed by hydropower and MEF releases. Results thus indicate that prioritizing MEF releases improves can meet MEF requirements without significant compromises in other objectives. We hypothesize that similar investigations may reveal how simple modification of release order may improve ability of other reservoirs to meet environmental goals.

Random reshuffling with momentum (RRM) corresponds to the SGD optimizer with momentum option enabled, as found in popular machine learning libraries like PyTorch and TensorFlow. Despite its widespread use in practical applications, the understanding of its convergence properties in nonconvex scenarios remains limited. Under a Lipschitz smoothness assumption, this paper provides one of the first iteration complexities for RRM. Specifically, we prove that RRM achieves the iteration complexity $O(n^{-1/3}((1-\beta^n)T)^{-2/3})$ where $n$ denotes the number of component functions $f(\cdot;i)$ and $\beta \in [0,1)$ is the momentum parameter. Furthermore, every accumulation point of a sequence of iterates $\{x^k\}_k$ generated by RRM is shown to be a stationary point of the problem. In addition, under the Kurdyka-Lojasiewicz inequality - a local geometric property - the iterates $\{x^k\}_k$ provably converge to a unique stationary point $x^*$ of the objective function. Importantly, in our analysis, this last iterate convergence is obtained without requiring convexity nor a priori boundedness of the iterates. Finally, for polynomial step size schemes, convergence rates of the form $\|x^k - x^*\| = O(k^{-p})$, $\|\nabla f(x^k)\|^2 = O(k^{-q})$, and $|f(x^k) - f(x^*)| = O(k^{-q})$, $p \in (0,1]$, $q \in (0,2]$ are derived., Comment: 51 pages, 10 figures

Stochastic matching is the stochastic version of the well-known matching problem, which consists in maximizing the rewards of a matching under a set of probability distributions associated with the nodes and edges. In most stochastic matching problems, the probability distributions inherent in the nodes and edges are set a priori and are not controllable. However, many resource allocation platforms can control the probability distributions by changing prices. For example, a rideshare platform can control the distribution of the number of requesters by setting the fare to maximize the reward of a taxi-requester matching. Although several methods for optimizing price have been developed, optimizations in consideration of the matching problem are still in its infancy. In this paper, we tackle the problem of optimizing price in the consideration of the resulting bipartite graph matching, given the effect of the price on the probabilistic uncertainty in the graph. Even though our problem involves hard to evaluate objective values and is non-convex, we construct a (1-1/e)-approximation algorithm under the assumption that a convex min-cost flow problem can be solved exactly., Comment: 29 pages, 2 figures, 4 tables

Exploring novel strategies for the regulation of axon growth, we introduce a periodic event-triggered control (PETC) to enhance the practical implementation of the associated PDE backstepping control law. Neurological injuries may impair neuronal function, but therapies like Chondroitinase ABC (ChABC) have shown promise in improving axon elongation by influencing the extracellular matrix. This matrix, composed of extracellular macromolecules and minerals, regulates tubulin protein concentration, potentially aiding in neuronal recovery. The concentration and spatial distribution of tubulin influence axon elongation dynamics. Recent research explores feedback control strategies for this model, leading to the development of an event-triggering control (CETC) approach. In this approach, the control law updates when the monitored triggering condition is met, reducing actuation resource consumption. Through the meticulous redesign of the triggering mechanism, we introduce a periodic event-triggering control (PETC), updating control inputs at specific intervals, but evaluating the event-trigger only periodically, an ideal tool for standard time-sliced actuators like ChABC. PETC is a step forward to the design of practically feasible feedback laws for the neuron growth process. The PETC strategy establishes an upper bound on event triggers between periodic examinations, ensuring convergence and preventing Zeno behavior. Through Lyapunov analysis, we demonstrate the local exponential convergence of the system with the periodic event-triggering mechanism in the $L^2$-norm sense. Numerical examples are presented to confirm the theoretical findings., Comment: Submitted to 2024 Conference on Decision and Control

We consider a two-stage distributionally robust optimization (DRO) model with multimodal uncertainty, where both the mode probabilities and uncertainty distributions could be affected by the first-stage decisions. To address this setting, we propose a generic framework by introducing a $\phi$-divergence based ambiguity set to characterize the decision-dependent mode probabilities and further consider both moment-based and Wasserstein distance-based ambiguity sets to characterize the uncertainty distribution under each mode. We identify two special $\phi$-divergence examples (variation distance and $\chi^2$-distance) and provide specific forms of decision dependence relationships under which we can derive tractable reformulations. Furthermore, we investigate the benefits of considering multimodality in a DRO model compared to a single-modal counterpart through an analytical analysis. We provide a computational study over the facility location problem to illustrate our results, which demonstrate that omission of multimodality and decision-dependent uncertainties within DRO frameworks result in inadequately performing solutions with worse in-sample and out-of-sample performances under various settings.

The compactness of routes in distribution plans is a criterion that has not been sufficiently explored in the literature related to logistics distribution but has shown to have a significant impact on the practical implementation of routing plans, for example in solid waste collection problems. In this regard, this article presents a bi-objective model to optimize the vehicle routing problem with time constraints, considering the minimization of travel times and the compactness of routes. Experimental tests were conducted on small-scale instances using two exact solution methods for multi-objective problems: weighted sum and augmented {\epsilon}-constraint methods. The results obtained allowed us to explore the trade-off relationship between both objectives while evaluating the computational efficiency of both multi-objective solution methods., Comment: Conference paper submitted to XXI SEPROSUL

Using small noise limit approach, we study degenerate stochastic ergodic control problems and as a byproduct obtain error bounds for the $\varepsilon$-optimal controls. We also establish tunneling for a special ergodic control problem and give a representation of the ergodic value using the tunneled Markov chain.

We investigate global dynamic feedback stabilization from a topological viewpoint. In particular, we consider the general case of dynamic feedback systems, whereby the total space (which includes the state space of the system and of the controller) is a fibre bundle, and derive conditions on the topology of the bundle that are necessary for various notions of global stabilization to hold. This point of view highlight the importance of distinguishing trivial bundles and twisted bundles in the study of global dynamic feedback stabilization, as we show that dynamic feedback defined on a twisted bundle can stabilize systems that dynamic feedback on trivial bundles cannot.

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme and obtain a global reliability bound.

This paper studies moment and tensor recovery problems whose decomposing vectors are contained in some given semialgebraic sets. We propose Moment-SOS relaxations with generic objectives for recovering moments and tensors, whose decomposition lengths are expected to be low. This kind of problems have broad applications in various tensor decomposition questions. Numerical experiments are provided to demonstrate the efficiency of this approach.

Recently, the mysterious In-Context Learning (ICL) ability exhibited by Transformer architectures, especially in large language models (LLMs), has sparked significant research interest. However, the resilience of Transformers' in-context learning capabilities in the presence of noisy samples, prevalent in both training corpora and prompt demonstrations, remains underexplored. In this paper, inspired by prior research that studies ICL ability using simple function classes, we take a closer look at this problem by investigating the robustness of Transformers against noisy labels. Specifically, we first conduct a thorough evaluation and analysis of the robustness of Transformers against noisy labels during in-context learning and show that they exhibit notable resilience against diverse types of noise in demonstration labels. Furthermore, we delve deeper into this problem by exploring whether introducing noise into the training set, akin to a form of data augmentation, enhances such robustness during inference, and find that such noise can indeed improve the robustness of ICL. Overall, our fruitful analysis and findings provide a comprehensive understanding of the resilience of Transformer models against label noises during ICL and provide valuable insights into the research on Transformers in natural language processing. Our code is available at https://github.com/InezYu0928/in-context-learning., Comment: ICLR 2024 Workshop on Reliable and Responsible Foundation Models

This paper focuses on studying the evaluation index system for the production efficiency of tobacco enterprises. Considering the limitations of existing evaluation methods in accurately assessing the production quality of cigarette enterprises, a mathematical model based on the Analytic Hierarchy Process (AHP) is established. This model constructs an evaluation framework for the production efficiency of cigarette enterprises and subsequently analyzes the significance of each index within this framework. To comprehensively analyze the multi-index and feasibility aspects of the selected projects, the AHP method is employed to establish a comprehensive feasibility research and evaluation structure model. The result of this feasibility study provides the conclusion that the construction of an evaluation index system for the production efficiency of cigarette enterprises can indeed promote the enhancement of their production efficiency.