Continuous Problem Research Articles

Research on Unmanned Aerial Vehicle Intelligent Maneuvering Method Based on Hierarchical Proximal Policy Optimization

Improving decision-making in the autonomous maneuvering of unmanned aerial vehicles (UAVs) is of great significance to improving flight safety, the mission execution rate, and environmental adaptability. The method of deep reinforcement learning makes the autonomous maneuvering decision of UAVs possible. However, the current algorithm is prone to low training efficiency and poor performance when dealing with complex continuous maneuvering problems. In order to further improve the autonomous maneuvering level of UAVs and explore safe and efficient maneuvering methods in complex environments, a maneuvering decision-making method based on hierarchical reinforcement learning and ‌Proximal Policy Optimization (PPO) is proposed in this paper. By introducing the idea of hierarchical reinforcement learning into the PPO algorithm, the complex problem of UAV maneuvering and obstacle avoidance is separated into high-level macro-maneuver guidance and low-level micro-action execution, greatly simplifying the task of addressing complex maneuvering decisions using a single-layer PPO. In addition, by designing static/dynamic threat zones and varying their quantity, size, and location, the complexity of the environment is enhanced, thereby improving the algorithm’s adaptability and robustness to different conditions. The experimental results indicate that when the number of threat targets is five, the success rate of the H-PPO algorithm for maneuvering to the designated target point is 80%, which is significantly higher than the 58% rate achieved by the original PPO algorithm. Additionally, both the average maneuvering distance and time are lower than those of the PPO, and the network computation time is only 1.64 s, which is shorter than the 2.46 s computation time of the PPO. Additionally, as the complexity of the environment increases, the H-PPO algorithm outperforms other compared networks, demonstrating the effectiveness of the algorithm constructed in this paper for guiding intelligent agents to autonomously maneuver and avoid obstacles in complex and time-varying environments. This provides a feasible technical approach and theoretical support for realizing autonomous maneuvering decisions in UAVs.

A class of discontinuous Galerkin methods for nonlinear variational problems

In the context of discontinuous Galerkin methods, we study approximations of nonlinear variational problems. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using Γ \Gamma -convergence arguments we show that for convex energies the discrete minimisers converge to a minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. In addition, we provide error analysis in the case where the solution of the continuous minimization problem is smooth enough. Under appropriate assumptions we show local existence, uniqueness and corresponding optimal order error estimates for the discrete Euler-Lagrange equations of our scheme. We finally substantiate the feasibility of our methods by numerical examples.

Research on the rapid diagnosis method for hunting of high-speed trains

PurposeHigh-speed turnouts are more complex in structure and thus may cause abnormal vibration of high-speed train car body, affecting driving safety and passenger riding experience. Therefore, it is necessary to analyze the data characteristics of continuous hunting of high-speed trains passing through turnouts and propose a diagnostic method for engineering applications.Design/methodology/approachFirst, Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) is performed to determine the first characteristic component of the car body’s lateral acceleration. Then, the Short-Time Fourier Transform (STFT) is performed to calculate the marginal spectra. Finally, the presence of a continuous hunting problem is determined based on the results of the comparison calculations and diagnostic thresholds. To improve computational efficiency, permutation entropy (PE) is used as a fast indicator to identify turnouts with potential problems.FindingsUnder continuous hunting conditions, the PE is less than 0.90; the ratio of the maximum peak value of the signal component to the original signal peak value exceeded 0.7, and there is an energy band in the STFT time-frequency map, which corresponds to a frequency distribution range of 1–2 Hz.Originality/valueThe research results have revealed the lateral vibration characteristics of the high-speed train’s car body during continuous hunting when passing through turnouts. On this basis, an effective diagnostic method has been proposed. With a focus on practical engineering applications, a rapid screening index for identifying potential issues has been proposed, significantly enhancing the efficiency of diagnostic processes.

On Approximability of Steiner Tree in $\ell_p$-metrics

In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric. In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics.Comment: Abstract shortened. Journal version for TheoretiCS

Two new extragradient algorithms for solving non-monotone VIs in Hilbert spaces

In this work, we propose two new iterative schemes for finding an element of the set of solutions of a non-monotone, Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak convergence theorems of the sequence generated by the proposed algorithms are presented. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms.

ЗАДАЧА ПРОДОВЖЕННЯ ПЛОСКОМЕРИДІАННОГО МАГНІТОСТАТИЧНОГО ПОЛЯ З ПЛОСКОЇ ГРАНИЧНОЇ ПОВЕРХНІ ФЕРОГМАГНЕТИКА

The problem of axisymmetric steady magnetic field continuation from flat boundary surface of ferromagnetic is formulated for magnetic flux and scalar potential. Boundary conditions for magnetic flux are not classical because on boundary magnetic flux is unknown and its normal derivative equal zero. The formulation for scalar potential is Cauchy's problem for elliptical partial differential equation. Analytical solutions of the problem are obtained by method of partial solutions, which depend on parameter continuously, and Hankel's integral transformation. It is shown that there are similar properties in problems of axisymmetric fields continuation from flat boundaries of ideal conductor for magnetic field and conductor for steady electric field. It is fixed that field lines, which bound unknown profile, are determined directly by solution of Cauchy's problem for one from two functions. Equipotential lines are calculated to determine of electromagnetic pole profile. References 7, figures 2, table 1.

Almost-Sure Convergence of Iterates and Multipliers in Stochastic Sequential Quadratic Optimization

Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been limited to the asymptotic convergence of the expected value of a stationarity measure to zero. This is in contrast to the unconstrained setting in which almost-sure convergence guarantees (of the gradient of the objective to zero) can be proved for stochastic-gradient-based methods. In this paper, new almost-sure convergence guarantees for the primal iterates, Lagrange multipliers, and stationarity measures generated by a stochastic SQP algorithm in this subclass of methods are proved. It is shown that the error in the Lagrange multipliers can be bounded by the distance of the primal iterate to a primal stationary point plus the error in the latest stochastic gradient estimate. It is further shown that, subject to certain assumptions, this latter error can be made to vanish by employing a running average of the Lagrange multipliers that are computed during the run of the algorithm. The results of numerical experiments are provided to demonstrate the proved theoretical guarantees.

The Pareto Tracer for the treatment of degenerated multi-objective optimization problems

Multi-objective continuation algorithms are very powerful tools for the numerical treatment of continuous multi-objective optimization problems (MOPs). All of these methods, however, are based on certain regularity assumptions that imply that the solution set of the given MOP forms, at least locally, objects of below a certain dimension. While this is indeed the case for most problems, there exist examples where the Pareto set/front is lower-dimensional, which are called degenerated cases. This article presents and discusses a new predictor step designed for use within multi-objective continuation methods that automatically detects (numerical) degeneration and performs movements in essential directions. Furthermore, this predictor is integrated into the multi-objective continuation method ‘Pareto Tracer’. The effectiveness of this new continuation method is demonstrated on selected benchmark problems, illustrating its capability to handle both degenerated and non-degenerated MOPs efficiently.

Floorplanning with I/O Assignment via Feasibility-Seeking and Superiorization Methods.

The feasibility-seeking approach offers a systematic framework for managing and resolving intricate constraints in continuous problems, making it a promising avenue to explore in the context of floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be expressed as the union of convex sets. However, conventional projection-based algorithms for feasibility-seeking do not guarantee convergence in such situations, which are also heavily influenced by the initialization. We present a quantitative property about the choice of the initial point that helps good initialization and analyze the occurrence of the oscillation phenomena for bad initialization. In implementation, we introduce a resetting strategy aimed at effectively reducing the problem of algorithmic divergence in the projection-based method used for the feasibility-seeking formulation. Furthermore, we introduce the novel application of the superiorization method (SM) to floorplanning, which bridges the gap between feasibility-seeking and constrained optimization. The SM employs perturbations to steer the iterations of the feasibility-seeking algorithm towards feasible solutions with reduced (not necessarily minimal) total wirelength. Notably, the proposed algorithmic flow is adaptable to handle various constraints and variations of floorplanning problems, such as those involving I/O assignment. To evaluate the performance of Per-RMAP, we conduct comprehensive experiments on the MCNC benchmarks and GSRC benchmarks. The results demonstrate that we can obtain legal floorplanning results 166× faster than the branch-and-bound (B&B) method while incurring only a 5% wirelength increase compared to the optimal results. Furthermore, we evaluate the effectiveness of the algorithmic flow that considers the I/O assignment constraints, which achieves an 6% improvement in wirelength. Besides, considering the soft modules with a larger feasible solution space, we obtain 15% improved runtime compared with PeF, the state-of-the-art analytical method. Moreover, we compared our method with Parquet-4 and Fast-SA on GSRC benchmarks which include larger-scale instances. The results highlight the ability of our approach to maintain a balance between floorplanning quality and efficiency.

Polyvinyl chloride (PVC), its additives, microplastic and human health: Unresolved and emerging issues.

Polyvinyl chloride (PVC), a commonly used plastic across Europe, poses a number of risks at various stages of its life cycle. The carcinogenicity of PVC monomer, the need to use high number and volume of problematic additives, the easiness of fragmentation compared to other thermoplastics, the high volume of use in everyday products and the resulting extent to which European population is potentially exposed to both microplastics and chemicals and, finally, continuous problems during waste management, have raised concerns about impacts of PVC on human health and the environment for decades. As far back as in 2000, the European Commission recognized that PVC causes a wide range of serious problems for the environment and human health. More recently, in April 2022, PVC and its additives were included in the European Union's Restrictions Roadmap, and the European Chemicals Agency's investigation ruled that, to limit the use of some additives and to minimize releases of PVC microparticles, regulatory action would be necessary. Additionally, the Global Plastics Treaty discussions emphasise a need to ensure that plastics that remain in the economy are free of hazardous chemicals, including hazardous polymers. In this paper, we reviewed the available data on PVC microplastic, additives, the end of life options of products made of PVC, and how they all are connected. It is crucial to consider this polymer within the broader context of chemical pollution and circular economy, acknowledging that changes in how we manage our resources are necessary to achieve the goal for a truly non-toxic environment in the future.

Reinforcement learning-driven dual neighborhood structure artificial bee colony algorithm for continuous optimization problem

Reinforcement learning-driven dual neighborhood structure artificial bee colony algorithm for continuous optimization problem

Optimization design of multi-scale TPMS lattices based on geometric continuity fusion and strain energy driven

3D lattice structure is a kind of advanced metamaterial structure with excellent performance, such as lightweight, high specific strength and stiffness, shock damping, heat dissipation, and electromagnetic shielding capabilities. With the development of additive manufacturing technology, it has been widely concerned and developed rapidly in recent years. In order to further integrate the performance of lattice structures with mesoscale units into multi-scale optimization, this paper proposes an optimization design method for the multi-scale TPMS (Triply Periodic Minimal Surfaces) lattices based on geometric continuity fusion and strain energy driven. Regarding the challenge of geometric continuity problem of complex transition boundaries in multi-TPMS lattices, interface interpolation and density factor optimization methodologies are studied. At the same time, the mapping mechanism between lattice configuration and mechanical properties of multi-TPMS lattices is discussed in view of the requirements of multi-scale design from macro to micro configurations. Then, the multi-TPMS lattice density and configuration fields in the design domain are co-designed with the strain energy driven. Simulation and experimental results show that the mechanical properties of the multi-scale TPMS lattice designed by the proposed method are significantly improved compared with the traditional single-configuration gradient lattice, including the stiffness of the cantilever beam optimized structure is improved by 20.5% and the flexural stiffness of the three-point flexural beam optimized structure is improved by 51.3%.

Platon et la question de l’héritage

This paper examines the question of inheritance in Plato's thought. The aim is to show that in his major dialogues, such as the Laws and the Republic, Plato places the notion of inheritance at the heart of his thought, striving to provide an answer to the problem of continuity in the moment of rupture that is death. He was the voice of the society of his day, proposing that inheritance be passed on to the preferred child. The founder of the academy adheres to a commonly accepted idea: the idea of passing on the assets we accumulate to our loved ones according to well-defined rules. We can deduce from his analysis that only human beings are entitled to a legacy. Only they deserve to be elevated to a higher level through inheritance. The reason is that they have abilities unknown to animals: abilities that enable them to safeguard and grow what has been passed on to them.

INNOVATIVE IMPROVED APPROXIMATE SOLUTION METHOD FOR THE INTEGER KNAPSACK PROBLEM, ERROR COMPRESSION AND COMPUTATIONAL EXPERIMENTS

Context. Mathematical models of many optimization problems encountered in economics and engineering are taken in the form of an integer knapsack problem. Since this problem belongs to the class of “NP-complete”, that is, “hard to solve” problems, the number of operations required by known methods to find its optimal solution is exponential. This does not allow solving large-scale problems in real time. Therefore, various and fast working approximate solution methods of this problem have been developed. However, it is known that the approximate solution provided by those methods can differ significantly from the optimal solution in most cases. Therefore, after taking any approximate solution as a starting point, there is a demand to develop methods for its further improvement. Development of such methods has both theoretical and great practical importance. Objective. The main purpose solving of this issue is as follows. The main purpose in performing this work is to first find an initial approximate solution of the problem using any known method, and then work out an algorithm for successively further improvement of this solution. For this purpose, the set of numbers with which the coordinates of the optimal solution and the found approximate solution can differ should be determined. After that, new solutions should be constructed by assigning possible values to the unknowns corresponding to the numbers in that set, and the best among these solutions should be selected. However, the algorithm for constructing such a solution should be simple, require a small number of operations, not cause difficulties from the point of view of programming, be new and be applicable to practical issues. Method. The essence of the proposed method consists of the following. First, the initial approximate solution of the considered problem and the value of the objective function corresponding to this solution are found by a known rule. After that, the optimal solution of the problem is easily found by a known method, without taking into account the condition that the unknowns are integers. Obviously, this solution can take at most one coordinate fractional value. It is assumed that the coordinates of the optimal solution of the integer knapsack problem and the initial approximate solution may differ around a certain fractional coordinate of the optimal solution of the continuous problem. Then, the minimum number of non-zero coordinates and zero coordinates in the optimal solution is found. Corresponding theorems have been proved for this. It is assumed that the different coordinates of the optimal solution and the initial approximate solution located between those minimal numbers. Therefore, the best solution can be selected by successively changing the coordinates between those minimum numbers one by one. Results. Extensive calculation experiments were conducted with the application of the proposed method.To have a high quality of this method was confirmed once again through experiments. Conclusions. The proposed method is new, simple in nature, easy to consider from the programming point of view, and has important practical importance. Thus, we call this solution the innovative improved approximate solution.

Manifestations of the phatic principle in the 1920s–1930s national literatures of the North Caucasus

The work contains analytical information about the processes of aesthetic unification and national identity of the North Caucasian literatures of the early twentieth century. The consideration is based on reflections on the nature of the ideological factors that influenced the division of national literatures into "traditional" and "newly written". The problem of continuity and isolation of the national literatures of the North Caucasus, the issues of ideological and artistic enrichment of Balkar literature with the principles of Soviet literature are analyzed. The methodology of the article is based on the Marxist-Leninist concept of the legacy of the past, which formed new judgments about national literatures. It is about the preservation of archetypes of different levels and comparing the degree of this preservation in the North Caucasian and Russian literatures.

Continuity of educational programs between organizations of preschool and general education (part 1)

This article is devoted to the analysis of the main approaches to the problem of ensuring the continuity of the organization of educational activities between organizations of preschool and general education. Based on the analysis of regulatory documents on this topic, the Federal Work Program of education, which is a structural element of the Federal Educational Program of preschool education, is analyzed. The article provides an overview of the directions, goals, values and targets of preschool education. The analysis of partial programs aimed at the implementation of different directions of preschool education is proposed. The presented material is the first part of the review of the problem of continuity of educational activities between preschool and general education organizations.

Reduced Order Modelling of Fully Coupled Electro‐Mechanical Systems Through Invariant Manifolds With Applications to Microstructures

ABSTRACTThis article presents the first application of the direct parametrisation method for invariant manifolds to a fully coupled multiphysics problem involving the nonlinear vibrations of deformable structures subjected to an electrostatic field. The formulation proposed is intended for model order reduction of electrostatically actuated resonating Micro‐Electro‐Mechanical Systems (MEMS). The continuous problem is first rewritten in a manner that can be directly handled by the parametrisation method, which relies upon automated asymptotic expansions. A new mixed fully Lagrangian formulation is thus proposed, which contains only explicit polynomial nonlinearities, which is then discretised in the framework of finite element procedures. Validation is performed on the classical parallel plate configuration, where different formulations using either the general framework or an approximation of the electrostatic field due to the geometric configuration selected are compared. Reduced‐order models along these formulations are also compared to full‐order simulations operated with a time integration approach. Numerical results show a remarkable performance both in terms of accuracy and the wealth of nonlinear effects that can be accounted for. In particular, the transition from hardening to softening behaviour of the primary resonance while increasing the constant voltage component of the electric actuation is recovered. Secondary resonances leading to superharmonic and parametric resonances are also investigated with the reduced‐order model.

High-dimensional experiments for the downward continuation using the LRFMP algorithm

AbstractTime-dependent gravity data from satellite missions like GRACE-FO reveal mass redistribution in the system Earth at various time scales: long-term climate change signals, inter-annual phenomena like El Niño, seasonal mass transports and transients, e. g. due to earthquakes. For this contemporary issue, a classical inverse problem has to be considered: the gravitational potential has to be modelled on the Earth’s surface from measurements in space. This is also known as the downward continuation problem. Thus, it is important to further develop current mathematical methods for such inverse problems. For this, the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed within the last decade. Their unique feature is the combination of local as well as global trial functions in the approximative solution of an inverse problem such as the downward continuation of the gravitational potential. In this way, they harmonize the ideas of a traditional spherical harmonic ansatz and the radial basis function approach. Previous publications on these methods showed proofs of concept. In this paper, we report on the progress of our developments towards more realistic scenarios. In particular, we consider the methods for high-dimensional experiment settings with more than 500 000 grid points which yields a resolution of 20 km at best on a realistic satellite geometry. We also explain the changes in the methods that had to be done to work with such a large amount of data. The corresponding code (updated for big data use) is available at https://doi.org/10.5281/zenodo.8223771 under the licence CC BY-NC-SA 3.0 Germany.

SPECTRAL METHODS FOR SOLVING DIFFERENTIAL AND FUNCTIONAL EQUATIONS

The operator approach previously developed for the spectral method using Legendre polynomials is generalized here to any systems of basis functions (not necessarily orthogonal) that satisfy two conditions: the result of the operation of multiplication by x or differentiation with respect to x is expressed in the same functions. All systems of classical orthogonal polynomials meet these conditions. In particular, a spectral method utilizing Chebyshev polynomials is constructed, which is most efficient for numerical calculations. This method is applied for the numerical solution of linear functional equations that arise in generalized series summation problems, aswell as in problems of analytic continuation of discrete mappings. It is also shown how these methods solve nonstandard and nonlinear boundary value problems for which conventional algorithms are not applicable.

An Evolutionary Algorithm for Expensive Mixed-Integer Black-Box Optimization with Explicit Constraints

Determining the feasibility of a candidate solution to a constrained black-box optimization problem may be similarly expensive compared to the process of determining its quality, or it may be much cheaper. Constraints that allow obtaining degrees of feasibility or constraint violation without incurring significant computational costs are referred to as explicit. We present an evolutionary algorithm for solving mixed-integer black-box optimization problems where objective function evaluations are expensive but constraints are explicit. We do not assume relaxability of the constraints. The method wraps active-set evolution strategies, an algorithm for solving continuous black-box problems with explicit constraints, in a branching mechanism that allows enforcing integrality constraints. In computer experiments we demonstrate that the algorithm solves a set of mixed-integer problems with significantly fewer objective function evaluations than several algorithms that do not exploit the explicitness of the constraints.