Continuous Problem Research Articles

Code and Data Repository for Ranking Model Averaging: Ranking Based on Model Averaging

Ranking problems are commonly encountered in practical applications, including order priority ranking, wine quality ranking, and piston slap noise performance ranking. The responses of these ranking applications are often considered as continuous responses and there is uncertainty on which scoring function is used to model the responses. In this paper, we address the scoring function uncertainty of continuous response ranking problems by proposing a Ranking Model Averaging (RMA) method. With a set of candidate models varied by scoring functions, RMA assigns weights for each model determined by a K-fold cross-validation criterion based on pairwise loss.

Neural quantile optimization for edge–cloud networking

We seek the best traffic allocation scheme for the edge–cloud networking subject to SD-WAN architecture and burstable billing. First, we formulate a family of quantile-based integer programming problems for a fixed network topology with random parameters describing the traffic demands. Then, to overcome the difficulty caused by the discrete feature, we generalize the Gumbel-softmax reparameterization method to induce an unconstrained continuous optimization problem as a regularized continuation of the discrete problem. Finally, we introduce the Gumbel-softmax sampling neural network to solve optimization problems via unsupervised learning. The neural network structure reflects the edge–cloud networking topology and is trained to minimize the expectation of the cost function for unconstrained continuous optimization problems. The trained network works as an efficient traffic allocation scheme sampler, outperforming the random strategy in feasibility and cost value. Besides testing the quality of the output allocation scheme, we examine the generalization property of the network by increasing the time steps and the number of users. We also feed the solution to existing integer optimization solvers as initial conditions and verify the warm-starts can accelerate the short-time iteration process. The framework is general, and the decoupled feature of the random neural networks is adequate for practical implementations.

Improving quality of service for Internet of Things(IoT) in real life application: A novel adaptation based Hybrid Evolutionary Algorithm

In this paper, we address critical challenges in IoT sensor lifespan, service latency, and coverage area, all impacting energy consumption in smart agriculture applications. To enhance the quality of service (QoS) while prolonging the energy efficiency of smart sensors, a novel optimization algorithm is introduced. Referred to as the ”Adaptation-Based Hybrid Evolutionary Algorithm,” this innovative approach combines the strengths of Grey Wolf Optimizers (GWO) and Differential Evolution (DE) algorithms. The methodology involves a new adaptation-based strategy and incorporates a hybrid algorithm that synergizes the exploratory and exploitative capabilities of both GWO and DE algorithms. This hybrid approach is leveraged to meticulously select optimal mutation new adaptation services, drawing from the GWO and DE algorithm frameworks. Notably, the algorithm’s control parameters autonomously adjust through insights gained from prior evolutionary searches. Furthermore, we enhance the DE-based crossover technique by integrating the proficient search capabilities of the GWO algorithm, renowned for tackling continuous global optimization problems. To validate our approach, we apply it to IoT scenarios and optimize QoS through a fitness function that comprehensively accounts for energy consumption, coverage rate, lifespan, and latency. Comparative evaluations against standard algorithms underscore the superior performance of our proposed methodology, particularly evident in its application to IoT-smart agriculture settings.

FATA: An efficient optimization method based on geophysics

An efficient swarm intelligence algorithm is proposed to solve continuous multi-type optimization problems, named the fata morgana algorithm (FATA). By mimicking the process of mirage formation, FATA designs the mirage light filtering principle (MLF) and the light propagation strategy (LPS), respectively. The MLF strategy, combined with the definite integration principle, drives the algorithmic population to enhance FATA’s exploration capability. The LPS strategy, combined with the trigonometric principle, drives the algorithmic individual to improve the algorithm's convergence speed and exploitation capability. These two search strategies can better use FATA’s population and individual search capabilities. The FATA is compared with a broad array of competitive optimizers on 23 benchmark functions and IEEE CEC 2014 to verify the optimization capability. This work is designed separately for qualitative analysis, exploration and exploitation competence analysis, the analysis of avoiding locally optimal solutions, and comprehensive comparison experiments. The experimental results demonstrate the comprehensiveness and competitiveness of FATA for solving multi-type functions. Meanwhile, FATA is applied to three practical engineering optimization problems to evaluate its performance. Then, the algorithm obtains better results than its counterparts in engineering problems. According to the results, FATA has excellent potential to be used as an efficient computer-aided tool for dealing with practical optimization tasks. Source codes and related files are available at https://aliasgharheidari.com/FATA.html and other websites.

Viscosity solutions to the inhomogeneous reaction–diffusion equation involving the infinity Laplacian

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian: [Formula: see text] where the continuous function [Formula: see text] satisfies [Formula: see text] a positive function [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text]. Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For [Formula: see text] and the non-positive inhomogeneous term [Formula: see text] we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on [Formula: see text] and [Formula: see text] we obtain the optimal [Formula: see text] regularity across the free boundary [Formula: see text] Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case [Formula: see text]

Convergence analysis of an efficient scheme for the steady state second grade fluid model

We are interested in studying the stationary second grade fluid model in a bounded domain in R2. To approximate the solution of the continuous model, we propose a fully decoupled numerical scheme based on a splitting method combined with the use of the Grad–Div operator. This approach allows the complete decoupling of the three variables: velocity, pressure and vorticity. Each variable is computed using an iterative procedure, with the pressure step involving a simple L2-projection.We provide a proof of the convergence of the scheme to the continuous problem under smallness assumptions on the data. This theoretical analysis ensures the reliability of our method in approximating the behavior of the stationary second grade fluid model.Finally, we present several numerical tests to validate our approach. These tests illustrate the effectiveness and efficiency of our scheme in various scenarios, highlighting its potential applicability to a wide range of problems involving second grade fluids.

Application of speech recognition algorithm based on interactive artificial intelligence system in English video teaching system

Social robots are intelligent programs that can autonomously run, automatically publish information, and interact on social media platforms. This article focuses on the speech recognition algorithm of interactive artificial intelligence systems, combined with the Hanning window function and the Vitit algorithm, and explores its application value in English video teaching systems through detailed experiments. Adopting speech feature extraction technology provides strong support for the subsequent recognition process. The application of Hanning window function has a positive impact on speech signal preprocessing. The most indispensable algorithm in speech recognition models is the Vitit algorithm, which successfully solves the problem of time variation and continuity of speech signals. This study successfully applied speech recognition algorithms based on interactive artificial intelligence systems to English video teaching systems. By combining speech feature extraction, Hanning window function, and Vitit algorithm, a system with strong recognition ability and interactivity was constructed. The English video teaching system aims to improve students’ learning process. By combining interactive artificial intelligence speech recognition technology with English video teaching, it can improve students’ experience in learning English and also meet their personalized learning habits.

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

Finite element approximation of unique continuation of functions with finite dimensional trace

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

A new heuristic for the continuous fixed-radius disc cover problem

The continuous fixed-radius disc cover (CFDC) problem is a location model to cover the given demand points by finding the minimum number of fixed-radius discs in the plane. It is an NP-hard problem, and only a few approximate algorithms have been proposed over the past three decades. In the article, a new heuristic called dynamic reduction (DR) is introduced. The heuristic works by initiating an upper bound on the number of discs, and then reducing the number by removing redundant discs during updates to the disc centres. Experimental results, using different radius settings on some datasets of a travelling salesman problem library (TSP-Lib), show that the proposed heuristic greatly outperforms the recent approximation algorithm in accuracy. Notably, an enhancement of the heuristic is also provided.

An improved mountain gazelle optimizer based on chaotic map and spiral disturbance for medical feature selection.

Feature selection is an important solution for dealing with high-dimensional data in the fields of machine learning and data mining. In this paper, we present an improved mountain gazelle optimizer (IMGO) based on the newly proposed mountain gazelle optimizer (MGO) and design a binary version of IMGO (BIMGO) to solve the feature selection problem for medical data. First, the gazelle population is initialized using iterative chaotic map with infinite collapses (ICMIC) mapping, which increases the diversity of the population. Second, a nonlinear control factor is introduced to balance the exploration and exploitation components of the algorithm. Individuals in the population are perturbed using a spiral perturbation mechanism to enhance the local search capability of the algorithm. Finally, a neighborhood search strategy is used for the optimal individuals to enhance the exploitation and convergence capabilities of the algorithm. The superior ability of the IMGO algorithm to solve continuous problems is demonstrated on 23 benchmark datasets. Then, BIMGO is evaluated on 16 medical datasets of different dimensions and compared with 8 well-known metaheuristic algorithms. The experimental results indicate that BIMGO outperforms the competing algorithms in terms of the fitness value, number of selected features and sensitivity. In addition, the statistical results of the experiments demonstrate the significantly superior ability of BIMGO to select the most effective features in medical datasets.

Roulette wheel-based level learning evolutionary algorithm for feature selection of high-dimensional data

Feature selection in high-dimensional data is a large-scale sparse and discrete optimization problem. Most evolutionary algorithms are designed to tackle continuous optimization problems. However, when dealing with high-dimensional feature selection tasks, they often suffer from poor population diversity and are computationally expensive. To address these challenges, this work introduces a roulette wheel-based level learning evolutionary algorithm (RWLLEA). RWLLEA integrates two key components. Firstly, it employs a leveled population mode. Individuals from higher levels provide guidance to those at lower levels during the evolutionary process, thereby exploring the potential combinatorial effects among features. Secondly, recognizing the characteristics associated with the high-dimensional feature selection task, a roulette wheel-based update method is devised to dynamically reduce search space and harmonize the algorithm's exploitation and exploration capacities across different stages. The performance of the proposed method was evaluated by comparing it with six other feature selection techniques across a range of fifteen diverse datasets. The experimental findings demonstrate that the proposed method can achieve a reduced feature set with a shorter runtime and exhibit superior classification accuracy.

Metaheuristics for variable-size mixed optimization problems: A unified taxonomy and survey

Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle VMVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy or comprehensive survey for handling this important family of optimization problems. This paper presents an unified taxonomy for metaheuristic solutions for solving VMVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of VMVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

Advanced backtracking search for solving continuous optimization problems

Advanced backtracking search for solving continuous optimization problems

Opt2Vec - a continuous optimization problem representation based on the algorithm's behavior: A case study on problem classification

Characterization of the optimization problem is a crucial task in many recent optimization research topics (e.g., explainable algorithm performance assessment, and automated algorithm selection and configuration). The state-of-the-art approaches use exploratory landscape analysis to represent the optimization problem, where for each one, a set of features is extracted using a set of candidate solutions sampled by a sampling strategy over the whole decision space. This paper proposes a novel representation of continuous optimization problems by encoding the information found in the interaction between an algorithm and an optimization problem. The new problem representation is learned using the information from the states/positions in the optimization run trajectory (i.e., the candidate solutions visited by the algorithm). With the novel representation, the problem can be characterized dynamically during the optimization run, instead of using a set of candidate solutions from the whole decision space that have never been observed by the algorithm. The novel optimization problem representation is called Opt2Vec and uses an autoencoder type of neural network to encode the information found in the interaction between an optimization algorithm and optimization problem into an embedded subspace. The Opt2Vec representation efficiency is shown by enabling different optimization problems to be successfully identified using only the information obtained from the optimization run trajectory.

Optimized quantization parameter selection for video-based point cloud compression

Point clouds are sets of points used to visualize three-dimensional (3D) objects. Point clouds can be static or dynamic. Each point is characterized by its 3D geometry coordinates and attributes such as color. High-quality visualizations often require millions of points, resulting in large storage and transmission costs, especially for dynamic point clouds. To address this problem, the moving picture experts group has recently developed a compression standard for dynamic point clouds called video-based point cloud compression (V-PCC). The standard generates two-dimensional videos from the geometry and color information of the point cloud sequence. Each video is then compressed with a video coder, which converts each frame into frequency coefficients and quantizes them using a quantization parameter (QP). Traditionally, the QPs are severely constrained. For example, in the low-delay configuration of the V-PCC reference software, the quantization parameter values of all the frames in a group of pictures are set to be equal. We show that the rate-distortion performance can be improved by relaxing this constraint and treating the QP selection problem as a multi-variable constrained combinatorial optimization problem, where the variables are the QPs. To solve the optimization problem, we propose a variant of the differential evolution (DE) algorithm. Differential evolution is an evolutionary algorithm that has been successfully applied to various optimization problems. In DE, an initial population of randomly generated candidate solutions is iteratively improved. At each iteration, mutants are generated from the population. Crossover between a mutant and a parent produces offspring. If the performance of the offspring is better than that of the parent, the offspring replaces the parent. While DE was initially introduced for continuous unconstrained optimization problems, we adapt it for our constrained combinatorial optimization problem. Also, unlike standard DE, we apply individual mutation to each variable. Furthermore, we use a variable crossover rate to balance exploration and exploitation. Experimental results for the low-delay configuration of the V-PCC reference software show that our method can reduce the average bitrate by up to 43% compared to a method that uses the same QP values for all frames and selects them according to an interior point method.

On the combined inverse-square effect of multiple points in multidimensional space

The inverse-square law states that a source's effect is inversely proportional to the distance from that source squared. In continuous location problems, objective functions often obey the inverse-square law. This work shows that for any region in D dimensions, the maximum inverse-square effect is on the region's boundary if D<4, the minimum is on the boundary if D>4, and both the maximum and minimum are on the boundary if D=4.

Multi-Scale Feature Alignment for Continual Learning of Unlabeled Domains.

Methods for unsupervised domain adaptation (UDA) help to improve the performance of deep neural networks on unseen domains without any labeled data. Especially in medical disciplines such as histopathology, this is crucial since large datasets with detailed annotations are scarce. While the majority of existing UDA methods focus on the adaptation from a labeled source to a single unlabeled target domain, many real-world applications with a long life cycle involve more than one target domain. Thus, the ability to sequentially adapt to multiple target domains becomes essential. In settings where the data from previously seen domains cannot be stored, e.g., due to data protection regulations, the above becomes a challenging continual learning problem. To this end, we propose to use generative feature-driven image replay in conjunction with a dual-purpose discriminator that not only enables the generation of images with realistic features for replay, but also promotes feature alignment during domain adaptation. We evaluate our approach extensively on a sequence of three histopathological datasets for tissue-type classification, achieving state-of-the-art results. We present detailed ablation experiments studying our proposed method components and demonstrate a possible use-case of our continual UDA method for an unsupervised patch-based segmentation task given high-resolution tissue images. Our code is available at: https://github.com/histocartography/multi-scale-feature-alignment.

Joint Dual Feature Distillation and Gradient Progressive Pruning for BERT compression

The increasing size of pre-trained language models has led to a growing interest in model compression. Pruning and distillation are the primary methods employed to compress these models. Existing pruning and distillation methods are effective in maintaining model accuracy and reducing its size. However, they come with limitations. For instance, pruning is often suboptimal and biased by transforming it into a continuous optimization problem. Distillation relies primarily on one-to-one layer mappings for knowledge transfer, which leads to underutilization of the rich knowledge in teacher. Therefore, we propose a method of joint pruning and distillation for automatic pruning of pre-trained language models. Specifically, we first propose Gradient Progressive Pruning (GPP), which achieves a smooth transition of indicator vector values from real to binary by progressively converging the values of unimportant units’ indicator vectors to zero before the end of the search phase. This effectively overcomes the limitations of traditional pruning methods while supporting compression with higher sparsity. In addition, we propose the Dual Feature Distillation (DFD). DFD adaptively globally fuses teacher features and locally fuses student features, and then uses the dual features of global teacher features and local student features for knowledge distillation. This realizes a “preview-review” mechanism that can better extract useful information from multi-level teacher information and transfer it to student. Comparative experiments on the GLUE benchmark dataset and ablation experiments indicate that our method outperforms other state-of-the-art methods.

COVID-19 Pandemic Quarantines and Mental Health Among Adolescents in Norway

Adolescence is a critical developmental phase when mental health disorders, such as anxiety and depression, often emerge. Stringent public health measures and quarantine mandates during the COVID-19 pandemic could threaten adolescent mental health. To investigate the associations of public health measures and quarantine experiences with mental distress among Norwegian adolescents and to explore if certain vulnerability factors moderate these associations. This longitudinal cohort study used repeated measures to capture variations in mental distress explained by the stringency of public health measures and quarantine experiences. Data from the Norwegian Mother, Father, and Child cohort study were linked to national health registries and a national stringency index from April 1, 2020, to February 17, 2021. Participant included 7787 Norwegian adolescents aged 16 to 18 years. Data were analyzed from October 2022 to October 2023. Stringency index of public health measures and quarantine experiences including recent quarantine (within the last 2 weeks) and quarantine frequency (cumulative number of quarantine episodes). Mental distress was measured using the Hopkins Symptom Checklist across 6 data collection waves. In this study, 7787 participants were included in the analysis (4473 female [57%]; mean [SD] age, 17.0 [0.6] years). Stringent public health measures (β = 0.18; SE, 0.02; P < .001), recent quarantine (β = 0.11; SE, 0.02; P < .001), and frequent quarantine (β = 0.08; SE, 0.01; P < .001) were associated with higher levels of mental distress. The associations between public health measures and mental distress were not moderated by sex, age, prepandemic anxiety or depression, or genetic liability for mental health conditions. Frequency of quarantine appeared to be more strongly associated with mental distress among younger adolescents (β = -0.04; SE, 0.01; P = .008), those with parents with lower education (β = -0.04; SE, 0.01; P = .007), and those with lower genetic risk for depression (β = -0.03; SE, 0.01; P = .006). In this study, younger adolescents, those with parents with lower education, or those with low genetic liability for depression appeared more vulnerable when being quarantined several times. These findings emphasize the need for targeted support strategies to better protect adolescent well-being during future crises. Adolescents who experienced increased mental distress during the COVID-19 pandemic may be at risk of continued mental health problems and in need of ongoing support.