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We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

Chimp optimization algorithm (CHOA) is a recently developed nature-inspired technique that mimics the swarm intelligence of chimpanzee colonies. However, the original CHOA suffers from slow convergence and a tendency to reach local optima when dealing with multidimensional problems. To address these limitations, we propose TASR-CHOA, a twofold adaptive stochastic reinforced variant. The TASR-CHOA algorithm integrates two novel methodologies: a stochastic approach to improve the speed at which convergence is achieved and a dual adaptive weighting approach to optimize the exploration of early patterns, which refer to initial trends or behaviors in the algorithm’s convergence process during the early stages of iterations and the exploitation of subsequent tendencies, indicating how these initial trends develop over time as the algorithm iterates and refines its search. To evaluate TASR-CHOA, we apply it to 29 conventional optimization benchmark functions, 10 IEEE CEC-06 benchmarks, 30 complicated IEEE CEC-BC benchmark functions, and ten well-known benchmark real-world challenges. We evaluate TASR-CHOA against 4 categorical optimization techniques as well as 18 top IEEE CEC-BC algorithms. Based on our broad investigation done using three statistical tests, we claim that TASR-CHOA outperforms the majority of the algorithms since within a position takes the best place, 54 out of 73 evaluation functions and engineering problems. In other cases, the results are almost the same as those of SHADE and CMA-ES over several comparisons. As an illustrative application of this joint approach, a computer-aided fire detection task is performed using a deep convolutional neural network combined with TASR-CHOA. We also outline the algorithm executed in steps, indicating computational complexity, which is O( NI × NV ) + O( NI × NV ×6) + O( NI × NV  + 2× NI × NV ) as a function of number of individuals ( NI ) and dimensions ( NV ).

This study proposes a novel approach to constructing composite indicators, utilizing discriminant analysis to identify the determining factors for the diagnosis of multidimensional depression and to provide an index that represents the multidimensionality of this construct. By focusing solely on factors that significantly correlate with the diagnosis of multidimensional depression, this method provides a more precise and objective representation of the problem. The application of the method to the 2019 Brazilian Health Survey data demonstrated its effectiveness, resulting in a composite indicator that separates individuals who self-declare as having depression from individuals who self-declare as not having depression. The results highlight individuals who have a limiting chronic condition, high levels of education, less support from friends and family, perform unhealthy work, and are male. In contrast, individuals with the opposite characteristics are associated with a negative multidimensional depression diagnosis. The proposed composite indicator not only improves the measurement accuracy but also offers a new means of visualizing and understanding the multidimensional nature of depression diagnosis, providing valuable information for the formulation of targeted public health policies aimed at reducing the time for which people show symptoms of depression.

We study a truncated two-dimensional moment problem in terms of the Stieltjes transform. The set of the solutions is described by the Schur step-by-step algorithm, which is based on the continued fraction expansion of the solution. In particular, the obtained results are applicable to the two-dimensional moment problem for atomic measures.

Some iterative processes in Krylov subspaces are considered for solving systems of linear algebraic equations (SLAE) with high-order sparse matrices that arise in grid approximations of multidimensional boundary value problems. The SLAE are preconditioned by a uniform combined method that includes domain decomposition and recursive application of a two-grid algorithm, which are implemented by forming block-tridiagonal algebraic and grid structures inverted by using incomplete factorization and diagonal compensation. Stability and convergence of iterations are studied for some Stieltjes systems. Parallelization and generalization of the methods to wider classes of relevant practical problems are discussed.

In this paper, we present a solution method for the multidimensional knapsack problem (MKP) and the knapsack problem with forfeit sets (KPFS) using a population-based matheuristic approach. Specifically, the learning mechanism of the fixed set search (FSS) metaheuristic is combined with the use of integer programming for solving subproblems. This is achieved by introducing a new ground set of elements that can be used for both the MKP and the KPFS that aim to maximize the information provided by the fixed set. The method for creating fixed sets is also adjusted to enhance the diversity of generated solutions. Compared to state-of-the-art methods for the MKP and the KPFS, the proposed approach offers an implementation that can be easily extended to other variants of the knapsack problem. Computational experiments indicate that the matheuristic FSS is highly competitive to best-performing methods from the literature. The proposed approach is robust in the sense of having a good performance for a wide range of parameter values of the method.

Urban infrastructure, such as water supply networks, sewage systems, and electricity networks, is essential for the functioning of cities and, consequently, for the well-being of citizens. Despite its essentiality, the distribution of infrastructure in urban areas is not homogeneous, especially in cities in developing countries. Socially vulnerable areas often face significant deficiencies in sewage and road paving, exacerbating urban inequalities. In this regard, urban planners must consider the multiple elements of urban infrastructure and assess the compensation levels between them to reduce inequality effectively. In particular, the complexity of the problem necessitates considering the multidimensionality and heterogeneity of urban infrastructure. This complexity qualifies the operational framework of composite indicators as the natural solution to the problem. This study develops a new weighting system for the balanced expansion of urban infrastructures through composite indicators constructed by the Ordered Weighted Average operator. Implementing these weighting systems provides an opportunity to analyze urban infrastructure from different perspectives, offering transparency regarding the weaknesses and strengths of each perspective. This prevents unreliable representations from being used in decision-making and provides a solid basis for allocating investments in urban infrastructure. In particular, the study suggests that adopting weighting systems that prioritize intermediate values and avoid extreme values can lead to better resource allocation, helping to identify areas with deficient infrastructure and promoting more equitable urban development.

The multidimensional knapsack problem (MKP) is one of the widely known integer programming problems. The MKP has received significant attention from the operational research community for its large number of applications. Solving this NP-hard problem remains a very interesting challenge, especially when the number of constraints increases. In this paper we present a k-means transition ranking (KMTR) framework to solve the MKP. This framework has the property to binarize continuous population-based metaheuristics using a data mining k-means technique. In particular we binarize a Cuckoo Search and Black Hole metaheuristics. These techniques were chosen by the difference between their iteration mechanisms. We provide necessary experiments to investigate the role of key ingredients of the framework. Finally to demonstrate the efficiency of our proposal, MKP benchmark instances of the literature show that KMTR competes with the state-of-the-art algorithms.

Machine Learning (ML) has gained much importance in recent years as many of its effective applications are involved in different fields, healthcare, banking, trading, gaming, etc. Similarly, Combinatorial Optimisation (CO) keeps challenging researchers by new problems with more complex constraints. Merging both fields opens new horizons for development in many areas. This study investigates how effective is to solve CO problems by ML methods. The work considers the Multidimensional Knapsack Problem (MKP) as a study case, which is an np-hard CO problem well-known for its multiple applications. The proposed approach suggests to use solutions of small-size MKP to build models with different ML methods; then, to apply the obtained models on large-size MKP to predict their solutions. The features consist of scores calculated based on information about items while the labels consist of decision variables of optimal solutions calculated from applying CPLEX Solver on small-size MKP. Supervised ML methods build models that help to predict structures of large-size MKP solutions and build them accordingly. A comparison of five ML methods is conducted on standard data set. The experiments showed that the tested methods were able to reach encouraging results. In addition, the study proposes a Genetic Algorithm (GA) that exploits ML outputs essentially in initialisation operator and to repair unfeasible solutions. The algorithm denoted GaPR explores the ML solution neighbourhood as a way of intensification to approach optimal solutions. The carried out experiments indicated that the approach was effective and competitive.

We study the recent metaheuristic search algorithm for the multidimensional assignment problem (MAP) using fitness landscape theory. The analyzed algorithm performs a very large-scale neighborhood search on a set of feasible solutions to the problem. We derive properties of the landscape graphs that represent these very large-scale search algorithms acting on the solutions of the MAP. In particular, we show that the search graph is a generalization of a hypercube. We extend and generalize the original very large-scale neighborhood search to develop the variable neighborhood search. The new search is capable of searching even larger large-scale neighborhoods. We perform numerical analyses of the search graph structures for various problem instances of the MAP and different neighborhood structures of the MAP algorithm based on a very large-scale search. We also investigate the correlation between fitness (i.e., objective values) and distance (i.e., path lengths) of the local minima (i.e., sinks of the landscape). Our results can be used to design improved search-based metaheuristics for the MAP.