Partial Order Relation Research Articles

An algorithm to solve polytopic set optimization problem based on a partial set order relation

An algorithm to solve polytopic set optimization problem based on a partial set order relation

The Continuity and Convexity of a Nonlinear Scalarization Function with Applications in Set Optimization Problems Involving a Partial Order Relation

In this paper, we deal with the properties and applications of a nonlinear scalarization function for sets by using the Minkowski difference. Under some suitable assumptions, the continuity and convexity concerned with the nonlinear scalarization function for sets are presented. As applications, the path connectedness of the solution sets to set optimization problems and the continuity of the solution mappings of parametric set optimization problems are established. The results achieved do not impose the monotonicity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

Assessment of primary school in Changsha based on AISM method

To tackle the challenge of a one-size-fits-all and inconsistent approach to evaluating school development across districts, we have zeroed in on five key urban districts in Changsha City. Fourteen secondary indicators, spanning four critical categories — campus environment, teaching facilities, financial investment, and teacher resources — were selected to comprehensively assess the development of elementary schools in 2021. This assessment was conducted using the Advanced Interpretive Structural Model (AISM). Drawing on the principles of partial order set theory, AISM delves into the partial relationships among these indicators, considering the raw data matrix, optimistic positive and negative distances, and program fit degree. Subsequently, Haas map technology is employed to graphically represent the partial order relationship of elementary school development in each region, providing a clear picture of the hierarchical structure and merit ranking of the evaluation objects. The evaluation results reveal stark differences in the development of elementary schools in Changsha's districts, primary schools in Furong and Kaifu districts stand out for their superior development quality, while Yuhua and Yuelu districts lag behind in terms of elementary school development. Based on these findings, we propose policy recommendations aimed at achieving balanced primary education across districts and ensuring the adequacy of teacher resources.

A fusion model for multiple partial-order quotient spaces based on three-way decision

With a quotient space, a complex problem can be transformed into an appropriate granularity for processing. However, within a quotient space, there are no partial order relations between information granules. In practical decision-making, it may be necessary to deal with quotient spaces that have partial order relations between information granules. In this paper, first, the concept of a partial-order quotient space is proposed and some of its properties are studied. Second, a fusion model for multiple partial-order quotient spaces based on three-way decision is proposed, and some related theorems are proved. Finally, a case study of teacher evaluation, an experiment on real data, and several simulation experiments are presented to demonstrate the practicality of the proposed model in this paper.

Multi-Behavior Recommendation with Personalized Directed Acyclic Behavior Graphs

A well-developed recommendation system can not only leverage multi-typed interactions (such as page view, add-to-cart, and purchase) to better identify user preferences, but also demonstrate high performance, low complexity, and strong interpretability. However, many existing solutions for multi-behavior recommendation fall short of intuitive modeling of real-world scenarios, leading to overly complex models with massive parameters and cumbersome components. In particular, they share two critical limitations: (1) Some pioneering models are built upon the strict assumption of cascade effects across behaviors, which contradicts multifarious behavior paths in practical applications. (2) Existing approaches fail to explicitly capture the unique idiosyncrasies of users and even neglect the inherent nature of items involved in the multi-behavior interactions. To this end, we propose a novel Directed Acyclic Graph Convolutional Network (DA-GCN) for the multi-behavior recommendation task. Specifically, we pinpoint the partial order relations within the monotonic behavior chain and extend it to personalized directed acyclic behavior graphs to exploit behavior dependencies. Then, a GCN-based directed edge encoder is employed to distill rich collaborative signals embodied by each directed edge. In light of the information flows over the directed acyclic structure, we propose an attentive aggregation module to gather messages from all potential antecedent behaviors, representing distinct perspectives to understand the terminated behavior. Thus, we obtain comprehensive representations for the follow-up behavior through learnable distributions over its preceding behaviors, explicitly reflecting personalized interactive patterns of users and underlying properties of items simultaneously. Finally, we design a customized multi-task learning objective for flexible joint optimization. Extensive experiments on public benchmarking datasets fully demonstrate the superiority of DA-GCN with significant performance improvement and computational efficiency over a wide range of state-of-the-art methods. Our code is available at https://github.com/xizhu1022/DA-GCN .

On Ekeland Variational Principle and Its Applications Through Fuzzy Quasi Metric Spaces with Non-Archimedean t-norm

The aim of this article is to introduce Ekeland variational principle (EVP) and some results in fuzzy quasi metric space (FQMS) under the non-Archimedean \(t\)-norms. In this article the basic topological properties and a partial order relation are defined on FQMS. Utilizing Brézis-Browder principle on a partial order set, we extend the EVP to FQMS also. Moreover, we derive Takahashi’s minimization theorem, which ensures the existence of a solution of an optimal problem without taking the help of compactness and convexity properties on the underlying space. Furthermore, we give an equivalence chain between these two theorems. Finally, two fixed point results namely the Banach fixed point and the Caristi-Kirk fixed point theorems are described extensively.

Theorem of Alternative and Scalarization of Optimization Problems in Topological Vector Spaces

In this paper, we introduce the definitions of affinelike, preaffinelike, generalized affinelike, and generalized preaffinelike functions by use of "pointed convex cone", and prove that those definitions of generalized affine functions are all different. We discuss optimization problems in topological vector spaces and obtain a theorem of alternative and a scalarization theorem. Our inequalities are given by partial order relations. Our generalized affineness may be used for many discussions in mathematics or applied mathematics wherever the affineness is a condition.

A Recurrent Neural Network Approach for Constrained Distributed Fuzzy Convex Optimization.

This article investigates a class of constrained distributed fuzzy convex optimization problems, where the objective function is the sum of a set of local fuzzy convex objective functions, and the constraints include partial order relation and closed convex set constraints. In undirected connected node communication network, each node only knows its own objective function and constraints, and the local objective function and partial order relation functions may be nonsmooth. To solve this problem, a recurrent neural network approach based on differential inclusion framework is proposed. The network model is constructed with the help of the idea of penalty function, and the estimation of penalty parameters in advance is eliminated. Through theoretical analysis, it is proven that the state solution of the network enters the feasible region in finite time and does not escape again, and finally reaches consensus at an optimal solution of the distributed fuzzy optimization problem. Furthermore, the stability and global convergence of the network do not depend on the selection of the initial state. A numerical example and an intelligent ship output power optimization problem are given to illustrate the feasibility and effectiveness of the proposed approach.

A fault‐tolerant and scalable boosting method over vertically partitioned data

AbstractVertical federated learning (VFL) can learn a common machine learning model over vertically partitioned datasets. However, VFL are faced with these thorny problems: (1) both the training and prediction are very vulnerable to stragglers; (2) most VFL methods can only support a specific machine learning model. Suppose that VFL incorporates the features of centralised learning, then the above issues can be alleviated. With that in mind, this paper proposes a new VFL scheme, called FedBoost, which makes private parties upload the compressed partial order relations to the honest but curious server before training and prediction. The server can build a machine learning model and predict samples on the union of coded data. The theoretical analysis indicates that the absence of any private party will not affect the training and prediction as long as a round of communication is achieved. Our scheme can support canonical tree‐based models such as Tree Boosting methods and Random Forests. The experimental results also demonstrate the availability of our scheme.

J-selfadjoint matrix means and their indefinite inequalities

AbstractLet J be a non trivial involutive Hermitian matrix. Consider $${\mathbb {C}}^n$$ C n equipped with the indefinite inner product induced by J, $$[x,y]=y^*J x$$ [ x , y ] = y ∗ J x for all $$x,y\in {{\mathbb {C}}}^n,$$ x , y ∈ C n , which endows the matrix algebra $${\mathbb {C}}^{n\times n}$$ C n × n with a partial order relation $$\le ^J$$ ≤ J between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint $$\alpha $$ α -weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type.

Isomorphism on Complex Fuzzy Graph

Objective: To investigate isomorphism between two complex fuzzy graphs and prove it is an equivalence relation. The major objective of this research paper is to elucidate weak and strong isomorphism, and the study also endeavours to look at the complement of complex fuzzy graphs. Methods: Isomorphism is examined by comparing the membership values of nodes and arcs (both amplitude and phase). The same technique also proves further validation of the equivalence relation, which is also proven by the same technique. This criterion helps us to identify and formalise the isomorphic relationship between complex fuzzy graphs. Findings: This study demonstrates that isomorphism in complex fuzzy graphs allows for consideration of graph structures' differences and similarities. It provides for a better understanding of significant connections between complex fuzzy graphs. Novelty: By offering an elaborate perception of the idea of isomorphism in complex fuzzy graphs, this work advances the discipline. It seeks a better understanding of the complexities of equivalency determination for complicated structures, as well as the application of isomorphism theory to complex fuzzy graphs. Keywords: Complex fuzzy graph, Homomorphism, Isomorphism, Partial order relation, Self complementary

Personalized multi-attribute recommendation method based on partial order set

With the rapid development of big data and continuous optimization of online shopping platforms, personalized recommendation has become a standard feature of recommendation methods. In order to effectively provide personalized recommendations to customers, improve recommendation accuracy, and customer satisfaction, it is necessary to consider customers’ preferences for multiple product attributes when making product recommendations. However, existing recommendation methods require precise calculation of product attribute weights, which is computationally expensive, complex, and often results in unstable weight values. This paper proposes a multi-attribute recommendation method based on consumer decision preference information that overcomes the need for weights and reflects personalized customer preferences. Based on the acquisition of customer product attribute preference sequences, a partial order relation for recommended products is constructed using partial order set theory. Finally, the recommended products are determined through the partial order Hasse diagram, where the top layer elements of the Hasse diagram represent the recommended product set. This method addresses challenges that traditional content-based recommendations cannot overcome. The experiment in this paper uses a dataset of 30,000 records from Beeradvocate beer reviews. The experimental results show that, compared to traditional multi-attribute recommendation methods, this method only requires decision-maker preference information to complete product recommendations, requiring less information and having lower computational costs, resulting in more robust results.

Necessary and sufficient conditions of the enforcement problem for various argumentation semantics

Abstract In the research on dynamics of argumentation frameworks (AFs), the enforcement problem deals with changing an AF for the purpose of ensuring that a certain set of desired arguments becomes (part of) an extension. In this paper we focus on expansions of an AF where solely the addition of new arguments and attacks is allowed and the original framework remains unchanged. Existing results about the enforcement problem are all sufficient conditions, and we argue that necessary and sufficient conditions are essential concerning the solvability of the enforcement problem. A suite of jointly necessary and sufficient conditions are identified for the enforcement problem under normal expansion. These conditions are presented from multiple perspectives including the sub-strict versus non-strict problem, constructive versus eliminative conditions and arbitrary AFs versus even(odd)-length cycle free AFs, determined by the nature of various argumentation semantics. Such necessary and sufficient conditions are of the essence to the enforcement problem by defining requirements that are both minimum and absolute, which can be used to determine when the enforcement problem is (un)solvable. We further infer solely sufficient and solely necessary conditions based on the partial order relation (w.r.t. set-inclusion) among various types of AF extensions.

On Soft Bitopological Ordered Spaces

On Soft Bitopological Ordered Spaces

Teaching Partial Order Relations: A Programming Approach

Teaching Partial Order Relations: A Programming Approach

Community relations discovery methods for users in Fancircle based on sentiment analysis in China

PurposeThe identification of network user relationship in Fancircle contributes to quantifying the violence index of user text, mining the internal correlation of network behaviors among users, which provides necessary data support for the construction of knowledge graph.Design/methodology/approachA correlation identification method based on sentiment analysis (CRDM-SA) is put forward by extracting user semantic information, as well as introducing violent sentiment membership. To be specific, the topic of the implementation of topology mapping in the community can be obtained based on self-built field of violent sentiment dictionary (VSD) by extracting user text information. Afterward, the violence index of the user text is calculated to quantify the fuzzy sentiment representation between the user and the topic. Finally, the multi-granularity violence association rules mining of user text is realized by constructing violence fuzzy concept lattice.FindingsIt is helpful to reveal the internal relationship of online violence under complex network environment. In that case, the sentiment dependence of users can be characterized from a granular perspective.Originality/valueThe membership degree of violent sentiment into user relationship recognition in Fancircle community is introduced, and a text sentiment association recognition method based on VSD is proposed. By calculating the value of violent sentiment in the user text, the annotation of violent sentiment in the topic dimension of the text is achieved, and the partial order relation between fuzzy concepts of violence under the effective confidence threshold is utilized to obtain the association relation.

Partial order relation-based gene ontology embedding improves protein function prediction.

Protein annotation has long been a challenging task in computational biology. Gene Ontology (GO) has become one of the most popular frameworks to describe protein functions and their relationships. Prediction of a protein annotation with proper GO terms demands high-quality GO term representation learning, which aims to learn a low-dimensional dense vector representation with accompanying semantic meaning for each functional label, also known as embedding. However, existing GO term embedding methods, which mainly take into account ancestral co-occurrence information, have yet to capture the full topological information in the GO-directed acyclic graph (DAG). In this study, we propose a novel GO term representation learning method, PO2Vec, to utilize the partial order relationships to improve the GO term representations. Extensive evaluations show that PO2Vec achieves better outcomes than existing embedding methods in a variety of downstream biological tasks. Based on PO2Vec, we further developed a new protein function prediction method PO2GO, which demonstrates superior performance measured in multiple metrics and annotation specificity as well as few-shot prediction capability in the benchmarks. These results suggest that the high-quality representation of GO structure is critical for diverse biological tasks including computational protein annotation.

Some well known inequalities on two dimensional convex mappings by means of Pseudo $ \mathcal{L-R} $ interval order relations via fractional integral operators having non-singular kernel

<abstract><p>Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fejér, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines.</p></abstract>

Generalized conditional spacings and their stochastic properties

<p>This paper introduces the concept of generalized conditional spacings and establishes partial order relations between different generalized spacings. First, we derive the survival function of the generalized conditional spacings. Second, we construct the stochastic and hazard rate order relationships between different generalized conditional spacings and generalized normal conditional spacings, considering parent distributions that belong to the decreasing failure rate (DFR) and increasing likelihood rate (ILR) classes. Finally, for parent distributions within the DFR class, we obtain the dispersive order between different conditional spacings, along with an inequality for the variance. Additionally, we present illustrative examples involving Pareto and Gamma distributions.</p>

FUZZY GENERALISED LATTICEs (fuzzy gls) Based on fuzzy partial ordering relation (fuzzy porel)

A generalised lattice is an intermediate structure between directed poset and lattice. Every generalised lattice is a directed poset but the converse is not true. Every lattice is a generalised lattice but the converse is not true. A fuzzy relation defined on a non-empty set is called a fuzzy partial ordered relation if it satisfies reflexive, anti-symmetric and transitive. A non-empty set together with a fuzzy partial ordered relation is called a fuzzy poset. This article deals with the concept of FUZZY GENERALISED LATTICE (fuzzy gl) Based on fuzzy partial ordering relation (fuzzy porel).