Optimal Set Partitioning Research Articles

Theory of Continuous Optimal Set Partitioning Problems as a Universal Mathematical Formalism for Constructing Voronoi Diagrams and Their Generalizations. I. Theoretical Foundations

This article considers a method for constructing Voronoi diagrams and their generalizations on the basis of the unified approach that consists of formulating a continuous optimal set partitioning problem with a partition quality criterion that provides appropriate types of Voronoi diagrams and applying the mathematical and algorithmic apparatus to solve such problems. This approach provides the ability not only to construct well-known Voronoi diagrams but also to design new ones.

Multi-view Ensemble Learning Using Optimal Feature Set Partitioning: An Extended Experiments and Analysis in Low Dimensional Scenario

Abstract Multi-view ensemble learning (MEL) has successfully addressed the issue related to high dimensionality of the data. It exploits the information of views of the data. To obtain views of data, an optimal feature set partitioning (OFSP) method [1] has been shown performance enhancement of MEL. Results of the experiments carried out on datasets and their statistical analysis show the effectiveness for classification problem in high dimensional scenario. In this work, classification performance of MEL using OFSP method has been analyzed in low dimensional situations. Therefore, experiments are performed on low dimension datasets using K-Nearest Neighbor (KNN), Naive Bayesian (NB) and Support Vector Machine (SVM) classification algorithms. The experimental results and their statistical analysis show that OFSP method is also effective in low dimensional environment.

Solving the Dynamic Optimal Set Partitioning Problem with Arrangement of Centers of Subsets

A mathematical model is presented for the dynamic problem of optimal partitioning of a set from a space Rn with arrangement of centers of subsets under joint constraints on the partition and phase variable. A method is described that solves this problem and synthesizes the essentials of the theory of continuous partitioning problems and optimal control theory for dynamic systems. A numerical algorithm for solving the problem and an analysis of the results of computational experiments are presented.

SUPER ORTHOGONAL SPACE TIME TRELLIS CODES OVER NAKAGAMI FADING MODEL

Performance evaluation of super orthogonal space-time trellis codes for non-frequency selective fading channels & frequency selective fading channels. The analysis is done in presence of fast fading, block fading and quasi-static fading in Rayleigh, and Nakhagami fast fading channels along with comparison. While providing full diversity and full rate, the structure of our new codes allows an increase in the coding gain. Not only does our new SOSTTC outperform the space-time trellis codes in the literature, but it also provides a systematic method for designing space time trellis codes at different rates and for different trellises. Since we have used orthogonal designs as the building blocks in our new SOSTTCs, the complexity of the decoding remains low while full diversity is guaranteed. Codes operating at different rates, up to the highest theoretically possible rate, for different number of states, can be designed by using our optimal set partitioning. In general, new SOSTTCs can provide a tradeoff between rate and coding gain while achieving full diversity.

On Dynamical Problem of Optimal Set Partition with Integral Constraints

On Dynamical Problem of Optimal Set Partition with Integral Constraints

On Solving and Properties of the Simplest Dynamical Problem of Optimal Set Partition Problem

On Solving and Properties of the Simplest Dynamical Problem of Optimal Set Partition Problem

Comparison of economics and mathematical modeling of economic structures in health area

The article describes the results of the analysis of economic and mathematical modeling. The method of optimal partitioning of sets for modeling economic structures in the field of health care.

Modeling broken characters recognition as a set-partitioning problem

This paper presents a novel technique for recognizing broken characters found in degraded text documents by modeling it as a set-partitioning problem (SPP). The proposed technique searches for the optimal set-partition of the connected components by which each subset yields a reconstructed character. Given the non-linear nature of the objective function needed for optimal set-partitioning, we design an algorithm that we call Heuristic Incremental Integer Programming (HIIP). The algorithm employs integer programming (IP) with an incremental approach using heuristics to hasten the convergence. The objective function is formulated as probability functions that reflect common OCR measurements – pattern resemblance, sizing conformity and distance between connected components. We applied the HIIP technique to Thai and English degraded text documents and achieved accuracy rates over 90%. We also compared HIIP against three competing algorithms and achieved higher comparative accuracy in each case.

General Scheme to Obtain Necessary Optimality Conditions for Continuous Optimal Set Partitioning Problems

General Scheme to Obtain Necessary Optimality Conditions for Continuous Optimal Set Partitioning Problems

Algorithm of Solving of Nonlinear Continuous Multicomponent Problem of Optimal Set Partitioning with Placement of Subsets Centers

Algorithm of Solving of Nonlinear Continuous Multicomponent Problem of Optimal Set Partitioning with Placement of Subsets Centers

Tolerance analysis in linear systems and linear programming

In many practical linear programming problems, it is often important to know how different optimality criteria (optimal solution, optimal basis, optimal partition, etc.) change under input data perturbations. Our aim is to compute tolerances (intervals) for the objective function and the right-hand side coefficients such that these coefficients can independently and simultaneously vary inside their tolerances while preserving the corresponding optimality criterion. We put tolerance analysis in a unified framework that is convenient for algorithmic processing and that is applicable not only in linear programming but for other linear systems as well. We survey the known results (pioneered by R.E. Wendell) and propose an improvement that is optimal in some sense (the resulting tolerances are maximal and they take into account proportionality). We apply our approach to several optimality invariancies: optimal basis, support set and optimal partition invariancy. Thus, the approach is useful not only for simplex method solvers, but for the interior points methods, too. We also discuss time complexity and show that it is NP-hard to determine the maximal tolerances.

Privacy-preserving data mining: A feature set partitioning approach

In privacy-preserving data mining (PPDM), a widely used method for achieving data mining goals while preserving privacy is based on k-anonymity. This method, which protects subject-specific sensitive data by anonymizing it before it is released for data mining, demands that every tuple in the released table should be indistinguishable from no fewer than k subjects. The most common approach for achieving compliance with k-anonymity is to replace certain values with less specific but semantically consistent values. In this paper we propose a different approach for achieving k-anonymity by partitioning the original dataset into several projections such that each one of them adheres to k-anonymity. Moreover, any attempt to rejoin the projections, results in a table that still complies with k-anonymity. A classifier is trained on each projection and subsequently, an unlabelled instance is classified by combining the classifications of all classifiers. Guided by classification accuracy and k-anonymity constraints, the proposed data mining privacy by decomposition (DMPD) algorithm uses a genetic algorithm to search for optimal feature set partitioning. Ten separate datasets were evaluated with DMPD in order to compare its classification performance with other k-anonymity-based methods. The results suggest that DMPD performs better than existing k-anonymity-based algorithms and there is no necessity for applying domain dependent knowledge. Using multiobjective optimization methods, we also examine the tradeoff between the two conflicting objectives in PPDM: privacy and predictive performance.

Solution of continuous problems of optimal covering with spheres using optimal set-partition theory

The paper considers a continuous problem of optimal c-sphere covering of a compact set from ? from E n with a given number of spheres of minimum radius and a problem of covering a set with the minimum number of spheres of given radius. Algorithms are proposed and substantiated to solve the problems using optimal set-partition theory and Shor's r-algorithm.

Solving a continuous nonlinear problem of optimal set partition with arrangement of subset centers in the case of a convex objective functional

A continuous nonlinear single-commodity problem of optimal partition of a set ? in an n-measurable Euclidean space into disjoint subsets with arrangement of their centers is analyzed using equality and inequality constraints in the case of a convex objective functional. A method and algorithm are proposed to solve this problem.

Super-orthogonal space-time trellis codes

We introduce a new class of space-time codes called super-orthogonal space-time trellis codes. These codes combine set partitioning and a super set of orthogonal space-time block codes in a systematic way to provide full diversity and improved coding gain over earlier space-time trellis code constructions. We also study the optimality of our set partitioning and provide coding gain analysis. Codes operating at different rates, up to the highest theoretically possible rate, for different number of states can be designed by using our optimal set partitioning. Super-orthogonal space-time trellis codes can provide a tradeoff between rate and coding gain. Simulation results show more than 2-dB improvements over the codes presented in the literature while providing a systematic design methodology.

On the Efficiency of a Global Non-differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning

The examined algorithm for global optimization of the multiextremal non-differentiable function is based on the following idea: the problem of determination of the global minimum point of the function f(x) on the set Ω(f(x) has a finite number of local minima in this domain) is reduced to the problem of finding all local minima and their attraction spheres with a consequent choice of the global minimum point among them. This reduction is made by application of the optimal set partitioning method. The proposed algorithm is evaluated on a set of well-known one-dimensional, two-dimensional and three-dimensional test functions. Recommendations for choosing the algorithm parameters are given.

Search for the Global Minimum of a Nondifferentiable Function by the Method of Optimal Set Partitioning

For finding the global minimum of a nondifferentiable function of several variables, which has a finite number of local minima in its domain of definition, an algorithm based on optimal set partitioning method is suggested. The algorithm enables one to simultaneously find all local minima and their attraction zones with subsequent selection of the global minimum point. As the optimality criterion of partitioning the definition domain of the minimized function into attraction zones of local minima, we take the minimal error admitted by incorrect referring the point x of attraction zone of one local minimum to the attraction zone of another local minimum. The algorithm is evaluated on a set of test one-dimensional functions. Recommendations for selecting parameters of the algorithm are given.

An Integral Simplex Algorithm for Solving Combinatorial Optimization Problems

In this paper a local integral simplex algorithm will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex algorithm creates and solves the problems in a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc. The solution to at least one of these subproblems is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is an optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n. A method for programming the algorithms in this paper to run on parallel computers is discussed briefly.

Methods of Optimal Set Partitioning for Some Start Control Problems with Quadratic Functional

Methods of Optimal Set Partitioning for Some Start Control Problems with Quadratic Functional

On Application of the Method of Potential Functions to the Stochastic Problem of the Optimal Partitioning of Sets

On Application of the Method of Potential Functions to the Stochastic Problem of the Optimal Partitioning of Sets