scholarly journals On stochastic recursive equations of sum and max type

2006 ◽  
Vol 43 (03) ◽  
pp. 687-703 ◽  
Author(s):  
Ludger Rüschendorf
Keyword(s):  
Optimization Problems ◽  
Analysis Of Algorithms ◽  
Homogeneous Equation ◽  
Inhomogeneous Equation ◽  
Recursive Structure ◽  
Combinatorial Optimization Problems ◽  
Uniqueness Results ◽  
Nonnegative Solutions ◽  
Recursive Equations ◽  
Stochastic Recursive Equations

In this paper we consider stochastic recursive equations of sum type,, and of max type,, whereAi,bi, andbare random, (Xi) are independent, identically distributed copies ofX, anddenotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimalLs-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

scholarly journals On stochastic recursive equations of sum and max type

2006 ◽  
Vol 43 (3) ◽  
pp. 687-703 ◽  
Author(s):  
Ludger Rüschendorf
Keyword(s):  
Optimization Problems ◽  
Analysis Of Algorithms ◽  
Homogeneous Equation ◽  
Inhomogeneous Equation ◽  
Recursive Structure ◽  
Combinatorial Optimization Problems ◽  
Uniqueness Results ◽  
Nonnegative Solutions ◽  
Recursive Equations ◽  
Stochastic Recursive Equations

In this paper we consider stochastic recursive equations of sum type, , and of max type, , where Ai, bi, and b are random, (Xi) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

scholarly journals An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 225
Author(s):  
José García ◽  
Gino Astorga ◽  
Víctor Yepes
Keyword(s):  
Transfer Functions ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Set Covering ◽  
Covering Problem ◽  
Perturbation Operator ◽  
K Nearest Neighbors ◽  
Combinatorial Optimization Problems ◽  
Box Plots ◽  
Metaheuristic Techniques

The optimization methods and, in particular, metaheuristics must be constantly improved to reduce execution times, improve the results, and thus be able to address broader instances. In particular, addressing combinatorial optimization problems is critical in the areas of operational research and engineering. In this work, a perturbation operator is proposed which uses the k-nearest neighbors technique, and this is studied with the aim of improving the diversification and intensification properties of metaheuristic algorithms in their binary version. Random operators are designed to study the contribution of the perturbation operator. To verify the proposal, large instances of the well-known set covering problem are studied. Box plots, convergence charts, and the Wilcoxon statistical test are used to determine the operator contribution. Furthermore, a comparison is made using metaheuristic techniques that use general binarization mechanisms such as transfer functions or db-scan as binarization methods. The results obtained indicate that the KNN perturbation operator improves significantly the results.

scholarly journals ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1456
Author(s):  
Stefka Fidanova ◽  
Krassimir Todorov Atanassov
Keyword(s):  
Decision Making ◽  
Knapsack Problem ◽  
Propositional Logic ◽  
Optimization Problems ◽  
Real Life ◽  
Combinatorial Optimization Problems ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Multiple Constraint

Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between the best methods, that solves combinatorial optimization problems. The method mimics behavior of the ants in the nature, when they look for a food. One of the algorithm parameters is called pheromone, and it is updated every iteration according quality of the achieved solutions. The intuitionistic fuzzy (propositional) logic was introduced as an extension of Zadeh’s fuzzy logic. In it, each proposition is estimated by two values: degree of validity and degree of non-validity. In this paper, we propose two variants of intuitionistic fuzzy pheromone updating. We apply our ideas on Multiple-Constraint Knapsack Problem (MKP) and compare achieved results with traditional ACO.

scholarly journals Optimization of the composition in a composite material for microelectronics application using the Ising model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yoshihiko Imanaka ◽  
Toshihisa Anazawa ◽  
Fumiaki Kumasaka ◽  
Hideyuki Jippo
Keyword(s):  
Thermal Conductivity ◽  
Thermal Expansion ◽  
Ising Model ◽  
Specific Gravity ◽  
Simulated Annealing Algorithm ◽  
Optimization Problems ◽  
Industrial Applications ◽  
Thermal Dissipation ◽  
Materials Informatics ◽  
Combinatorial Optimization Problems

AbstractTailored material is necessary in many industrial applications since material properties directly determine the characteristics of components. However, the conventional trial and error approach is costly and time-consuming. Therefore, materials informatics is expected to overcome these drawbacks. Here, we show a new materials informatics approach applying the Ising model for solving discrete combinatorial optimization problems. In this study, the composition of the composite, aimed at developing a heat sink with three necessary properties: high thermal dissipation, attachability to Si, and a low weight, is optimized. We formulate an energy function equation concerning three objective terms with regard to the thermal conductivity, thermal expansion and specific gravity, with the composition variable and two constrained terms with a quadratic unconstrained binary optimization style equivalent to the Ising model and calculated by a simulated annealing algorithm. The composite properties of the composition selected from ten constituents are verified by the empirical mixture rule of the composite. As a result, an optimized composition with high thermal conductivity, thermal expansion close to that of Si, and a low specific gravity is acquired.

scholarly journals Service Restoring Reconfiguration for Distribution NetworksConsidering Uncertainty in Available Information

2021 ◽  
Vol 11 (9) ◽  
pp. 4169
Author(s):  
Hirotaka Takano ◽  
Junichi Murata ◽  
Kazuki Morishita ◽  
Hiroshi Asano
Keyword(s):  
Power Distribution ◽  
Solution Method ◽  
Optimization Problems ◽  
Problem Formulation ◽  
Distribution Networks ◽  
Accurate Information ◽  
Power Distribution Networks ◽  
Combinatorial Optimization Problems ◽  
Service Restoration ◽  
Available Information

The recent growth in the penetration of photovoltaic generation systems (PVs) has brought new difficulties in the operating and planning of electric power distribution networks. This is because operators of the distribution networks normally cannot monitor or control the output of the PVs, which introduces additional uncertainty into the available information that operations must rely on. This paper focuses on the service restoration of the distribution networks, and the authors propose a problem framework and its solution method that finds the optimal restoration configuration under extensive PV installation. The service restoration problems have been formulated as combinatorial optimization problems. They do, however, require accurate information on load sections, which is impractical in distribution networks with extensively installed PVs. A combined framework of robust optimization and two-stage stochastic programming adopted in the proposed problem formulation enables us to deal with the PV-originated uncertainty using readily available information only. In addition, this problem framework can be treated by a traditional solution method with slight extensions. The validity of the authors’ proposal is verified through numerical simulations on a real-scale distribution network model and includes a discussion of their results.

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