scholarly journals Improving Lagrange Dual Bounds for Quadratic Extremal Problems

2020 ◽  
pp. 15-22
Author(s):  
Oleg Berezovskyi
Keyword(s):  
Lagrangian Relaxation ◽  
Optimization Problems ◽  
Problem Formulation ◽  
Extremal Problems ◽  
General Concept ◽  
Duality Gap ◽  
Sdp Relaxation ◽  
Dual Bound ◽  
Dual Bounds

Introduction. Due to the fact that quadratic extremal problems are generally NP-hard, various convex relaxations to find bounds for their global extrema are used, namely, Lagrangian relaxation, SDP-relaxation, SOCP-relaxation, LP-relaxation, and others. This article investigates a dual bound that results from the Lagrangian relaxation of all constraints of quadratic extremal problem. The main issue when using this approach for solving quadratic extremal problems is the quality of the obtained bounds (the magnitude of the duality gap) and the possibility to improve them. While for quadratic convex optimization problems such bounds are exact, in other cases this issue is rather complicated. In non-convex cases, to improve the dual bounds (to reduce the duality gap) the techniques, based on ambiguity of the problem formulation, can be used. The most common of these techniques is an extension of the original quadratic formulation of the problem by introducing the so-called functionally superfluous constraints (additional constraints that result from available constraints). The ways to construct such constraints can be general in nature or they can use specific features of the concrete problems. The purpose of the article is to propose methods for improving the Lagrange dual bounds for quadratic extremal problems by using technique of functionally superfluous constraints; to present examples of constructing such constraints. Results. The general concept of using functionally superfluous constraints for improving the Lagrange dual bounds for quadratic extremal problems is considered. Methods of constructing such constraints are presented. In particular, the method proposed by N.Z. Shor for constructing functionally superfluous constraints for quadratic problems of general form is presented in generalized and schematized forms. Also it is pointed out that other special techniques, which employ the features of specific problems for constructing functionally superfluous constraints, can be used. Conclusions. In order to improve dual bounds for quadratic extremal problems, one can use various families of functionally superfluous constraints, both of general and specific type. In some cases, their application can improve bounds or even provide an opportunity to obtain exact values of global extrema.

scholarly journals Dissecting the duality gap: the supporting hyperplane interpretation revisited

2021 ◽  
Author(s):  
Nils-Hassan Quttineh ◽  
Torbjörn Larsson
Keyword(s):  
Convex Optimization ◽  
Lagrangian Relaxation ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Duality Gap ◽  
Supporting Hyperplane ◽  
Solution Approach ◽  
Convex Optimization Problems ◽  
Near Optimality

AbstractWe revisit the classic supporting hyperplane illustration of the duality gap for non-convex optimization problems. It is refined by dissecting the duality gap into two terms: the first measures the degree of near-optimality in a Lagrangian relaxation, while the second measures the degree of near-complementarity in the Lagrangian relaxed constraints. We also give an example of how this dissection may be exploited in the design of a solution approach within discrete optimization.

scholarly journals Evolutionary Machine Learning: A Survey

2022 ◽  
Vol 54 (8) ◽  
pp. 1-35
Author(s):  
Akbar Telikani ◽  
Amirhessam Tahmassebi ◽  
Wolfgang Banzhaf ◽  
Amir H. Gandomi
Keyword(s):  
Machine Learning ◽  
Feature Selection ◽  
Optimization Problems ◽  
Ensemble Methods ◽  
Problem Formulation ◽  
Fitness Value ◽  
Future Work ◽  
Rule Optimization

Evolutionary Computation (EC) approaches are inspired by nature and solve optimization problems in a stochastic manner. They can offer a reliable and effective approach to address complex problems in real-world applications. EC algorithms have recently been used to improve the performance of Machine Learning (ML) models and the quality of their results. Evolutionary approaches can be used in all three parts of ML: preprocessing (e.g., feature selection and resampling), learning (e.g., parameter setting, membership functions, and neural network topology), and postprocessing (e.g., rule optimization, decision tree/support vectors pruning, and ensemble learning). This article investigates the role of EC algorithms in solving different ML challenges. We do not provide a comprehensive review of evolutionary ML approaches here; instead, we discuss how EC algorithms can contribute to ML by addressing conventional challenges of the artificial intelligence and ML communities. We look at the contributions of EC to ML in nine sub-fields: feature selection, resampling, classifiers, neural networks, reinforcement learning, clustering, association rule mining, and ensemble methods. For each category, we discuss evolutionary machine learning in terms of three aspects: problem formulation, search mechanisms, and fitness value computation. We also consider open issues and challenges that should be addressed in future work.

SDP RELAXATIONS FOR QUADRATIC OPTIMIZATION PROBLEMS DERIVED FROM POLYNOMIAL OPTIMIZATION PROBLEMS

2010 ◽  
Vol 27 (01) ◽  
pp. 15-38 ◽  
Author(s):  
MARTIN MEVISSEN ◽  
MASAKAZU KOJIMA
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Convergent Sequence ◽  
Quadratic Optimization ◽  
Optimization Techniques ◽  
Multivariate Polynomial ◽  
Sdp Relaxation ◽  
Polynomial Optimization Problem

Based on the convergent sequence of SDP relaxations for a multivariate polynomial optimization problem (POP) by Lasserre (2006), Waki et al. (2006) constructed a sequence of sparse SDP relaxations to solve sparse POPs efficiently. Nevertheless, the size of the sparse SDP relaxation is the major obstacle in order to solve POPs of higher degree. This paper proposes an approach to transform general POPs to quadratic optimization problems (QOPs), which allows to reduce the size of the SDP relaxation substantially. We introduce different heuristics resulting in equivalent QOPs and show how sparsity of a POP is maintained under the transformation procedure. As the most important issue, we discuss how to increase the quality of the SDP relaxation for a QOP. Moreover, we increase the accuracy of the solution of the SDP relaxation by applying additional local optimization techniques. Finally, we demonstrate the high potential of this approach through numerical results for large scale POPs of higher degree.

Software suite for determining the tilts of tower-type structures according to terrestrial laser scanning

2020 ◽  
Vol 961 (7) ◽  
pp. 2-7
Author(s):  
A.V. Zubov ◽  
N.N. Eliseeva
Keyword(s):  
Laser Scanning ◽  
Optimization Problems ◽  
Optimization Methods ◽  
User Participation ◽  
Software Suite ◽  
Current State ◽  
Visual Basic For Applications ◽  
Operational Data ◽  
Theoretical Developments

The authors describe a software suite for determining tilt degrees of tower-type structures according to ground laser scanning indication. Defining the tilt of the pipe is carried out with a set of measured data through approximating the sections by circumferences. They are constructed using one of the simplest search engine optimization methods (evolutionary algorithm). Automatic filtering the scan of the current section from distorting data is performed by the method of assessing the quality of models constructed with that of least squares. The software was designed using Visual Basic for Applications. It contains several blocks (subprograms), with each of them performing a specific task. The developed complex enables obtaining operational data on the current state of the object with minimal user participation in the calculation process. The software suite is the result of practical implementing theoretical developments on the possibilities of using search methods at solving optimization problems in geodetic practice.

scholarly journals Multi-Task Optimization and Multi-Task Evolutionary Computation in the Past Five Years: A Brief Review

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 864
Author(s):  
Qingzheng Xu ◽  
Na Wang ◽  
Lei Wang ◽  
Wei Li ◽  
Qian Sun
Keyword(s):  
Evolutionary Computation ◽  
Explicit Knowledge ◽  
Optimization Problems ◽  
Effective Means ◽  
Typical Application ◽  
The Past ◽  
Current State ◽  
Evolution Algorithms ◽  
Application Fields

Traditional evolution algorithms tend to start the search from scratch. However, real-world problems seldom exist in isolation and humans effectively manage and execute multiple tasks at the same time. Inspired by this concept, the paradigm of multi-task evolutionary computation (MTEC) has recently emerged as an effective means of facilitating implicit or explicit knowledge transfer across optimization tasks, thereby potentially accelerating convergence and improving the quality of solutions for multi-task optimization problems. An increasing number of works have thus been proposed since 2016. The authors collect the abundant specialized literature related to this novel optimization paradigm that was published in the past five years. The quantity of papers, the nationality of authors, and the important professional publications are analyzed by a statistical method. As a survey on state-of-the-art of research on this topic, this review article covers basic concepts, theoretical foundation, basic implementation approaches of MTEC, related extension issues of MTEC, and typical application fields in science and engineering. In particular, several approaches of chromosome encoding and decoding, intro-population reproduction, inter-population reproduction, and evaluation and selection are reviewed when developing an effective MTEC algorithm. A number of open challenges to date, along with promising directions that can be undertaken to help move it forward in the future, are also discussed according to the current state. The principal purpose is to provide a comprehensive review and examination of MTEC for researchers in this community, as well as promote more practitioners working in the related fields to be involved in this fascinating territory.

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 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.

scholarly journals Utilizing Model Knowledge for Design Developed Genetic Algorithm to Solving Problem

2018 ◽  
Vol 3 (2) ◽  
pp. 172
Author(s):  
Hamidreza Salmani mojaveri
Keyword(s):  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Parallel Machines ◽  
Job Shop ◽  
Optimization Problems ◽  
Premature Convergence ◽  
Local Optimum ◽  
Scheduling Problems ◽  
Algorithm Implementation

One of the discussed topics in scheduling problems is Dynamic Flexible Job Shop with Parallel Machines (FDJSPM). Surveys show that this problem because of its concave and nonlinear nature usually has several local optimums. Some of the scheduling problems researchers think that genetic algorithms (GA) are appropriate approach to solve optimization problems of this kind. But researches show that one of the disadvantages of classical genetic algorithms is premature convergence and the probability of trap into the local optimum. Considering these facts, in present research, represented a developed genetic algorithm that its controlling parameters change during algorithm implementation and optimization process. This approach decreases the probability of premature convergence and trap into the local optimum. The several experiments were done show that the priority of proposed procedure of solving in field of the quality of obtained solution and convergence speed toward other present procedure.

scholarly journals An improved arithmetic optimization algorithm with forced switching mechanism for global optimization problems

2022 ◽  
Vol 19 (1) ◽  
pp. 473-512
Author(s):  
Rong Zheng ◽  
◽  
Heming Jia ◽  
Laith Abualigah ◽  
Qingxin Liu ◽  
...  
Keyword(s):  
Optimization Algorithm ◽  
Optimization Problems ◽  
Heuristic Method ◽  
Population Diversity ◽  
Test Functions ◽  
Design Problems ◽  
Local Optima ◽  
Switching Mechanism ◽  
Engineering Design Problems

<abstract> <p>Arithmetic optimization algorithm (AOA) is a newly proposed meta-heuristic method which is inspired by the arithmetic operators in mathematics. However, the AOA has the weaknesses of insufficient exploration capability and is likely to fall into local optima. To improve the searching quality of original AOA, this paper presents an improved AOA (IAOA) integrated with proposed forced switching mechanism (FSM). The enhanced algorithm uses the random math optimizer probability (<italic>RMOP</italic>) to increase the population diversity for better global search. And then the forced switching mechanism is introduced into the AOA to help the search agents jump out of the local optima. When the search agents cannot find better positions within a certain number of iterations, the proposed FSM will make them conduct the exploratory behavior. Thus the cases of being trapped into local optima can be avoided effectively. The proposed IAOA is extensively tested by twenty-three classical benchmark functions and ten CEC2020 test functions and compared with the AOA and other well-known optimization algorithms. The experimental results show that the proposed algorithm is superior to other comparative algorithms on most of the test functions. Furthermore, the test results of two training problems of multi-layer perceptron (MLP) and three classical engineering design problems also indicate that the proposed IAOA is highly effective when dealing with real-world problems.</p> </abstract>

scholarly journals EVALUASI KURIKULUM PROGRAM KEAHLIAN TEKNIK KOMPUTER DAN JARINGAN SMKN 2 WATAMPONE DI KABUPATEN BONE SULAWESI SELATAN Hasbullah

2019 ◽  
Vol 8 (1) ◽  
pp. 761-783
Author(s):  
HASBULLAH Ulla
Keyword(s):  
Program Implementation ◽  
Quantitative Research ◽  
Sampling Technique ◽  
Problem Formulation ◽  
Theory And Practice ◽  
South Sulawesi ◽  
Network Engineering ◽  
Teachers And Students ◽  
Specific Skill

Abstract. This research is motivated by the fact that graduates of SMK are considered not flexible in adjusting to changes in the workplace, only having a single or specific skill that is quickly obsolete. In addition, there is a gap between schools and industry in terms of curriculum that makes students rigid when involved in the world of work and the industrial world. The general problem formulation in this research is how is the quality of the implementation of the computer and network engineering expertise program at SMKN Bone Regency of South Sulawesi? Specifically, (1) What is the quality of the implementation of the process aspects of the computer and network engineering expertise program? (2) What is the quality of the product aspects of the computer and network engineering expertise program? The purpose of this research is to find out the quality of the implementation of computer and network engineering expertise programs in the Bone Regency of South Sulawesi. The method in this study uses descriptive evaluative with quantitative research approaches. The population is the SMKN majoring in computer and network engineering in the Bone District of South Sulawesi consisting of principals, teachers and students. There were 52 students taken using stratification random sampling technique. The instruments used were questionnaire, interview, documentation study. This research uses descriptive data analysis. The results showed that in general the quality of program implementation was going well. This can be seen from the two aspects that were specifically examined namely the aspects of the process obtained from the questionnaire relating to the perceptions of students and teachers have been going well because it is in accordance with curriculum documents. In the aspect of product / results obtained from the study of documents relating to student learning outcomes data both theory and practice have been going well because it is in accordance with the planned objectives.K

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