Variety Of Discrete Optimization Problems Research Articles

A Scalable Distributed Dynamical Systems Approach to Learn the Strongly Connected Components and Diameter of Networks

Finding strongly connected components (SCCs) and the diameter of a directed network play a key role in a variety of discrete optimization problems, and subsequently, machine learning and control theory problems. On the one hand, SCCs are used in solving the 2-satisfiability problem, which has applications in clustering, scheduling, and visualization. On the other hand, the diameter has applications in network learning and discovery problems enabling efficient internet routing and searches, as well as identifying faults in the power grid. In this paper, we leverage consensus-based principles to find the SCCs in a scalable and distributed fashion with a computational complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(Dd_{\rm{in-degree}}^{\max })$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula> is the (finite) diameter of the network and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d_{\rm{in-degree}}^{\max }$</tex-math></inline-formula> is the maximum in-degree of the network. Additionally, we prove that our algorithm terminates in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D+1$</tex-math></inline-formula> iterations, which allows us to retrieve the diameter of the network. We illustrate the performance of our algorithm on several random networks, including Erdö-Rényi, Barabási-Albert, and Watts-Strogatz networks.

Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory

This paper presents an optimal sensor placement method for vibration signal acquisition in the field of industrial robot health monitoring and fault diagnosis. Based on the general formula of Bayes and relative entropy, the evaluation function of sensor placement is deduced, and the modal confidence matrix is used to express the redundancy of sensor placement. The optimal placement of vibration sensors is described as a discrete variable optimization problem, which is defined as whether the existing sensor layout can obtain joint state information efficiently. The initial layout of the sensor was obtained from the structural simulation results of the industrial robots, and the initial layout was optimized by the derived objective function. The efficiency of the optimized layout in capturing joint state information is proven by the validation experiment with a simulation model. The problem of popularizing the optimization method in engineering is solved by a verification experiment without a simulation model. The optimal sensor placement method provides a theoretical basis for industrial robots to acquire vibration data effectively.

Design of curved composite panels for maximum buckling load using sequential permutation search algorithm

New buckling optimal solutions for curved composite panels are presented in terms of principle of virtual work and sequential permutation search (SPS) algorithm. By utilizing the principle of virtual displacements, the buckling load is calculated and the accuracy of the derived buckling solutions is verified by comparison with experimental data and numerical results. SPS utilizes the inherent mechanical properties of composite laminates in which the bending stiffness of the outer layers is more sensitive than those close to the mid-surface. Curved composite panels exhibit a variety of length-to-arc ratios and radius-to-length ratios, under axial compression, pure shear, and combined loadings, are optimized. Some of the results are compared with the global optima obtained by enumeration method, and SPS shows good robustness in locating the global optima. In addition, the buckling load carrying capacity of optimized curved composite panels can be improved by as much as twice over cross-ply laminates. Low computational costs of SPS algorithm imply its potential for solving high-dimensional discrete variable optimization problems of composite structures.

Submodularity in Action: From Machine Learning to Signal Processing Applications

Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of well-known convexity/concavity properties in the continuous domain. Submodular functions exhibit strong structure that lead to efficient optimization algorithms with provable near-optimality guarantees. These characteristics, namely, efficiency and provable performance bounds, are of particular interest for signal processing (SP) and machine learning (ML) practitioners, as a variety of discrete optimization problems are encountered in a wide range of applications. Conventionally, two general approaches exist to solve discrete problems: 1) relaxation into the continuous domain to obtain an approximate solution or 2) the development of a tailored algorithm that applies directly in the discrete domain. In both approaches, worst-case performance guarantees are often hard to establish. Furthermore, they are often complex and thus not practical for large-scale problems. In this article, we show how certain scenarios lend themselves to exploiting submodularity for constructing scalable solutions with provable worst-case performance guarantees. We introduce a variety of submodular-friendly applications and elucidate the relation of submodularity to convexity and concavity, which enables efficient optimization. With a mixture of theory and practice, we present different flavors of submodularity accompanying illustrative real-world case studies from modern SP and ML. In all of the cases, optimization algorithms are presented along with hints on how optimality guarantees can be established.

The application of improved adaptive genetic algorithm in the optimization of discrete variables

In view of the problem of optimization design of complex discrete variables in practical application, the optimization design method of discrete variables and the processing methods of discrete variables are analyzed. The shortcomings of these methods are summarized, and the characteristics of genetic algorithms and discrete variables are combined to propose a self-adaptive discrete non-uniform crossover operator and adaptive discrete mapping mutation operator, so that the genetic operation can be implemented in discrete space for higher efficiency search. Combined with the optimal individual protection strategy and adaptive crossover and mutation strategy execution sequence, accelerate the population rate of global optimization, overall improve the efficiency of the algorithm of discrete optimization, through the design of a practical discrete variable optimization problems research, verified the effectiveness of the algorithm to solve the problem of discrete optimization design.

Finance-based scheduling using meta-heuristics: discrete versus continuous optimization problems

Purpose – The purpose of this paper is to compare the performance of the genetic algorithm (GA), simulate annealing (SA) and shuffled frog-leaping algorithm (SFLA) in solving discrete versus continuous-variable optimization problems of the finance-based scheduling. This involves the minimization of the project duration and consequently the time-related cost components of construction contractors including overheads, finance costs and delay penalties. Design/methodology/approach – The meta-heuristics of the GA, SA and SFLA have been implemented to solve non-deterministic polynomial-time hard (NP-hard) finance-based scheduling problem employing the objective of minimizing the project duration. The traditional problem of generating unfeasible solutions in scheduling problems is adequately tackled in the implementations of the meta-heuristics in this paper. Findings – The obtained results indicated that the SA outperformed the SFLA and GA in terms of the quality of solutions as well as the computational cost based on the small-size networks of 30 activities, whereas it exhibited the least total duration based on the large-size networks of 120 and 210 activities after prolonged processing time. Research limitations/implications – From researchers’ perspective, finance-based scheduling is one of the few domain problems which can be formulated as discrete and continuous-variable optimization problems and, thus, can be used by researchers as a test bed to give more insight into the performance of new developments of meta-heuristics in solving discrete and continuous-variable optimization problems. Practical implications – Finance-based scheduling discrete-variable optimization problem is of high relevance to the practitioners, as it allows schedulers to devise finance-feasible schedules of minimum duration. The minimization of project duration is focal for the minimization of time-related cost components of construction contractors including overheads, finance costs and delay penalties. Moreover, planning for the expedient project completion is a major time-management aspect of construction contractors towards the achievement of the objective of client satisfaction through the expedient delivery of the completed project for clients to start reaping the anticipated benefits. Social implications – Planning for the expedient project completion is a major time-management aspect of construction contractors towards the achievement of the objective of client satisfaction. Originality/value – SFLA represents a relatively recent meta-heuristic that proved to be promising, based on its limited number of applications in the literature. This paper is to implement SFLA to solve the discrete-variable optimization problem of the finance-based scheduling and assess its performance by comparing its results against those of the GA and SA.

Classifier-guided sampling for discrete variable, discontinuous design space exploration: Convergence and computational performance

A classifier-guided sampling (CGS) method is introduced for solving engineering design optimization problems with discrete and/or continuous variables and continuous and/or discontinuous responses. The method merges concepts from metamodel-guided sampling and population-based optimization algorithms. The CGS method uses a Bayesian network classifier for predicting the performance of new designs based on a set of known observations or training points. Unlike most metamodelling techniques, however, the classifier assigns a categorical class label to a new design, rather than predicting the resulting response in continuous space, and thereby accommodates non-differentiable and discontinuous functions of discrete or categorical variables. The CGS method uses these classifiers to guide a population-based sampling process towards combinations of discrete and/or continuous variable values with a high probability of yielding preferred performance. Accordingly, the CGS method is appropriate for discrete/discontinuous design problems that are ill suited for conventional metamodelling techniques and too computationally expensive to be solved by population-based algorithms alone. The rates of convergence and computational properties of the CGS method are investigated when applied to a set of discrete variable optimization problems. Results show that the CGS method significantly improves the rate of convergence towards known global optima, on average, compared with genetic algorithms.

SYNTHESIS OF THINNED ARRAY WITH SIDE LOBE LEVELS REDUCTION USING IMPROVED BINARY INVASIVE WEED OPTIMIZATION

As a very powerful optimization algorithm, invasive weed optimization has been widely applied to continuous optimization problems in electromagnetic (EM) fleld. However, the optimization of a thinned array can be formulated as a discrete-variable optimization problem with solutions encoded as binary strings. Therefore, in this paper, an improved binary invasive weed optimization (IBIWO) is proposed to design a thinned array with minimum side lobe levels. To evaluate the performance of the proposed algorithm, two examples have been presented and solved. Simulation results of the proposed thinned arrays obtained by IBIWO are compared with published results to verify the efiectiveness of the proposed method.

Development of Optimization Design Software for Bevel Gear Based on Integer Serial Number Encoding Genetic Algorithm

Bevel gear drive is widely used, quality of which not only affects its own transmission performance, size and weight, but also has some impact on the machine's performance. This paper introduces optimization design software for bevel gear, in which automatic optimization design is realized. In the paper mathematical model, programming of design data and realization of optimization design based on genetic algorithm are described in detail. The paper proposed integer serial number encoding genetic algorithm, which effectively deals with continuous and discrete variable optimization problem and reduces the code length of the string to improve the encoding and decoding efficiency, no invalid solution or duplicate solutions.

Thinned Planar Array Design Using Boolean PSO With Velocity Mutation

The design of thinned planar microstrip arrays under specific constraints concerning the impedance-matching condition of the array elements and the radiation pattern is presented. The radiation characteristics of the structure are extracted by applying the method-of-moments. The array design is based on a novel optimization method, which is a modified version of the boolean particle swarm optimization that employs velocity mutation (BPSO-vm). Apart from the optimization of the array geometry, the proposed method is applicable to other discrete-variable optimization problems. Moreover, the planar array design is coped with by means of other techniques, namely, a binary coded Genetic Algorithm, the binary Particle Swarm Optimization, and the Boolean PSO. The comparison of the above methods and the BPSO-vm shows the efficiency of the proposed technique.

Ant colony optimization of clustering models

The original ant system algorithm is simplified leading to a generalized ant colony optimization algorithm that can be used to solve a wide variety of discrete optimization problems. It is shown how objective function based clustering models such as hard and fuzzy c-means can be optimized using particular extensions of this simplified ant optimization algorithm. Experiments with artificial and real datasets show that ant clustering produces better results than alternating optimization because it is less sensitive to local extrema. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1233–1251, 2005.

Optimal Design of Latticed Towers Subjected to Earthquake Loading

The problem of optimal design of latticed transmission towers subjected to normal operating loads and an earthquake load is formulated. A time history analysis of the structure is performed. The constraints given in a design code are imposed. Many of the constraints are time dependent; therefore methods to treat them are incorporated in the solution process. All members are treated as equal- ~unequal-! legged angle sections. The members are selected from the sections available in a manufacturer's catalog. Two methods are presented and evaluated for such discrete variable optimization problems. The first method, called the two-phase method, uses a combi- nation of continuous and discrete optimization algorithms, and the second one, called an adaptive discrete assignment method ~ADAM!, uses only a continuous optimization algorithm. The proposed methods are applicable to any discrete variable problem that can be formulated as a continuous problem. In addition, the problem is solved using a genetic algorithm ~GA!. The efficiency and practicality of the proposed methods are compared to those of the GA. From the results, it is concluded that the GA is a very straightforward method to use for discrete problems; however, it requires a significant amount of Central Processing Unit time. On the other hand, the two-phase method and the ADAM do not require as much computational effort but the discrete designs found with them are slightly more expensive. Also, numerical implementation of the two methods requires more effort.

Coping with Anomalies in Parallel Branch-and-Bound Algorithms

A general technique that can be used to solve a wide variety of discrete optimization problems is the branch-and-bound algorithm. We have adapted and extended branch-and-bound algorithms for parallel processing. The computational efficiency of these algorithms depends on the allowance function, the data structure, and the search strategies. Anomalies owing to parallelism may occur. In this correspondence, anomalies of parallel branch-and-bound algorithms using the same search strategy as the corresponding serial algorithms are studied. Sufficient conditions to guarantee no degradation in performance due to parallelism and necessary conditions for allowing parallelism to have a speedup greater than the number of processors are presented.

UNCONSTRAINED DISCRETE NONLINEAR PROGRAMMING

In this research several search methods for unconstrained nonlinear discrete variable optimization problems have been developed. Many of these new methods are modifications of effective continuous variable search techniques including gradient–free and gradient–based methods. In order to search only over a set of discrete points, the concepts of integer search direction and the subsequential search procedure are introduced. Other developments include regeneration/ acceleration procedures for gradient–based methods and a second level acceleration procedure applicable to both gradient–free and gradient–based methods. These new methods have been compared with each other and existing techniques using test problems with various characteristics, including penalty functions from constrained problems. In all cases, the best results have been obtained from one of the new methods. Moreover, the success of these new methods in finding good solutions in penalty function problems indicates their usefulness in solving engineering design problems.

Minimizing the Number of Operations in Certain Discrete-Variable Optimization Problems

This paper deals with the solution to a discrete optimization problem by decomposition. It develops a criterion for ranking the decomposition procedures, discusses the properties of the optimal decompositions, and gives an algorithm for finding the best decomposition in the case of no storage limitation.