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11 pages, 292 KiB

Open AccessArticle

Cyclic Structure, Vertex Degree and Number of Linear Vertices in Minimal Strong Digraphs

by Miguel Arcos-Argudo, Jesús Lacalle and Luis M. Pozo-Coronado

Mathematics 2024, 12(23), 3657; https://doi.org/10.3390/math12233657 - 22 Nov 2024

Minimal Strong Digraphs (MSDs) can be regarded as a generalization of the concept of tree to directed graphs. Their cyclic structure and some spectral properties have been studied in several articles. In this work, we further study some properties of MSDs that have [...] Read more.

Minimal Strong Digraphs (MSDs) can be regarded as a generalization of the concept of tree to directed graphs. Their cyclic structure and some spectral properties have been studied in several articles. In this work, we further study some properties of MSDs that have to do with bounding the length of the longest cycle (regarding the number of linear vertices, or the maximal in- or outdegree of vertices); studying whatever consequences from the spectral point of view; and giving some insight about the circumstances in which an efficient algorithm to find the longest cycle contained in an MSD can be formulated. Among other properties, we show that the number of linear vertices contained in an MSD is greater than or equal to the maximal (respectively minimal) in- or outdegree of any vertex of the MSD and that the maximal length of a cycle contained in an MSD is lesser than or equal to

2 n - m

where

n, m

are the order and the size of the MSD, respectively; we find a bound for the coefficients of the characteristic polynomial of an MSD, and finally, we prove that computing the longest cycle contained in an MSD is an NP-hard problem. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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12 pages, 3444 KiB

Open AccessArticle

Enhancing Parameters Tuning of Overlay Models with Ridge Regression: Addressing Multicollinearity in High-Dimensional Data

by Aris Magklaras, Christos Gogos, Panayiotis Alefragis and Alexios Birbas

Mathematics 2024, 12(20), 3179; https://doi.org/10.3390/math12203179 - 11 Oct 2024

Viewed by 893

Abstract

The extreme ultraviolet (EUV) photolithography process is a cornerstone of semiconductor manufacturing and operates under demanding precision standards realized via nanometer-level overlay (OVL) error modeling. This procedure allows the machine to anticipate and correct OVL errors before impacting the wafer, thereby facilitating near-optimal [...] Read more.

The extreme ultraviolet (EUV) photolithography process is a cornerstone of semiconductor manufacturing and operates under demanding precision standards realized via nanometer-level overlay (OVL) error modeling. This procedure allows the machine to anticipate and correct OVL errors before impacting the wafer, thereby facilitating near-optimal image exposure while simultaneously minimizing the overall OVL error. Such models are usually high dimensional and exhibit rigorous statistical phenomena such as collinearities that play a crucial role in the process of tuning their parameters. Ordinary least squares (OLS) is the most widely used method for parameters tuning of overlay models, but in most cases it fails to compensate for such phenomena. In this paper, we propose the usage of ridge regression, a widely known machine learning (ML) algorithm especially suitable for datasets that exhibit high multicollinearity. The proposed method was applied in perturbed data from a 300 mm wafer fab, and the results show reduced residuals when ridge regression is applied instead of OLS. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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16 pages, 2402 KiB

Open AccessArticle

A New Hybrid Descent Algorithm for Large-Scale Nonconvex Optimization and Application to Some Image Restoration Problems

by Shuai Wang, Xiaoliang Wang, Yuzhu Tian and Liping Pang

Mathematics 2024, 12(19), 3088; https://doi.org/10.3390/math12193088 - 2 Oct 2024

Viewed by 436

Abstract

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance [...] Read more.

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance of the Hestenes–Stiefel method, a new hybrid descent conjugate gradient method is proposed in this paper. The proposed method satisfies the sufficient descent property independent of the accuracy of the line search strategies. Under the standard conditions, the trust region property and the global convergence are established, respectively. Numerical results of 61 problems with 9 large-scale dimensions and 46 ill-conditioned matrix problems reveal that the proposed method is more effective, robust, and reliable than the other methods. Additionally, the hybrid method also demonstrates reliable results for some image restoration problems. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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39 pages, 5553 KiB

Open AccessArticle

Weight Vector Definition for MOEA/D-Based Algorithms Using Augmented Covering Arrays for Many-Objective Optimization

by Carlos Cobos, Cristian Ordoñez, Jose Torres-Jimenez, Hugo Ordoñez and Martha Mendoza

Mathematics 2024, 12(11), 1680; https://doi.org/10.3390/math12111680 - 28 May 2024

Viewed by 1717

Abstract

Many-objective optimization problems are today ever more common. The decomposition-based approach stands out among the evolutionary algorithms used for their solution, with MOEA/D and its variations playing significant roles. MOEA/D variations seek to improve weight vector definition, improve the dynamic adjustment of weight [...] Read more.

Many-objective optimization problems are today ever more common. The decomposition-based approach stands out among the evolutionary algorithms used for their solution, with MOEA/D and its variations playing significant roles. MOEA/D variations seek to improve weight vector definition, improve the dynamic adjustment of weight vectors during the evolution process, improve the evolutionary operators, use alternative decomposition methods, and hybridize with other metaheuristics, among others. Although an essential topic for the success of MOEA/D depends on how well the weight vectors are defined when decomposing the problem, not as much research has been performed on this topic as on the others. This paper proposes using a new mathematical object called augmented covering arrays (ACAs) that enable a better sampling of interactions of M objectives using the least number of weight vectors based on an interaction level (strength), defined a priori by the user. The proposed method obtains better results, measured in inverted generational distance, using small to medium populations (up to 850 solutions) of 30 to 100 objectives over DTLZ and WFG problems against the traditional weight vector definition used by MOEA/D-DE and results obtained by NSGA-III. Other MOEA/D variations can include the proposed approach and thus improve their results. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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33 pages, 824 KiB

Open AccessArticle

Optimizing Retaining Walls through Reinforcement Learning Approaches and Metaheuristic Techniques

by José Lemus-Romani, Diego Ossandón, Rocío Sepúlveda, Nicolás Carrasco-Astudillo, Victor Yepes and José García

Mathematics 2023, 11(9), 2104; https://doi.org/10.3390/math11092104 - 28 Apr 2023

Cited by 2 | Viewed by 2686

Abstract

The structural design of civil works is closely tied to empirical knowledge and the design professional’s experience. Based on this, adequate designs are generated in terms of strength, operability, and durability. However, such designs can be optimized to reduce conditions associated with the [...] Read more.

The structural design of civil works is closely tied to empirical knowledge and the design professional’s experience. Based on this, adequate designs are generated in terms of strength, operability, and durability. However, such designs can be optimized to reduce conditions associated with the structure’s design and execution, such as costs, CO₂ emissions, and related earthworks. In this study, a new discretization technique based on reinforcement learning and transfer functions is developed. The application of metaheuristic techniques to the retaining wall problem is examined, defining two objective functions: cost and CO₂ emissions. An extensive comparison is made with various metaheuristics and brute force methods, where the results show that the S-shaped transfer functions consistently yield more robust outcomes. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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24 pages, 2041 KiB

Open AccessArticle

A Modified q-BFGS Algorithm for Unconstrained Optimization

by Kin Keung Lai, Shashi Kant Mishra, Ravina Sharma, Manjari Sharma and Bhagwat Ram

Mathematics 2023, 11(6), 1420; https://doi.org/10.3390/math11061420 - 15 Mar 2023

Cited by 7 | Viewed by 2779

Abstract

This paper presents a modification of the q-BFGS method for nonlinear unconstrained optimization problems. For this modification, we use a simple symmetric positive definite matrix and propose a new q-quasi-Newton equation, which is close to the ordinary q-quasi-Newton equation in [...] Read more.

This paper presents a modification of the q-BFGS method for nonlinear unconstrained optimization problems. For this modification, we use a simple symmetric positive definite matrix and propose a new q-quasi-Newton equation, which is close to the ordinary q-quasi-Newton equation in the limiting case. This method uses only first order q-derivatives to build an approximate q-Hessian over a number of iterations. The q-Armijo-Wolfe line search condition is used to calculate step length, which guarantees that the objective function value is decreasing. This modified q-BFGS method preserves the global convergence properties of the q-BFGS method, without the convexity assumption on the objective function. Numerical results on some test problems are presented, which show that an improvement has been achieved. Moreover, we depict the numerical results through the performance profiles. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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25 pages, 2399 KiB

Open AccessArticle

A Distributed Blocking Flowshop Scheduling with Setup Times Using Multi-Factory Collaboration Iterated Greedy Algorithm

by Chenyao Zhang, Yuyan Han, Yuting Wang, Junqing Li and Kaizhou Gao

Mathematics 2023, 11(3), 581; https://doi.org/10.3390/math11030581 - 22 Jan 2023

Cited by 6 | Viewed by 1801

Abstract

As multi-factory production models are more widespread in modern manufacturing systems, a distributed blocking flowshop scheduling problem (DBFSP) is studied in which no buffer between adjacent machines and setup time constraints are considered. To address the above problem, a mixed integer linear programming [...] Read more.

As multi-factory production models are more widespread in modern manufacturing systems, a distributed blocking flowshop scheduling problem (DBFSP) is studied in which no buffer between adjacent machines and setup time constraints are considered. To address the above problem, a mixed integer linear programming (MILP) model is first constructed, and its correctness is verified. Then, an iterated greedy-algorithm-blending multi-factory collaboration mechanism (mIG) is presented to optimize the makespan criterion. In the mIG algorithm, a rapid evaluation method is designed to reduce the time complexity, and two different iterative processes are selected by a certain probability. In addition, collaborative interactions between cross-factory and inner-factory are considered to further improve the exploitation and exploration of mIG. Finally, the 270 tests showed that the average makespan and RPI values of mIG are 1.93% and 78.35% better than the five comparison algorithms on average, respectively. Therefore, mIG is more suitable to solve the studied DBFSP_SDST. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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13 pages, 333 KiB

Open AccessArticle

Application of the ADMM Algorithm for a High-Dimensional Partially Linear Model

by Aifen Feng, Xiaogai Chang, Youlin Shang and Jingya Fan

Mathematics 2022, 10(24), 4767; https://doi.org/10.3390/math10244767 - 15 Dec 2022

Cited by 3 | Viewed by 2411

Abstract

This paper focuses on a high-dimensional semi-parametric regression model in which a partially linear model is used for the parametric part and the B-spline basis function approach is used to estimate the unknown function for the non-parametric part. Within the framework of this [...] Read more.

This paper focuses on a high-dimensional semi-parametric regression model in which a partially linear model is used for the parametric part and the B-spline basis function approach is used to estimate the unknown function for the non-parametric part. Within the framework of this model, the constrained least squares estimation is investigated, and the alternating-direction multiplier method (ADMM) is used to solve the model. The convergence is proved under certain conditions. Finally, numerical simulations are performed and applied to workers’ wage data from CPS85. The results show that the ADMM algorithm is very effective in solving high-dimensional partially linear models. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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27 pages, 2251 KiB

Open AccessArticle

An Improved Arithmetic Optimization Algorithm for Numerical Optimization Problems

by Mengnan Chen, Yongquan Zhou and Qifang Luo

Mathematics 2022, 10(12), 2152; https://doi.org/10.3390/math10122152 - 20 Jun 2022

Cited by 13 | Viewed by 2916

Abstract

The arithmetic optimization algorithm is a recently proposed metaheuristic algorithm. In this paper, an improved arithmetic optimization algorithm (IAOA) based on the population control strategy is introduced to solve numerical optimization problems. By classifying the population and adaptively controlling the number of individuals [...] Read more.

The arithmetic optimization algorithm is a recently proposed metaheuristic algorithm. In this paper, an improved arithmetic optimization algorithm (IAOA) based on the population control strategy is introduced to solve numerical optimization problems. By classifying the population and adaptively controlling the number of individuals in the subpopulation, the information of each individual can be used effectively, which speeds up the algorithm to find the optimal value, avoids falling into local optimum, and improves the accuracy of the solution. The performance of the proposed IAOA algorithm is evaluated on six systems of nonlinear equations, ten integrations, and engineering problems. The results show that the proposed algorithm outperforms other algorithms in terms of convergence speed, convergence accuracy, stability, and robustness. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

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