Algorithms

Editorial

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3 pages, 138 KiB

Open AccessEditorial

Editorial for the Special Issue on “Recent Advances in Nonsmooth Optimization and Analysis”

by Sorin-Mihai Grad

Algorithms 2024, 17(8), 349; https://doi.org/10.3390/a17080349 - 9 Aug 2024

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Abstract

In recent years, nonsmooth optimization and analysis have seen remarkable advancements, significantly impacting various scientific and engineering disciplines [...] Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

Research

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27 pages, 510 KiB

Open AccessArticle

Lipschitz Continuity Results for a Class of Parametric Variational Inequalities and Applications to Network Games

by Mauro Passacantando and Fabio Raciti

Algorithms 2023, 16(10), 458; https://doi.org/10.3390/a16100458 - 26 Sep 2023

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Abstract

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider [...] Read more.

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider the problem of computing the mean value of the solution with respect to the parameter and, to this end, adapt an algorithm devised to approximate a Lipschitz function whose analytic expression is unknown, but can be evaluated in arbitrarily chosen sample points. Finally, we apply our results to a class of Nash equilibrium problems, and generalized Nash equilibrium problems on networks. Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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16 pages, 449 KiB

Open AccessArticle

Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components

by Quanjiang Yu, Ola Carlson and Serik Sagitov

Algorithms 2023, 16(7), 334; https://doi.org/10.3390/a16070334 - 13 Jul 2023

Cited by 3 | Viewed by 1962

Abstract

Wind power is one of the most important sources of renewable energy available today. A large part of the cost of wind energy is due to the cost of maintaining wind power equipment. When a wind turbine component fails to function, it might [...] Read more.

Wind power is one of the most important sources of renewable energy available today. A large part of the cost of wind energy is due to the cost of maintaining wind power equipment. When a wind turbine component fails to function, it might need to be replaced under circumstances that are less than ideal. This is known as corrective maintenance. To minimize unnecessary costs, a more active maintenance policy based on the life expectancy of the key components is preferred. Optimal scheduling of preventive maintenance activities requires advanced mathematical modeling. In this paper, an optimal preventive maintenance algorithm is designed using the renewal-reward theorem. In the multi-component setting, our approach involves a new idea of virtual maintenance that allows us to treat each replacement event as a renewal event even if some components are not replaced by new ones. The proposed optimization algorithm is applied to a four-component model of a wind turbine, and the optimal maintenance plans are computed for various initial conditions. The modeling results clearly show the benefit of PM planning compared to a pure CM strategy (about 30% lower maintenance cost). Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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14 pages, 406 KiB

Open AccessArticle

On the Adaptive Penalty Parameter Selection in ADMM

by Serena Crisci, Valentina De Simone and Marco Viola

Algorithms 2023, 16(6), 264; https://doi.org/10.3390/a16060264 - 25 May 2023

Cited by 1 | Viewed by 1900

Abstract

Many data analysis problems can be modeled as a constrained optimization problem characterized by nonsmooth functionals, often because of the presence of

ℓ_{1}

-regularization terms. One of the most effective ways to solve such problems is through the Alternate Direction Method of [...] Read more.

Many data analysis problems can be modeled as a constrained optimization problem characterized by nonsmooth functionals, often because of the presence of

ℓ_{1}

-regularization terms. One of the most effective ways to solve such problems is through the Alternate Direction Method of Multipliers (ADMM), which has been proved to have good theoretical convergence properties even if the arising subproblems are solved inexactly. Nevertheless, experience shows that the choice of the parameter

τ

penalizing the constraint violation in the Augmented Lagrangian underlying ADMM affects the method’s performance. To this end, strategies for the adaptive selection of such parameter have been analyzed in the literature and are still of great interest. In this paper, starting from an adaptive spectral strategy recently proposed in the literature, we investigate the use of different strategies based on Barzilai–Borwein-like stepsize rules. We test the effectiveness of the proposed strategies in the solution of real-life consensus logistic regression and portfolio optimization problems. Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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14 pages, 674 KiB

Open AccessArticle

A Linearly Involved Generalized Moreau Enhancement of ℓ_2,1-Norm with Application to Weighted Group Sparse Classification

by Yang Chen, Masao Yamagishi and Isao Yamada

Algorithms 2021, 14(11), 312; https://doi.org/10.3390/a14110312 - 27 Oct 2021

Cited by 2 | Viewed by 2266

Abstract

This paper proposes a new group-sparsity-inducing regularizer to approximate

ℓ_{2, 0}

pseudo-norm. The regularizer is nonconvex, which can be seen as a linearly involved generalized Moreau enhancement of

ℓ_{2, 1}

-norm. Moreover, the overall convexity of the corresponding group-sparsity-regularized [...] Read more.

This paper proposes a new group-sparsity-inducing regularizer to approximate

ℓ_{2, 0}

pseudo-norm. The regularizer is nonconvex, which can be seen as a linearly involved generalized Moreau enhancement of

ℓ_{2, 1}

-norm. Moreover, the overall convexity of the corresponding group-sparsity-regularized least squares problem can be achieved. The model can handle general group configurations such as weighted group sparse problems, and can be solved through a proximal splitting algorithm. Among the applications, considering that the bias of convex regularizer may lead to incorrect classification results especially for unbalanced training sets, we apply the proposed model to the (weighted) group sparse classification problem. The proposed classifier can use the label, similarity and locality information of samples. It also suppresses the bias of convex regularizer-based classifiers. Experimental results demonstrate that the proposed classifier improves the performance of convex

ℓ_{2, 1}

regularizer-based methods, especially when the training data set is unbalanced. This paper enhances the potential applicability and effectiveness of using nonconvex regularizers in the frame of convex optimization. Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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14 pages, 1812 KiB

Open AccessArticle

A Convex Optimization Algorithm for Electricity Pricing of Charging Stations

by Jing Zhang, Xiangpeng Zhan, Taoyong Li, Linru Jiang, Jun Yang, Yuanxing Zhang, Xiaohong Diao and Sining Han

Algorithms 2019, 12(10), 208; https://doi.org/10.3390/a12100208 - 1 Oct 2019

Cited by 3 | Viewed by 4113

Abstract

The problem of electricity pricing for charging stations is a multi-objective mixed integer nonlinear programming. Existing algorithms have low efficiency in solving this problem. In this paper, a convex optimization algorithm is proposed to get the optimal solution quickly. Firstly, the model is [...] Read more.

The problem of electricity pricing for charging stations is a multi-objective mixed integer nonlinear programming. Existing algorithms have low efficiency in solving this problem. In this paper, a convex optimization algorithm is proposed to get the optimal solution quickly. Firstly, the model is transformed into a convex optimization problem by second-order conic relaxation and Karush–Kuhn–Tucker optimality conditions. Secondly, a polyhedral approximation method is applied to construct a mixed integer linear programming, which can be solved quickly by branch and bound method. Finally, the model is solved many times to obtain the Pareto front according to the scalarization basic theorem. Based on an IEEE 33-bus distribution network model, simulation results show that the proposed algorithm can obtain an exact global optimal solution quickly compared with the heuristic method. Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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21 pages, 937 KiB

Open AccessArticle

A Cross-Layer Optimization QoS Scheme in Wireless Multimedia Sensor Networks

by Shu Fan

Algorithms 2019, 12(4), 68; https://doi.org/10.3390/a12040068 - 30 Mar 2019

Cited by 4 | Viewed by 4653

Abstract

There are two main challenges in wireless multimedia sensors networks: energy constraints and providing DiffServ. In this paper, a joint flow control, routing, scheduling, and power control scheme based on a Lyapunov optimization framework is proposed to increase network lifetime and scheduling fairness. [...] Read more.

There are two main challenges in wireless multimedia sensors networks: energy constraints and providing DiffServ. In this paper, a joint flow control, routing, scheduling, and power control scheme based on a Lyapunov optimization framework is proposed to increase network lifetime and scheduling fairness. For an adaptive distribution of transmission opportunities, a differentiated queueing services (DQS) scheme is adopted for maintaining data queues. In the Lyapunov function, different types of queues are normalized for a unified dimension. To prolong network lifetime, control coefficients are designed according to the characteristics of the wireless sensor networks. The power control problem is proved to be a convex optimization problem and two optimal algorithms are discussed. Simulation results show that, compared with existing schemes, the proposed scheme can achieve a better trade-off between QoS performances and network lifetime. The simulation results also show that the scheme utilizing the distributed media access control scheme in scheduling performs best in the transmission of real-time services. Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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Review

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30 pages, 1491 KiB

Open AccessReview

A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems

by Enrique Castillo

Algorithms 2023, 16(8), 356; https://doi.org/10.3390/a16080356 - 25 Jul 2023

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Abstract

This work is a short review of the state of the art aiming to contribute to the use, disclosure, and propagation of systems of linear inequalities in real life, teaching, and research. It shows that the algebraic structure of their solutions consists of [...] Read more.

This work is a short review of the state of the art aiming to contribute to the use, disclosure, and propagation of systems of linear inequalities in real life, teaching, and research. It shows that the algebraic structure of their solutions consists of the sum of a linear subspace, an acute cone, and a polytope, and that adequate software exists to obtain, in their simplest forms, these three components. The work describes, based on orthogonality and polarity, homogeneous and complete systems of inequalities, the associated compatibility problems, and their relations with convex polyhedra and polytopes, which are the only possible solution for bounded problems, the most common in real practice. The compatibility and the observability problems, including their symbolic forms, are analyzed and solved, identifying the subsets of unknowns with unique solutions and those unbounded, important items of information with practical relevance in artificial intelligence and automatic learning. Having infinitely many solutions of a given problem allows us to find solutions when some of the assumptions fail and unexpected constraints come into play, a common situation for engineers. The linear programming problem becomes trivial when the set of all solutions is available and all solutions are obtained, contrary to the case of standard programs that provide only one solution. Several examples of applications to several areas of knowledge are presented, illustrating the advantages of solving these systems of inequalities. Full article

(This article belongs to the Special Issue Recent Advances in Nonsmooth Optimization and Analysis)

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