Table of Content

30 March 2022, Volume 10 Issue 1

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A Survey on Some Recent Developments of Alternating Direction Method of Multipliers

De-Ren Han

2022, 10(1):  1-52.  doi:10.1007/s40305-021-00368-3

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Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for different models. Moreover, its theoretical studies such as rate of convergence and extensions to nonconvex problems also achieve much progress. In this paper, we give a survey on some recent developments of ADMM and its variants.

On Iteration Complexity of a First-Order Primal-Dual Method for Nonlinear Convex Cone Programming

Lei Zhao, Dao-Li Zhu

2022, 10(1):  53-87.  doi:10.1007/s40305-021-00344-x

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Nonlinear convex cone programming (NCCP) models have found many practical applications. In this paper, we introduce a flexible first-order primal-dual algorithm, called the variant auxiliary problem principle (VAPP), for solving NCCP problems when the objective function and constraints are convex but may be nonsmooth. At each iteration, VAPP generates a nonlinear approximation of the primal augmented Lagrangian model. The approximation incorporates both linearization and a distance-like proximal term, and then the iterations of VAPP are shown to possess a decomposition property for NCCP. Motivated by recent applications in big data analytics, there has been a growing interest in the convergence rate analysis of algorithms with parallel computing capabilities for large scale optimization problems. We establish O(1/t) convergence rate towards primal optimality, feasibility and dual optimality. By adaptively setting parameters at different iterations, we show an O(1/t²) rate for the strongly convex case. Finally, we discuss some issues in the implementation of VAPP.

Intuitionistic Fuzzy Laplacian Twin Support Vector Machine for Semi-supervised Classification

Jia-Bin Zhou, Yan-Qin Bai, Yan-Ru Guo, Hai-Xiang Lin

2022, 10(1):  89-112.  doi:10.1007/s40305-021-00354-9

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In general, data contain noises which come from faulty instruments, flawed measurements or faulty communication. Learning with data in the context of classification or regression is inevitably affected by noises in the data. In order to remove or greatly reduce the impact of noises, we introduce the ideas of fuzzy membership functions and the Laplacian twin support vector machine (Lap-TSVM). A formulation of the linear intuitionistic fuzzy Laplacian twin support vector machine (IFLap-TSVM) is presented. Moreover, we extend the linear IFLap-TSVM to the nonlinear case by kernel function. The proposed IFLap-TSVM resolves the negative impact of noises and outliers by using fuzzy membership functions and is a more accurate reasonable classifier by using the geometric distribution information of labeled data and unlabeled data based on manifold regularization. Experiments with constructed artificial datasets, several UCI benchmark datasets and MNIST dataset show that the IFLap-TSVM has better classification accuracy than other state-of-the-art twin support vector machine (TSVM), intuitionistic fuzzy twin support vector machine (IFTSVM) and Lap-TSVM.

Online Scheduling on a Parallel Batch Machine with Delivery Times and Limited Restarts

Hai-Ling Liu, Xi-Wen Lu

2022, 10(1):  113-131.  doi:10.1007/s40305-021-00356-7

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The online scheduling on an unbounded parallel batch machine with delivery times and limited restarts is studied in this paper. Here, online means that jobs arrive over time and the characteristics of a job become known until it arrives. Limited restarts mean that once a running batch contains at least one restarted job, it cannot be restarted again. The goal is to minimize the time by which all jobs have been delivered. We consider a restricted model that the delivery time of each job is no more than its processing time. We present a best possible online algorithm with a competitive ratio of 3/2 for the problem.

The Odd Log-Logistic Weibull-G Family of Distributions with Regression and Financial Risk Models

Mahdi Rasekhi, Emrah Altun, Morad Alizadeh, Haitham M. Yousof

2022, 10(1):  133-158.  doi:10.1007/s40305-021-00349-6

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A new generalization of the Weibull-G family is proposed with two extra shape parameters. The mathematical properties are derived in great detail. Using the Weibull and normal distributions as baseline distributions, two models are introduced. The first model is a location-scale regression model based on a new extension of the Weibull distribution. The second model is a new two-step financial risk model to forecast the daily value at risk. The flexibility and applicability of the proposed models are investigated by means of five real data sets on the lifetime and financial returns. Empirical findings of the study show that proposed models work well and produce better results than other well-known models for financial risk modeling and censored lifetime data analysis.

Ordering and Pricing Decisions of a Retailer in the Presence of Multiple-Time Ordering

Lian-Ju Ning, Yong Han, Zhen-Kai Lou, Xue-Feng Xia

2022, 10(1):  159-171.  doi:10.1007/s40305-021-00342-z

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This paper studies ordering and pricing issues of a retailer who faces a linear-sensitive demand. The definition of a reasonable retail price is put forward. After that, a deterministic model for trading off sales revenue and total cost is constructed, in which the retailer determines the optimal ordering schedule and the optimal retail price simultaneously for the sake of maximizing its total profit. It is shown that the acquired retail prices are all reasonable. During the process of derivation, both ordering time points and the optimal retail price are expressed as functions of ordering times. By solving a quadratic programming model with an undetermined parameter, we demonstrate that the optimal ordering time points of the retailer are equidistant time points on the given selling periods. Further, the sensitivity of each parameter is analyzed and some meaningful conclusions are drawn. Finally, the ordering times is obtained by analyzing the derivatives of the profit function.

A Cost-Sharing Scheme for the k-Level Facility Location Game with Penalties

Feng-Min Wang, Jia-Jia Wang, Na Li, Yan-Jun Jiang, Shi-Cheng Li

2022, 10(1):  173-182.  doi:10.1007/s40305-021-00345-w

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In the k-level facility location problem with penalties,each client will be either serviced or rejected completely. And if the client is planned to be serviced, then it must be connected to asequence of k different kinds of facilities located in k levels of hierarchy. The total cost including the facility cost, connection cost and penalty cost will be jointly paid by all the clients. In the corresponding game of the k-level facility location problem with penalties, called the k-level facility location game with penalties, the total cost should be allocated to different clients. This work set out a cost-sharing scheme for the k-level facility location game with penalties that is cross-monotonic, competitive, and the approximate cost recovery is 6.

An Approximation Algorithm for the Generalized Prize-Collecting Steiner Forest Problem with Submodular Penalties

Xiao-Dan Jia, Bo Hou, Wen Liu

2022, 10(1):  183-192.  doi:10.1007/s40305-021-00355-8

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In this paper, we consider the generalized prize-collecting Steiner forest problem with submodular penalties (GPCSF-SP problem). In this problem, we are given an undirected connected graph G=(V, E) and a collection of disjoint vertex subsets V={V₁, V₂, · · ·, V_l}. Assume c:E → $\mathbb{R}$⁺ is an edge cost function and π:2^V → $\mathbb{R}$⁺ is a submodular penalty function. The objective of the GPCSF-SP problem is to find an edge subset F such that the total cost including the edge cost in F and the penalty cost of the subcollection S containing these V_i not connected by F is minimized. By using the primal-dual technique, we give a 3-approximation algorithm for this problem.