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Challenges faced by water resource systems are multi-faceted. The problem can be even more pronounced in a dry continent like Australia where the water resources can often be afflicted by high salinity and turbidity. Therefore, modern water resource systems require to appropriately manage both water quality and quantity. This study aims to illustrate the trade-offs between water quantity and quality in a reservoir, based on decisions to harvest different inflow sources. Taylors Lake of the Grampians reservoir system in Western Victoria, Australia was chosen as the case study for this research as it is sufficiently complex and includes many of the contemporary water resources challenges seen around the world. Different operational scenarios were analysed which included increasingly stringent water quality criteria before the water was harvested or otherwise allowed to by-pass the storage. The study suggests that selective harvesting of water can be an option to improve the overall and long-term water quality within a reservoir, but stringent water quality measures can lead to an associated loss of overall water quantity. This research study provides useful insight to water planners and stakeholders in similar catchment settings around the world, to identify water harvesting regimes with competing water quality constraints.

A method, called an augmented subgradient method, is developed to solve unconstrained nonsmooth difference of convex (DC) optimization problems. At each iteration of this method search directions are found by using several subgradients of the first DC component and one subgradient of the second DC component of the objective function. The developed method applies an Armijo-type line search procedure to find the next iteration point. It is proved that the sequence of points generated by the method converges to a critical point of the unconstrained DC optimization problem. The performance of the method is demonstrated using academic test problems with nonsmooth DC objective functions and its performance is compared with that of two general nonsmooth optimization solvers and five solvers specifically designed for unconstrained DC optimization. Computational results show that the developed method is efficient and robust for solving nonsmooth DC optimization problems.

Piecewise linear L 1 -regression problem is formulated as an unconstrained difference of convex (DC) optimization problem and an algorithm for solving this problem is developed. Auxiliary problems ...

In clusterwise linear regression (CLR), the aim is to simultaneously partition data into a given number of clusters and to find regression coefficients for each cluster. In this paper, we propose a novel approach to model and solve the CLR problem. The main idea is to utilize the support vector machine (SVM) approach to model the CLR problem by using the SVM for regression to approximate each cluster. This new formulation of the CLR problem is represented as an unconstrained nonsmooth optimization problem, where we minimize a difference of two convex (DC) functions. To solve this problem, a method based on the combination of the incremental algorithm and the double bundle method for DC optimization is designed. Numerical experiments are performed to validate the reliability of the new formulation for CLR and the efficiency of the proposed method. The results show that the SVM approach is suitable for solving CLR problems, especially, when there are outliers in data.

The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers.

An algorithm is developed for solving clustering problems with the similarity measure defined using the L1and L? norms. It is based on an incremental approach and applies nonsmooth optimization methods to find cluster centers. Computational results on 12 data sets are reported and the proposed algorithm is compared with the X-means algorithm. © 2020, JOURNAL OF INDUSTRIAL AND JOURNAL OF INDUSTRIAL AND, Chongqing Normal University, CNU Australian Government Australian Research Council, ARC: DP190100580, The authors would like to thank two anonymous referees for their comments that helped to improve the quality of the paper. This research by Dr. A. Bagirov was funded by the Australian Government through the Australian Research Council?s Discovery Projects scheme (Project No.: DP190100580). The paper was completed when Dr. A. Bagirov visited Chongqing Normal University, China., Acknowledgements. The authors would like to thank two anonymous referees for their comments that helped to improve the quality of the paper. This research by Dr. A. Bagirov was funded by the Australian Government through the Australian Research Council’s Discovery Projects scheme (Project No.: DP190100580). The paper was completed when Dr. A. Bagirov visited Chongqing Normal University, China.

Using a nonconvex nonsmooth optimization approach, we introduce a model for semisupervised clustering (SSC) with pairwise constraints. In this model, the objective function is represented as a sum of three terms: the first term reflects the clustering error for unlabeled data points, the second term expresses the error for data points with must-link (ML) constraints, and the third term represents the error for data points with cannot-link (CL) constraints. This function is nonconvex and nonsmooth. To find its optimal solutions, we introduce an adaptive SSC (A-SSC) algorithm. This algorithm is based on the combination of the nonsmooth optimization method and an incremental approach, which involves the auxiliary SSC problem. The algorithm constructs clusters incrementally starting from one cluster and gradually adding one cluster center at each iteration. The solutions to the auxiliary SSC problem are utilized as starting points for solving the nonconvex SSC problem. The discrete gradient method (DGM) of nonsmooth optimization is applied to solve the underlying nonsmooth optimization problems. This method does not require subgradient evaluations and uses only function values. The performance of the A-SSC algorithm is evaluated and compared with four benchmarking SSC algorithms on one synthetic and 12 real-world datasets. Results demonstrate that the proposed algorithm outperforms the other four algorithms in identifying compact and well-separated clusters while satisfying most constraints.

A clusterwise linear regression problem consists of finding a number of linear functions each approximating a subset of the given data. In this paper, the limited memory bundle method is modified and combined with the incremental approach to solve this problem using its nonsmooth optimization formulation. The main contribution of the proposed method is to obtain a fast solution time for large-scale clusterwise linear regression problems. The proposed algorithm is tested on small and large real-world data sets and compared with other algorithms for clusterwise linear regression. Numerical results demonstrate that the proposed algorithm is especially efficient in data sets with large numbers of data points and input variables.

Intrusion detection systems (IDSs) are devices or software applications that monitor networks or systems for malicious activities and signals alerts/alarms when such activity is discovered. However, an IDS may generate many false alerts which affect its accuracy. In this paper, we develop a cyberattack triage algorithm to detect these alerts (so-called outliers). The proposed algorithm is designed using the clustering, optimization and distance-based approaches. An optimization-based incremental clustering algorithm is proposed to find clusters of different types of cyberattacks. Using a special procedure, a set of clusters is divided into two subsets: normal and stable clusters. Then, outliers are found among stable clusters using an average distance between centroids of normal clusters. The proposed algorithm is evaluated using the well-known IDS data sets—Knowledge Discovery and Data mining Cup 1999 and UNSW-NB15—and compared with some other existing algorithms. Results show that the proposed algorithm has a high detection accuracy and its false negative rate is very low.

This paper analyzes the single-source capacitated facility location problem (SSCFLP) with soft capacity constraints. Hence, the maximum capacity at each facility can be potentially exceed by incurr...

The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed based on these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.

Real-time detection of cyber threats is a challenging task in cyber security. With the advancement of technology and ease of access to the internet, more and more individuals and organizations are becoming the target for various cyber attacks such as malware, ransomware, spyware. The target of these attacks is to steal money or valuable information from the victims. Signature-based detection methods fail to keep up with the constantly evolving new threats. Machine learning based detection has drawn more attention of researchers due to its capability of detecting new and modified attacks based on previous attack’s behaviour. The number of malicious activities in a certain domain is significantly low compared to the number of normal activities. Therefore, cyber threats detection data sets are imbalanced. In this paper, we proposed a partial undersampling method to deal with imbalanced data for detecting cyber threats.

In this chapter, we discuss the implementations of the incremental clustering algorithms described in previous chapters. We also provide some recommendations on the choice of parameters of the algorithms. In addition, the choice of parameters in the algorithm for finding starting cluster centers, which is an important part of all nonsmooth optimization based clustering algorithms, is discussed. Finally, data sets used in numerical experiments are described and grouped into different subclasses according to their size.

In the classical team orienteering problem (TOP), a fixed fleet of vehicles is employed, each of them with a limited driving range. The manager has to decide about the subset of customers to visit, as well as the visiting order (routes). Each customer offers a different reward, which is gathered the first time that it is visited. The goal is then to maximize the total reward collected without exceeding the driving range constraint. This paper analyzes a more realistic version of the TOP in which the driving range limitation is considered as a soft constraint: every time that this range is exceeded, a penalty cost is triggered. This cost is modeled as a piece-wise function, which depends on factors such as the distance of the vehicle to the destination depot. As a result, the traditional reward-maximization objective becomes a non-smooth function. In addition, a second objective, regarding the design of balanced routing plans, is considered as well. A mathematical model for this non-smooth and bi-objective TOP is provided, and a biased-randomized algorithm is proposed as a solving approach.

This chapter is devoted to the clustering algorithms based on DC optimization approaches. The incremental DC clustering algorithm is studied first. Then the DC diagonal bundle clustering algorithm is described. Finally, the incremental DCA for clustering is presented. We provide the detailed descriptions and flowcharts of these algorithms.

This chapter presents different formulations of the clustering problem using various optimization approaches. They include the mixed integer programming, the general nonsmooth, and the nonsmooth DC formulations of the clustering problem. We introduce the auxiliary clustering problem and study optimality conditions for both the clustering and the auxiliary clustering problems. Finally, we discuss the smoothing of the clustering and the auxiliary clustering problems.

This chapter is devoted to the description of some general methods of nonsmooth optimization. The first two—the standard subgradient and the proximal bundle methods—form the basis for numerical nonsmooth optimization. In addition, we will focus on the methods that will be utilized in the clustering algorithms given in Part II of this book. They are the limited memory bundle method, DC diagonal bundle method, nonsmooth DC method, DC algorithm, discrete gradient method, and the method based on the smoothing techniques. For each of these methods we present a flowchart, give some clarifying explanations, and study convergence properties.

This chapter is devoted to the most popular heuristic partitional clustering algorithms such as k-means, k-medians, and k-medoids. In addition, we give an overview of some clustering algorithms based on mixture models, self-organizing map, and fuzzy clustering. The description of these algorithms as well as their flowcharts is presented. The convergence results for the k-means and the k-medians algorithms using nonsmooth optimization techniques are discussed.

Metaheuristic algorithms are well-known optimization tools which have been used in many applications including those in data mining. In particular, these algorithms are well suited for solving nonconvex clustering problems since they are able to find global solutions whereas the traditional clustering algorithms such as k-means can only guarantee convergence to a local solution. The metaheuristic algorithms applied to solve clustering problems include the tabu search and the simulated annealing algorithms as well as evolutionary algorithms like the genetic algorithm, the artificial bee colony optimization, the particle swarm optimization, and the ant colony optimization algorithms. The description of these algorithms and their flowcharts are given in this chapter.

Performance and evaluation measures for clustering algorithms are described in this chapter. We discuss various evaluation measures in order to judge the quality of clustering solutions obtained by clustering algorithms. These measures include cluster validity indices, silhouette coefficients and plots, purity, and others. In addition, we introduce performance profiles to compare efficiency of different clustering algorithms using the accuracy of obtained solutions, the number of distance function evaluations, and computational time.

This chapter presents the general scheme for incremental clustering algorithms. These algorithms start with the calculation of the center of the whole data set and add one cluster center at each iteration. A procedure—that utilizes the incremental essence of clustering algorithms—for finding starting cluster centers is given. Furthermore, the multi-start incremental clustering algorithm that uses this procedure is discussed. Finally, the incremental k-medians algorithm is described, and the discussion on the decrease of its computational complexity is provided. The detailed descriptions of all algorithms as well as their flowcharts are given.

In this chapter, we first define the commonly used tasks and terminologies in data analysis and show the importance of data clustering. Then we give a mathematical formulation of the clustering problem and explain the frequently used similarity measures. Finally, we provide a short survey on different types of existing clustering algorithms and describe some of the most popular applications of data clustering.

In this chapter, we introduce two discrete gradient methods that can be considered as semi-derivative free methods in a sense that they do not use subgradient information and they do not approximate the subgradient but at the end of the solution process (i.e., near the optimal point). The introduced methods are the original discrete gradient method for small-scale nonsmooth optimization and its limited memory bundle version the limited memory discrete gradient bundle method for medium- and semi-large problems.

This chapter is devoted to algorithms for solving nonsmooth unconstrained difference of convex optimization problems. Different types of stationarity conditions are discussed and the relationship between sets of different stationary points (critical, Clarke stationary and inf-stationary) is established. Bundle methods are developed based on a nonconvex piecewise linear model of the objective function and the convergence of these methods is studied. Numerical results are presented to demonstrate the performance of the methods.