Hierarchical Clustering Based on Mathematical Optimization

DC approximation approaches for sparse optimization

It is undoubtedly that mathematical modelling and optimisation play a key role in the supply chain and the production management (SCPM). In this paper, we provide a survey on DC (Difference of Convex function) programming and DCA (DC Algorithm), a state-of-the-art optimisation approach for challenging problems in SCPM. DC programming and DCA constitute the backbone of non-convex programming and global optimisation. Whilst DC programming and DCA were widely and successfully investigated in many areas, it seems that they were not so much popular in the community of SCPM. There is therefore a need to further develop this efficient and scalable approach for SCPM applications, especially for large-scale problems in the context of Big data. For such purpose, this paper aims to present benchmark models and state-of-the-art DCA-based methods for solving challenging problems in SCPM systems. We prove that all the benchmark classes of optimisation models appeared in SCPM systems can be formulated/reformulated as a DC program and show how to solve these classes of problems by DCA-based algorithms. We offer the community of researchers in SCPM efficient algorithms in a unified DC programming framework to tackle various applications such as supply chain design, scheduling, multi-stage production/inventory system, vehicle routing, …

In this letter, we consider the nonnegative matrix factorization (NMF) problem and several NMF variants. Two approaches based on DC (difference of convex functions) programming and DCA (DC algorithm) are developed. The first approach follows the alternating framework that requires solving, at each iteration, two nonnegativity-constrained least squares subproblems for which DCA-based schemes are investigated. The convergence property of the proposed algorithm is carefully studied. We show that with suitable DC decompositions, our algorithm generates most of the standard methods for the NMF problem. The second approach directly applies DCA on the whole NMF problem. Two algorithms-one computing all variables and one deploying a variable selection strategy-are proposed. The proposed methods are then adapted to solve various NMF variants, including the nonnegative factorization, the smooth regularization NMF, the sparse regularization NMF, the multilayer NMF, the convex/convex-hull NMF, and the symmetric NMF. We also show that our algorithms include several existing methods for these NMF variants as special versions. The efficiency of the proposed approaches is empirically demonstrated on both real-world and synthetic data sets. It turns out that our algorithms compete favorably with five state-of-the-art alternating nonnegative least squares algorithms.

Time-index formulation for the earliness tardiness scheduling problem has received a great attention from many researchers because lower bound obtained by linear relaxation is rather good. Much work is devoted to tackle its upper bound. In this paper, we consider this formulation by additionally proposing a deadline for each job. We also propose an approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) to find upper bound efficiently for this problem. The results obtained are promising.

In this chapter, a so-called DCA method based on a DC (difference of convex functions) optimization approach for solving large-scale distance geometry problems is developed. Two main problems are considered: the exact and the general distance geometry problems. Different formulations of equivalent DC programs are introduced. Substantial subdifferential calculations permit to compute sequences of iterations in the DCA quite simply and allow exploiting sparsity in the large-scale setting. For improving the computational efficiency of the DCA schemes we investigate several techniques. A two-phase algorithm using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix, a spanning trees procedure, and a smoothing technique are investigated in order to compute a good starting point (SP) for the DCAs. An important issue in the DC optimization approach is well exploited, say the nice effect of DC decompositions of the objective functions. For this purpose we propose several equivalent DC formulations based on the stability of Lagrangian duality and the regularization techniques. Finally, many numerical simulations of the molecular optimization problems with up to 12,567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations, the globality of found solutions, the robustness, and the efficiency of our algorithms.

In this paper, a version of K-median problem, one of the most popular and best studied clustering measures, is discussed. The model using squared Euclidean distances terms to which the K-means algorithm has been successfully applied is considered. A fast and robust algorithm based on DC (Difference of Convex functions) programming and DC Algorithms (DCA) is investigated. Preliminary numerical solutions on real-world databases show the efficiency and the superiority of the appropriate DCA with respect to the standard K-means algorithm.

We investigate a nonconvex, nonsmooth optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) for the reduced-rank multitask linear regression problem with covariance estimation. The objective is to model the linear relationship between a multitask response and more explanatory variables by estimating a low-rank coefficient matrix and a covariance matrix. The problem is formulated as minimizing the constrained negative log-likelihood function of these two matrix variables. Then, we consider a reformulation of this problem which takes the form of a partial DC program i.e. it is a standard DC program for each variable when fixing the other variable. Next, an alternating version of a standard DCA scheme is developed. Numerical results on many synthetic multitask linear regression datasets and benchmark real datasets show the efficiency of our approach in comparison with the existing alternating/joint methods.

In this paper, we consider four optimization models for solving the Linear Complementarity (LCP) Problems. They are all formulated as DC (Difference of Convex functions) programs for which the unified DC programming and DCA (DC Algorithms) are applied. The resulting DCA are simple: they consist of solving either successive linear programs, or successive convex quadratic programs, or simply the projection of points on \(\mathbb{R}_{+}^{2n}\). Numerical experiments on several test problems illustrate the efficiency of the proposed approaches in terms of the quality of the obtained solutions, the speed of convergence, and so on. Moreover, the comparative results with Lemke algorithm, a well known method for the LCP, show that DCA outperforms the Lemke method.

This paper addresses the problem of feature selection for Multi-class Support Vector Machines (MSVM). Basing on the l 0 and the l 2-l 0 regularization we consider two models for this problem. The l 0-norm is approximated by a suitable way such that the resulting optimization problems can be expressed as DC (Difference of Convex functions) programs for which DC programming and DC Algorithms (DCA) are investigated. The preliminary numerical experiments on real-world datasets show the efficiency and the superiority of our methods versus one of the best standard algorithms on booth feature selection and classification.

A sparse extreme learning machine framework by continuous optimization algorithms and its application in pattern recognition

This letter proposes a novel approach using the [Formula: see text]-norm regularization for the sparse covariance matrix estimation (SCME) problem. The objective function of SCME problem is composed of a nonconvex part and the [Formula: see text] term, which is discontinuous and difficult to tackle. Appropriate DC (difference of convex functions) approximations of [Formula: see text]-norm are used that result in approximation SCME problems that are still nonconvex. DC programming and DCA (DC algorithm), powerful tools in nonconvex programming framework, are investigated. Two DC formulations are proposed and corresponding DCA schemes developed. Two applications of the SCME problem that are considered are classification via sparse quadratic discriminant analysis and portfolio optimization. A careful empirical experiment is performed through simulated and real data sets to study the performance of the proposed algorithms. Numerical results showed their efficiency and their superiority compared with seven state-of-the-art methods.

The weighted sum rate (WSR) maximizing linear precoding algorithm is studied in large correlated multiple-input single-output (MISO) interference broadcast channels (IBC). We consider an iterative WSR design via difference of convex functions (DC) programming as in [1], [2] and [3], focusing on the version in [3]. We propose an asymptotic approximation of the signal-to-interference plus noise ratio (SINR) at every iteration.

The security requirements of data transmission over wireless networks are energy-limited in many situations. In this paper, the secure energy efficiency (EE), is defined as the ratio of the secrecy rate to the total power, and is investigated in a systematic way considering a decode-and-forward (DF) relay network with a potential eavesdropper. We maximize the secure EE subject to the individual power constraint and the minimum decoding rate constraint of the relay. To deal with the nonconvexity of the formulated problem, a fractional programming approach embedded with DC (difference of convex functions) programming is proposed to solve the problem by two-layer iterations. The key point of the proposed algorithm is to translate the primal problem into a series of convex subproblems, which can be solved by convex programming. It is verified by simulation that the proposed algorithm achieves much better secure EE than the conventional secrecy rate maximization yet with a minor performance loss measured by average secrecy rate or secrecy outage probability.

The issue of physical layer security for single-input-multiple-output (SIMO) wiretap channel with uncoordinated cooperative jamming (UCJ) is addressed in this letter, and we focus on power allocation in secrecy rate maximization (SRM) problem. Differing from the existing works, a practical UCJ scheme is proposed by using multiple single-antenna helpers where each helper transmits a jamming signal independently to confound the eavesdropper. Assuming that statistical channel state information (CSI) concerning the eavesdropper is available, a modified DC (difference of convex function) programming method is provided to solve the SRM problem under secrecy outage probability constraint and an achievable solution of power allocation is obtained. Numerical results show that the proposed scheme has satisfactory secrecy performance especially when the number of helpers is large.

In this paper, the secure green communication over amplify-and-forward (AF) relaying channels is investigated by using the metric of secure energy efficiency (EE), defined as the number of bits securely delivered per unit energy consumption. We maximize the secure EE with the given maximum power and the minimum secrecy rate requirements. The difficulty of the problem comes from the nonconvexity of both the objective function and the secrecy rate constraint. Therefore, a suboptimal solution scheme, based on fractional programming, dual decomposition, and DC (difference of convex functions) programming, is addressed to solve this problem iteratively. The proposed solution scheme composed of three-layer iterations transforms the primal problem into easier subproblems at each iterative layer, such that the resulting subproblems can be solved by the corresponding optimization methods mentioned above. The simulation results demonstrate that the secure EE of the proposed scheme is superior than that obtained by the conventional schemes with a small decrease of the average secrecy rate.