Determination of trace elements in soils with laser-induced breakdown spectroscopy is significantly affected by the matrix effect, due to large variations in chemical composition and physical property of different soils. Spectroscopic data treatment with univariate models often leads to poor analytical performances. We have developed in this work a multivariate model using machine learning algorithms based on a back-propagation neural network (BPNN). Beyond the classical chemometry approach, machine learning, with tremendous progresses the last years especially for image processing, is offering an ensemble of powerful and constantly renewed algorithms and tools efficient for the different steps in the construction of a spectroscopic data treatment model, including feature selection and neural network training. Considering the matrix effect as the focus of this work, we have developed the concept of generalized spectrum, where the information about the soil matrix is explicitly included in the input vector of the model as an additional dimension. After a brief presentation of the experimental procedure and the results of regression with a univariate model, the development of the multivariate model will be described in detail together with its analytical performances, showing average relative errors of calibration ( REC ) and of prediction ( REP ) within the range of 5–6%.

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We are interested in developing DC (Difference-of-Convex) programming approach for solving higher-order moment (Mean-Variance-Skewness-Kurtosis) portfolio selection problem. The portfolio selection with higher moments can be formulated as a nonconvex quartic multivariate polynomial optimization. Based on the recent development in Difference-of-Convex-Sums-of-Squares (DCSOS) decomposition techniques for polynomial optimization, we can reformulate this problem as a DC program which can be solved by a well-known DC algorithm -DCA. We have also proposed an improved DC algorithm called Boosted-DCA (BDCA) based on an Armijo type line search to accelerate the convergence of DCA. We introduce this acceleration technique to both DC algorithm based on DCSOS decomposition proposed in this paper and the DC algorithm based on universal DC decomposition proposed in our previous paper. Results in numerical simulation show good performance of our proposed algorithms in portfolio optimization.

We are interested in solving the Asymmetric Eigenvalue Complementarity Problem (AEiCP) by accelerated Difference-of-Convex (DC) algorithms. Two novel hybrid accelerated DCA: the Hybrid DCA with Line search and Inertial force (HDCA-LI) and the Hybrid DCA with Nesterov's extrapolation and Inertial force (HDCA-NI), are established. We proposed three DC programming formulations of AEiCP based on Difference-of-Convex-Sums-of-Squares (DC-SOS) decomposition techniques, and applied the classical DCA and 6 accelerated variants (BDCA with exact and inexact line search, ADCA, InDCA, HDCA-LI and HDCA-NI) to the three DC formulations. Numerical simulations of 7 DCA-type methods against state-of-the-art optimization solvers IPOPT, KNITRO and FILTERSD, are reported.

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In this paper, we discuss the solution of a Quadratic Eigenvalue Complementarity Problem (QEiCP) by using Difference of Convex (DC) programming approaches. We first show that QEiCP can be represented as dc programming problem. Then we investigate different dc programming formulations of QEiCP and discuss their dc algorithms based on a well-known method -DCA. A new local dc decomposition is proposed which aims at constructing a better dc decomposition regarding to the specific feature of the target problem in some neighborhoods of the iterates. This new procedure yields faster convergence and better precision of the computed solution. Numerical results illustrate the efficiency of the new dc algorithms in practice.

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