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We propose a DC (Difference of two Convex functions) formulation approach for sparse optimization problems having a cardinality or rank constraint. With the largest-k norm, an exact DC representation of the cardinality constraint is provided. We then transform the cardinality-constrained problem into a penalty function form and derive exact penalty parameter values for some optimization problems, especially for quadratic minimization problems which often appear in practice. A DC Algorithm (DCA) is presented, where the dual step at each iteration can be efficiently carried out due to the accessible subgradient of the largest-k norm. Furthermore, we can solve each DCA subproblem in linear time via a soft thresholding operation if there are no additional constraints. The framework is extended to the rank-constrained problem as well as the cardinality- and the rank-minimization problems. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods which have other penalty terms. [ABSTRACT FROM AUTHOR]

In this article we propose a simple heuristic algorithm for approaching the maximally predictable portfolio, which is constructed so that return model of the resulting portfolio would attain the largest goodness-of-fit. It is obtained by solving a fractional program in which a ratio of two convex quadratic functions is maximized, and the number of variables associated with its nonconcavity has been a bottleneck in spite of continuing endeavour for its global optimization. The proposed algorithm can be implemented by simply solving a series of convex quadratic programs, and computational results show that it yields within a few seconds a (near) Karush–Kuhn–Tucker solution to each of the instances which were solved via a global optimization method in [H. Konno, Y. Takaya and R. Yamamoto,A maximal predictability portfolio using dynamic factor selection strategy, Int. J. Theor. Appl. Fin. 13 (2010) pp. 355–366]. In order to confirm the solution accuracy, we also pose a semidefinite programming relaxation approach, which succeeds in ensuring a near global optimality of the proposed approach. Our findings through computational experiments encourage us not to employ the global optimization approach, but to employ the local search algorithm for solving the fractional program of much larger size. [ABSTRACT FROM AUTHOR]

In this paper, we present a new formulation for constructing an n-dimensional ellipsoid by generalizing the computation of the minimum volume covering ellipsoid. The proposed ellipsoid construction is associated with a user-defined parameter β∈[0,1), and formulated as a convex optimization based on the CVaR minimization technique proposed by Rockafellar and Uryasev (J. Bank. Finance 26: 1443–1471, ). An interior point algorithm for the solution is developed by modifying the DRN algorithm of Sun and Freund (Oper. Res. 52(5):690–706, ) for the minimum volume covering ellipsoid. By exploiting the solution structure, the associated parametric computation can be performed in an efficient manner. Also, the maximization of the normal likelihood function can be characterized in the context of the proposed ellipsoid construction, and the likelihood maximization can be generalized with parameter β. Motivated by this fact, the new ellipsoid construction is examined through a multiclass discrimination problem. Numerical results are given, showing the nice computational efficiency of the interior point algorithm and the capability of the proposed generalization. [ABSTRACT FROM AUTHOR]

This paper studies the peer-to-peer lending and loan application processing of LendingClub. We tried to reproduce the existing loan application processing algorithm and find features used in this process. Loan application processing is considered a binary classification problem. We used the area under the ROC curve (AUC) for evaluation of algorithms. Features were transformed with splines for improving the performance of algorithms. We considered three classification algorithms: logistic regression, buffered AUC (bAUC) maximization, and AUC maximization. With only three features, Debt-to-Income Ratio, Employment Length, and Risk Score, we obtained an AUC close to 1. We have done both in-sample and out-of-sample evaluations. The codes for cross-validation and solving problems in a Portfolio Safeguard (PSG) format are in the Appendix. The calculation results with the data and codes are posted on the website and are available for downloading. [ABSTRACT FROM AUTHOR]