Minimization of the k-th maximum and its application on LMS regression and VaR optimization

• Published in: The Journal of the Operational Research Society Vol. 63; no. 11; pp. 1479 - 1491
• Main Authors: Huang, X, Xu, J, Wang, S, Xu, C
• Format: Journal Article
• Language: English
• Published: London Taylor & Francis 01.11.2012; Palgrave Macmillan; Palgrave Macmillan UK; Taylor & Francis Ltd
• Subjects: Algorithms; Applied sciences; Approximation; Business and Management; Datasets; Estimators; Exact sciences and technology; General Paper; General Papers; Integers; least median of squares; Linear programming; Linear regression; Management; Mathematical programming; Objective functions; Operational research and scientific management; Operational research. Management science; Operations research; Operations Research/Decision Theory; Optimal solutions; Optimization; piecewise linear optimization; Portfolio theory; Risk theory. Actuarial science; Statistical variance; value-at-risk; piecewise linear optimization; least median of squares; value-at-risk; Warranty; Necessary and sufficient condition; Local optimum; Financial management; Minimization; Linear programming; Risk analysis; Optimization; Least squares method; Risk management; Portfolio management; Quantile; Piecewise linearization; Piecewise-linear techniques
• ISSN: 0160-5682; 1476-9360
• DOI: 10.1057/jors.2011.163

Motivated by two important problems, the least median of squares (LMS) regression and value-at-risk (VaR) optimization, this paper considers the problem of minimizing the k-th maximum for linear functions. For this study, a sufficient and necessary condition of local optimality is given. From this condition and other properties, we propose an algorithm that uses linear programming technique. The algorithm is assessed on real data sets and the experiments for LMS regression and VaR optimization both show its effectiveness.