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This paper studies four families of polyhedral norms parametrized by a single parameter. The first two families consist of the CVaR norm (which is equivalent to the D-norm, or the largest- $$k$$ norm) and its dual norm, while the second two families consist of the convex combination of the $$\ell _1$$ - and $$\ell _\infty $$ -norms, referred to as the deltoidal norm, and its dual norm. These families contain the $$\ell _1$$ - and $$\ell _\infty $$ -norms as special limiting cases. These norms can be represented using linear programming (LP) and the size of LP formulations is independent of the norm parameters. The purpose of this paper is to establish a relation of the considered LP-representable norms to the $$\ell _p$$ -norm and to demonstrate their potential in optimization. On the basis of the ratio of the tight lower and upper bounds of the ratio of two norms, we show that in each dual pair, the primal and dual norms can equivalently well approximate the $$\ell _p$$ - and $$\ell _q$$ -norms, respectively, for $$p,q\in [1,\infty ]$$ satisfying $$1/p+1/q=1$$ . In addition, the deltoidal norm and its dual norm are shown to have better proximity to the $$\ell _p$$ -norm than the CVaR norm and its dual. Numerical examples demonstrate that LP solutions with optimized parameters attain better approximation of the $$\ell _{2}$$ -norm than the $$\ell _1$$ - and $$\ell _\infty $$ -norms do. [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]

An introduction is presented which discusses articles within the issue on topics including recent developments in modelling and optimization of engineering systems, Value-at-Risk, and Conditional Value-at-Risk.