The optimization of a quadratic function subject to linear constraints

Markowitz, Harry M.

doi:10.1002/nav.3800030110

Cited by 433 publications

(224 citation statements)

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“…Whereas, since Markowitz (1956), it has been possible to compute a nondominated frontier as in Figure 1, it has not been until recently, namely by Hirschberger et al (2013), that it has been possible to compute a tri-criterion nondominated surface as in Figure 2. Using the CIOS (Custom Investment Objective Solver)…”

Section: Modelmentioning

confidence: 99%

“…In the seminal work of Markowitz (1952Markowitz ( , 1956) and later in books (1959,1987,2000) only the expected returns and the covariances of the returns of all considered assets are taken into account when attempting to locate optimal portfolios. However, recent studies suggest that in many situations a more complex decision model may be at work (Abdelaziz et al, 2007;Ballestero et al, 2012;Bollen, 2007;Dorfleitner & Utz, 2012;Hallerbach et al, 2004;Steuer et al, 2007;.…”

Section: Introductionmentioning

confidence: 99%

“…In their description, one first computes the entire nondominated frontier, for instance with the critical line method of Markowitz (1956), and then notes that each portfolio along the nondominated frontier has it own risk tolerance (or risk aversion) parameter. The goal is to compute the implied risk tolerances of given portfolios (which are likely not to be on the nondominated frontier) by matching them with close-by portfolios that are on the nondominated frontier and then considering those portfolios' risk tolerances as proxies for the risk tolerances of the given portfolios.…”

Section: Introductionmentioning

confidence: 99%

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Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds

Utz

Wimmer

Hirschberger³

et al. 2014

European Journal of Operational Research

127

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We present a framework for inverse optimization in a Markowitz portfolio model that is extended to include a third criterion. The third criterion causes the traditional nondominated frontier to become a surface. Until recently, it had not been possible to compute such a surface. But by using a new method that is able to generate the nondominated surfaces of tri-criterion portfolio selection problems, we are able to compute via inverse optimization the implied risk tolerances of given funds that pursue an additional objective beyond risk and return. In applying this capability to a broad sample of conventional and socially responsible (SR) mutual funds, we find that after the screening process there appears to be no significant difference between how assets are allocated in socially responsible and conventional mutual funds, which is likely to be different from what most SR investors would expect.

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Section: Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds

Utz

Wimmer

Hirschberger³

et al. 2014

European Journal of Operational Research

127

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“…Applications of the concept of cone dominance are varied and include efficiency in economic planning [4], [11], mathematical statistics [31], maximizing utility vectors in exchange equilibrium [41,% [32], risk-return trade-offs in portfolio selection [33], [34], risk sharing and group decisions [35], and many others as suggested in [36] and the collection [37]. …”

Section: Introductionmentioning

confidence: 99%

The structure of admissible points with respect to cone dominance

Bitran

Magnanti

1979

J Optim Theory Appl

122

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We study the set of admissible (pareto-optimal) points of a closed convex set X when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions of X, we (i) extend a representation theorem of Arrow, Barankin and Blackwell by showing that all admissible points are either limit points of certain "strictly admissible" alternatives or translations of such limit points by rays in the closure of the preference cone, and (ii) show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed upon X, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element of X to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.

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“…The reason of this attitude is probably due to the widespread opinion that closed form formulae of the whole set of proportional efficient retentions may be obtained only under no correlation, while in the correlation case there is the need to make recourse to a sequential procedure which is the counterpart of the critical line algorithm of the classical portfolio problem (see Markowitz [23], [24], [25]). The issue has been treated e.g.…”

mentioning

confidence: 99%