Programming with linear fractional functionals

In problems of portfolio selection the reward-risk ratio criterion is optimized to search for a risky portfolio offering the maximum increase of the mean return, compared to the risk-free investment opportunities. In the classical model, following Markowitz, the risk is measured by the variance thus representing the Sharpe ratio optimization and leading to the quadratic optimization problems. Several polyhedral risk measures, being linear programming (LP) computable in the case of discrete random variables represented by their realizations under specified scenarios, have been introduced and applied in portfolio optimization. The reward-risk ratio optimization with polyhedral risk measures can be transformed into LP formulations. The LP models typically contain the number of constraints proportional to the number of scenarios while the number of variables (matrix columns) proportional to the total of the number of scenarios and the number of instruments. Real-life financial decisions are usually based on more advanced simulation models employed for scenario generation where one may get several thousands scenarios. This may lead to the LP models with huge number of variables and constraints thus decreasing their computational efficiency and making them hardly solvable by general LP tools. We show that the computational efficiency can be then dramatically improved by alternative models based on the inverse ratio minimization and taking advantages of the LP duality. In the introduced models the number of structural constraints (matrix rows) is proportional to the number of instruments thus not affecting seriously the simplex method efficiency by the number of scenarios and therefore guaranteeing easy solvability.

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“…The ratio model (21) can be linearized by the Charnes and Cooper substitutions (Charnes and Cooper 1962;Williams 2013)…”

Section: Smad Modelmentioning

confidence: 99%

Efficient optimization of the reward-risk ratio with polyhedral risk measures

Ogryczak

Śliwiński

2017

Math Meth Oper Res

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“…125-130) This transformation is due to Derman and his student Klein (see Derman 1962 andKlein 1962). See Charnes and Cooper (1962) and Wagner and Yuan (1968) for more general treatment of linear fractional programming.…”

Section: Theorem 17 Let the Assumptions (A1) (A2) And (A3) Hold Andmentioning

confidence: 99%

Derman’s book as inspiration: some results on LP for MDPs

Kallenberg

2012

Ann Oper Res

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In 1976 I was looking for a suitable subject for my PhD thesis. My thesis advisor Arie Hordijk and I found a lot of inspiration in Derman's book (Finite state Markovian decision processes, Academic Press, New York, 1970). Since that time I was interested in linear programming methods for Markov decision processes. In this article I will describe some results in this area on the following topics: (1) MDPs with the average reward criterion; (2) additional constraints; (3) applications. These topics are the main elements of Derman's book.

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“…Therefore available algorithms [4] for the gain problem cannot be directly applied to the present problem. Thus the approach considered here consists of transforming (6) through (8) to a linear program using the method of Charnes and Cooper [2] and exploiting the one-tree structure of the solution. Considering the relationship of the problem to the special case of the Markov ratio décision process, one has an obvious lemma as stated below.…”

Section: Statement Of the Problemmentioning

confidence: 99%