Browsing by Subject "Vector optimization"

Now showing 1 - 5 of 5

Open Access
A tri-objective reformulation for the dynamic mean-variance problem
(2024-07) Çolak, Muhammed Mustafa
The classical mean-variance problem aims to find a portfolio that minimizes a linear combination of the expectation and the variance of the terminal wealth. The dynamic version of the problem is known to be time-inconsistent in the classical sense, which makes the scalar dynamic programming approach inapplicable. By decomposing variance into two separate objectives, we introduce a tri-objective formulation in a discrete-time framework that generalizes the scalar problem and can reduce to the original setting. Using a less restrictive concept of time-consistency in a vector-valued sense, we show that the new formulation is time-consistent. Following the literature on set optimization, we develop a set-valued dynamic programming principle with the upper image of the vector-valued problem used as a value function. Finally, we reduce the generalized solutions of the formulation to the classical mean-variance problem using the minimal points of the three-dimensional upper images. We compute portfolios that are optimal for the initial mean-variance problem, and that remain time-consistent with respect to the tri-objective formulation.
Open Access
Computation of systemic risk measures: a mixed-integer programming approach
(INFORMS Inst.for Operations Res.and the Management Sciences, 2023-09-22) Ararat, Çaǧın; Meimanjan, N.
Systemic risk is concerned with the instability of a financial system whose members are interdependent in the sense that the failure of a few institutions may trigger a chain of defaults throughout the system. Recently, several systemic risk measures have been proposed in the literature that are used to determine capital requirements for the members subject to joint risk considerations. We address the problem of computing systemic risk measures for systems with sophisticated clearing mechanisms. In particular, we consider an extension of the Rogers-Veraart network model where the operating cash flows are unrestricted in sign. We propose a mixed-integer programming problem that can be used to compute clearing vectors in this model. Because of the binary variables in this problem, the corresponding (set-valued) systemic risk measure fails to have convex values in general. We associate nonconvex vector optimization problems with the systemic risk measure and provide theoretical results related to the weighted-sum and Pascoletti-Serafini scalarizations of this problem. Finally, we test the proposed formulations on computational examples and perform sensitivity analyses with respect to some model-specific and structural parameters. Copyright: © 2023 INFORMS.
Open Access
Dynamic mean-variance problem: recovering time-consistency
(2021-08) Düzoylum, Seyit Emre
As the foundation of modern portfolio theory, Markowitz’s mean-variance port-folio optimization problem is one of the fundamental problems of ﬁnancial math-ematics. The dynamic version of this problem in which a positive linear com-bination of the mean and variance objectives is minimized is known to be time-inconsistent, hence the classical dynamic programming approach is not applicable. Following the dynamic utility approach in the literature, we consider a less re-strictive notion of time-consistency, where the weights of the mean and variance are allowed to change over time. Precisely speaking, rather than considering a ﬁxed weight vector throughout the investment period, we consider an adapted weight process. Initially, we start by extending the well-known equivalence be-tween the dynamic mean-variance and the dynamic mean-second moment prob-lems in a general setting. Thereby, we utilize this equivalence to give a complete characterization of a time-consistent weight process, that is, a weight process which recovers the time-consistency of the mean-variance problem according to our deﬁnition. We formulate the mean-second moment problem as a biobjective optimization problem and develop a set-valued dynamic programming principle for the biobjective setup. Finally, we retrieve back to the dynamic mean-variance problem under the equivalence results that we establish and propose a backward-forward dynamic programming scheme by the methods of vector optimization. Consequently, we compute both the associated time-consistent weight process and the optimal solutions of the dynamic mean-variance problem.
Open Access
An iterative vertex enumeration method for objective space based vector optimization algorithms
(EDP Sciences, 2021-03-02) Kaya, İrfan Caner; Ulus, Firdevs
An application area of vertex enumeration problem (VEP) is the usage within objective space based linear/convex vector optimization algorithms whose aim is to generate (an approximation of) the Pareto frontier. In such algorithms, VEP, which is defined in the objective space, is solved in each iteration and it has a special structure. Namely, the recession cone of the polyhedron to be generated is the ordering cone. We consider and give a detailed description of a vertex enumeration procedure, which iterates by calling a modified “double description (DD) method” that works for such unbounded polyhedrons. We employ this procedure as a function of an existing objective space based vector optimization algorithm (Algorithm 1); and test the performance of it for randomly generated linear multiobjective optimization problems. We compare the efficiency of this procedure with another existing DD method as well as with the current vertex enumeration subroutine of Algorithm 1. We observe that the modified procedure excels the others especially as the dimension of the vertex enumeration problem (the number of objectives of the corresponding multiobjective problem) increases.
Open Access
Tractability of convex vector optimization problems in the sense of polyhedral approximations
(Springer New York LLC, 2018) Ulus, Firdevs
There are different solution concepts for convex vector optimization problems (CVOPs) and a recent one, which is motivated from a set optimization point of view, consists of finitely many efficient solutions that generate polyhedral inner and outer approximations to the Pareto frontier. A CVOP with compact feasible region is known to be bounded and there exists a solution of this sense to it. However, it is not known if it is possible to generate polyhedral inner and outer approximations to the Pareto frontier of a CVOP if the feasible region is not compact. This study shows that not all CVOPs are tractable in that sense and gives a characterization of tractable problems in terms of the well known weighted sum scalarization problems.