1,215 research outputs found
Optimization with multivariate conditional value-at-risk constraints
For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice.
As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to
finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the
proposed solution methods
Optimization with multivariate conditional value-at-risk constraints
For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice.
As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to
finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the
proposed solution methods
Artificial Noise-Aided Biobjective Transmitter Optimization for Service Integration in Multi-User MIMO Gaussian Broadcast Channel
This paper considers an artificial noise (AN)-aided transmit design for
multi-user MIMO systems with integrated services. Specifically, two sorts of
service messages are combined and served simultaneously: one multicast message
intended for all receivers and one confidential message intended for only one
receiver and required to be perfectly secure from other unauthorized receivers.
Our interest lies in the joint design of input covariances of the multicast
message, confidential message and artificial noise (AN), such that the
achievable secrecy rate and multicast rate are simultaneously maximized. This
problem is identified as a secrecy rate region maximization (SRRM) problem in
the context of physical-layer service integration. Since this bi-objective
optimization problem is inherently complex to solve, we put forward two
different scalarization methods to convert it into a scalar optimization
problem. First, we propose to prefix the multicast rate as a constant, and
accordingly, the primal biobjective problem is converted into a secrecy rate
maximization (SRM) problem with quality of multicast service (QoMS) constraint.
By varying the constant, we can obtain different Pareto optimal points. The
resulting SRM problem can be iteratively solved via a provably convergent
difference-of-concave (DC) algorithm. In the second method, we aim to maximize
the weighted sum of the secrecy rate and the multicast rate. Through varying
the weighted vector, one can also obtain different Pareto optimal points. We
show that this weighted sum rate maximization (WSRM) problem can be recast into
a primal decomposable form, which is amenable to alternating optimization (AO).
Then we compare these two scalarization methods in terms of their overall
performance and computational complexity via theoretical analysis as well as
numerical simulation, based on which new insights can be drawn.Comment: 14 pages, 5 figure
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Box-constrained vector optimization: a steepest descent method without “a priori” scalarization
In this paper a notion of descent direction for a vector function defined on a box is introduced. This concept is based on an appropriate convex combination of the “projected” gradients of the components of the objective functions. The proposed approach does not involve an “apriori” scalarization since the coefficients of the convex combination of the projected gradients are the solutions of a suitable minimization problem depending on the feasible point considered. Subsequently, the descent directions are considered in the formulation of a first order optimality condition for Pareto optimality in a box-constrained multiobjective optimization problem. Moreover, a computational method is proposed to solve box-constrained multiobjective optimization problems. This method determines the critical points of the box constrained multiobjective optimization problem following the trajectories defined through the descent directions mentioned above. The convergence of the method to the critical points is proved. The numerical experience shows that the computational method efficiently determines the whole local Pareto front.Multi-objective optimization problems, path following methods, dynamical systems, minimal selection.
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