For controlled discrete-time stochastic processes we introduce a new class of dynamic risk measures, which we call process-based. Their main features are that they measure risk of processes that are functions of the history of the base process. We introduce a new concept of conditional stochastic time consistency and we derive the structure of process-based … Read more
Bottom-up search methods for determining the efficient set of a multiple objective linear programming (MOLP) problem have a valuable advantage that they can quickly give efficient subsets of the MOLP problem to the decision makers. Main difficulties of the previously appeared bottom-up search methods are finding all efficient extreme points adjacent to and enumerating all … Read more
In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of \L ojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the H\”older type global weak sharp minima with explicitly calculated … Read more
We consider the information relaxation approach for calculating performance bounds for stochastic dynamic programs (DPs), following Brown, Smith, and Sun (2010). This approach generates performance bounds by solving problems with relaxed nonanticipativity constraints and a penalty that punishes violations of these constraints. In this paper, we study infinite horizon DPs with discounted costs and consider … Read more
We consider the directed Steiner tree problem (DSTP) with a constraint on the total number of arcs (hops) in the tree. This problem is known to be NP-hard, and therefore, only heuristics can be applied in the case of its large-scale instances. For the hop-constrained DSTP, we propose local search strategies aimed at improving any … Read more
We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. SA spectral theory on semi-simple complex Jordan algebras is used as atool to prove the convexity results. Possible applications to optimization and statistics are indicated CitationPreprint, University of Notre Dame, August 2014ArticleDownload View PDF
An extension of a proximal point algorithm for difference of two convex functions is presented in the context of Riemannian manifolds of nonposite sectional curvature. If the sequence generated by our algorithm is bounded it is proved that every cluster point is a critical point of the function (not necessarily convex) under consideration, even if … Read more
This paper presents a method for finding global optima to constrained nonlinear programs via slack variables. The method only applies if all functions involved are of class C1 but without any further qualification on the types of constraints allowed; it proceeds by reformulating the given program into a bi-objective program that is then solved for … Read more
We consider a class of multicriteria stochastic optimization problems that features benchmarking constraints based on conditional value-at-risk and second-order stochastic dominance. We develop alternative mixed-integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints. We give the complete linear description of two non-convex substructures appearing in these … Read more
We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states $\{\rho_1, \dots, \rho_k\}$ to another such set $\{\hat\rho_1, \dots, \hat\rho_k\}$. In other words, we must find a {\em completely positive linear map}, if it exists, that maps a given set of density matrices to … Read more