Weak differentiability of product measures

In this paper, we study cost functions over a finite collection of random variables. For these types of models, a calculus of differentiation is developed that allows us to obtain a closed-form expression for derivatives where "differentiation" has to be understood in the weak sense. The technique for proving the results is new and establishes an interesting link between functional analysis and gradient estimation. The key contribution of this paper is a product rule of weak differentiation. In addition, a product rule of weak analyticity is presented that allows for Taylor series approximations of finite products measures. In particular, from characteristics of the individual probability measures, a lower bound (i.e., domain of convergence) can be established for the set of parameter values for which the Taylor series converges to the true value. Applications of our theory to the ruin problem from insurance mathematics and to stochastic activity networks arising in project evaluation review techniques are provided. © 2010 INFORMS

Derivatives of Markov kernels and their Jordan decomposition

We study a particular class of transition kernels that stems from differentiating Markov kernels in the weak sense. Sufficient conditions are established for this type of kernels to admit a Jordan-type decomposition. The decomposition is explicitly constructed. © Heldermann Verlag

Measure-valued differentiation for stationary Markov chains

http://staff.feweb.vu.nl/bheidergot

Factorization of Tropical Matrices

In contrast to the situation in classical linear algebra, not every tropically non-singular matrix can be factored into a product of tropical elementary matrices. We do prove the factorizability of any tropically non-singular 2x2 matrix and, relating to the existing Bruhat decomposition, determine which 3x3 matrices are factorizable. Nevertheless, there is a closure operation, obtained by means of the tropical adjoint, which is always factorizable, generalizing the decomposition of the closure operation * of a matrix.Comment: This paper is part of the author's PhD thesis, which was written at Bar-Ilan University under the supervision of Prof. L. H. Rowe

Methods of tropical optimization in rating alternatives based on pairwise comparisons

We apply methods of tropical optimization to handle problems of rating alternatives on the basis of the log-Chebyshev approximation of pairwise comparison matrices. We derive a direct solution in a closed form, and investigate the obtained solution when it is not unique. Provided the approximation problem yields a set of score vectors, rather than a unique (up to a constant factor) one, we find those vectors in the set, which least and most differentiate between the alternatives with the highest and lowest scores, and thus can be representative of the entire solution.Comment: 9 pages, presented at the Annual Intern. Conf. of the German Operations Research Society (GOR), Helmut Schmidt University Hamburg, Germany, August 30 - September 2, 201

An approximation approach for the deviation matrix of continuous-time Markov processes with application to Markov decision theory

We present an update formula that allows the expression of the deviation matrix of a continuous-time Markov process with denumerable state space having generator matrix Q* through a continuous-time Markov process with generator matrix Q. We show that under suitable stability conditions the algorithm converges at a geometric rate. By applying the concept to three different examples, namely, the M/M/1 queue with vacations, the M/G/1 queue, and a tandem network, we illustrate the broad applicability of our approach. For a problem in admission control, we apply our approximation algorithm toMarkov decision theory for computing the optimal control policy. Numerical examples are presented to highlight the efficiency of the proposed algorithm. © 2010 INFORMS

Control of Uncertain (min,+)-Linear Systems

This paper deals with the control of uncertain (min,+)-linear systems which belong to an interval. Thanks to the residuation theory, a precompensator controller placed upstream of the studied system is given in such a way that even if the system’s behavior is not perfectly known, it has the property to delay the input as much as possible while keeping the input/output behavior unchanged. This precompensator is called neutral

Computation of the Transient in Max-Plus Linear Systems via SMT-Solving

This paper proposes a new approach, grounded in Satisfiability Modulo Theories (SMT), to study the transient of a Max-Plus Linear (MPL) system, that is the number of steps leading to its periodic regime. Differently from state-of-the-art techniques, our approach allows the analysis of periodic behaviors for subsets of initial states, as well as the characterization of sets of initial states exhibiting the same specific periodic behavior and transient. Our experiments show that the proposed technique dramatically outperforms state-of-the-art methods based on max-plus algebra computations for systems of large dimensions.Comment: The paper consists of 22 pages (including references and Appendix). It is accepted in FORMATS 2020 First revisio