TY - JOUR

T1 - Estimation of derivatives of nonsmooth performance measures in regenerative systems

AU - Homem-De-Mello, Tito

PY - 2001/11

Y1 - 2001/11

N2 - We investigate the problem of estimating derivatives of expected steady-state performance measures in parametric systems. Unlike most of the existing work in the area, we allow those functions to be nonsmooth and study the estimation of directional derivatives. For the class of regenerative Markovian systems we provide conditions under which we can obtain consistent estimators of those directional derivatives. An example illustrates that the conditions imposed must be different from those in the differentiable case. The result also allows us to derive necessary and sufficient conditions for differentiability of the expected steady-state function. We then analyze the process formed by the subdifferentials of the original process, and show that the subdifferential set of the expected steady-state function can be expressed as an average of integrals of multifunctions, which is the approach commonly found in the literature for integrals of sets. The latter result can also be viewed as a limit theorem for more general compact-convex multivalued processes.

AB - We investigate the problem of estimating derivatives of expected steady-state performance measures in parametric systems. Unlike most of the existing work in the area, we allow those functions to be nonsmooth and study the estimation of directional derivatives. For the class of regenerative Markovian systems we provide conditions under which we can obtain consistent estimators of those directional derivatives. An example illustrates that the conditions imposed must be different from those in the differentiable case. The result also allows us to derive necessary and sufficient conditions for differentiability of the expected steady-state function. We then analyze the process formed by the subdifferentials of the original process, and show that the subdifferential set of the expected steady-state function can be expressed as an average of integrals of multifunctions, which is the approach commonly found in the literature for integrals of sets. The latter result can also be viewed as a limit theorem for more general compact-convex multivalued processes.

KW - Convex analysis

KW - Derivative estimation

KW - Multi-functions

KW - Nonsmooth optimization

KW - Regenerative processes

KW - Steady-state systems

UR - http://www.scopus.com/inward/record.url?scp=0035518393&partnerID=8YFLogxK

U2 - 10.1287/moor.26.4.741.10010

DO - 10.1287/moor.26.4.741.10010

M3 - Article

AN - SCOPUS:0035518393

SN - 0364-765X

VL - 26

SP - 741

EP - 768

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

IS - 4

ER -