scholarly journals Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes

2018 ◽  
Author(s):  
Jinlong Lei ◽  
Uday V. Shanbhag
Keyword(s):  
Sample Size ◽  
Nash Games ◽  
Variable Sample Size ◽  
Best Response ◽  
Stochastic Nash Games

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

2020 ◽  
Vol 45 (1) ◽  
pp. 157-190 ◽  
Author(s):  
Jinlong Lei ◽  
Uday V. Shanbhag ◽  
Jong-Shi Pang ◽  
Suvrajeet Sen
Keyword(s):  
Optimization Problems ◽  
Almost Sure Convergence ◽  
Iteration Complexity ◽  
Two Stage ◽  
Nash Game ◽  
Nash Games ◽  
Mean Convergence ◽  
Best Response ◽  
Update Frequency ◽  
Stochastic Nash Games

In this paper, we consider a stochastic Nash game in which each player minimizes a parameterized expectation-valued convex objective function. In deterministic regimes, proximal best-response (BR) schemes have been shown to be convergent under a suitable spectral property associated with the proximal BR map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at each iteration. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the BR problem is computed in an expected-value sense via a stochastic approximation (SA) scheme. On the basis of this framework, we present three inexact BR schemes: (i) First, we propose a synchronous inexact BR scheme where all players simultaneously update their strategies. (ii) Second, we extend this to a randomized setting where a subset of players is randomly chosen to update their strategies while the other players keep their strategies invariant. (iii) Third, we propose an asynchronous scheme, where each player chooses its update frequency while using outdated rival-specific data in updating its strategy. Under a suitable contractive property on the proximal BR map, we proceed to derive almost sure convergence of the iterates to the Nash equilibrium (NE) for (i) and (ii) and mean convergence for (i)–(iii). In addition, we show that for (i)–(iii), the generated iterates converge to the unique equilibrium in mean at a linear rate with a prescribed constant rather than a sublinear rate. Finally, we establish the overall iteration complexity of the scheme in terms of projected stochastic gradient (SG) steps for computing an [Formula: see text]-NE2 (or [Formula: see text]-NE∞) and note that in all settings, the iteration complexity is [Formula: see text], where [Formula: see text] in the context of (i), and c > 0 represents the positive cost of randomization in (ii) and asynchronicity and delay in (iii). Notably, in the synchronous regime, we achieve a near-optimal rate from the standpoint of solving stochastic convex optimization problems by SA schemes. The schemes are further extended to settings where players solve two-stage stochastic Nash games with linear and quadratic recourse. Finally, preliminary numerics developed on a multiportfolio investment problem and a two-stage capacity expansion game support the rate and complexity statements.

Fuzzy U control chart based on fuzzy rules and evaluating its performance using fuzzy OC curve

2018 ◽  
Vol 30 (3) ◽  
pp. 232-247 ◽  
Author(s):  
Somayeh Fadaei ◽  
Alireza Pooya
Keyword(s):  
Fuzzy Control ◽  
Sample Size ◽  
Control Chart ◽  
Control Charts ◽  
Operating Characteristic ◽  
Fuzzy Rules ◽  
Type Ii ◽  
Type Ii Error ◽  
Content Type ◽  
Variable Sample Size

Purpose The purpose of this paper is to apply fuzzy spectrum in order to collect the vague and imprecise data and to employ the fuzzy U control chart in variable sample size using fuzzy rules. This approach is improved and developed by providing some new rules. Design/methodology/approach The fuzzy operating characteristic (FOC) curve is applied to investigate the performance of the fuzzy U control chart. The application of FOC presents fuzzy bounds of operating characteristic (OC) curve whose width depends on the ambiguity parameter in control charts. Findings To illustrate the efficiency of the proposed approach, a practical example is provided. Comparing performances of control charts indicates that OC curve of the crisp chart has been located between the FOC bounds, near the upper bound; as a result, for the crisp control chart, the probability of the type II error is of significant level. Also, a comparison of the crisp OC curve with OCavg curve and FOCα curve approved that the probability of the type II error for the crisp chart is more than the same amount for the fuzzy chart. Finally, the efficiency of the fuzzy chart is more than the crisp chart, and also it timely gives essential alerts by means of linguistic terms. Consequently, it is more capable of detecting process shifts. Originality/value This research develops the fuzzy U control chart with variable sample size whose output is fuzzy. After creating control charts, performance evaluation in the industry is important. The main contribution of this paper is to employs the FOC curve for evaluating the performance of the fuzzy control chart, while in prior studies in this area, the performance of fuzzy control chart has not been evaluated.

A variable sample size run sum coefficient of variation chart

2022 ◽  
Author(s):  
Wai Chung Yeong ◽  
Yen Yoon Tan ◽  
Sok Li Lim ◽  
Khai Wah Khaw ◽  
Michael Boon Chong Khoo
Keyword(s):  
Sample Size ◽  
Coefficient Of Variation ◽  
Variable Sample Size
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