The monotone Variational Inequality (VI) is a general model that has important applications in various engineering and scientific domains. In numerous instances, the VI problems are accompanied by function constraints which can possibly be data-driven, making the usual projection operator challenging to compute. In this paper, we present novel first-order methods for the function constrained VI (FCVI) problem under various settings, including smooth or nonsmooth problems with stochastic operator and/or stochastic constraints. First, we introduce the OpConEx method and its stochastic variants, which employ extrapolation of the operator and constraint evaluations to update the variables and the Lagrangian multipliers. These methods achieve optimal operator or sample complexities when the FCVI problem is either (i) deterministic nonsmooth, or (ii) stochastic, including smooth or nonsmooth stochastic constraints. Notably, our algorithms are simple single-loop procedures and do not require the knowledge of Lagrange multipliers to attain these complexities. Second, to obtain the optimal operator complexity for smooth deterministic problems, we present a novel single-loop Adaptive Lagrangian Extrapolation (AdLagEx) method that can adaptively search for and explicitly bound the Lagrange multipliers. Moreover, we show that all of our algorithms can be easily extended to saddle point problems with coupled function constraints, hence achieving similar complexity results for the aforementioned cases. To our best knowledge, these complexities are obtained for the first time in the literature.

S ervice level agreements (SLAs) are widely adopted performance-based contracts in operations management practice, and fill rate is the most common performance metric among all the measurements in SLAs. Traditional procedures characterizing the order-up-to level satisfying a specified fill rate implicitly assume an infinite performance review horizon. However, in practice, inventory managers are liable to maintain and report fill rates over a finite performance review horizon. This horizon discrepancy leads to deviation between the target fill rate and actual achieved fill rate. In this study, we first examine the behavior of the fill rate distribution over a finite horizon with positive lead time. We analytically prove that the expected fill rate assuming an infinite performance review horizon exceeds the expected fill rate assuming a finite performance review horizon, implying that there exists some inventory "waste" (i.e., overstocking) when the traditional procedure is used. Based on this observation and the complexity of the problem, we propose a simulation-based algorithm to reduce excess inventory while maintaining the contractual target fill rate. When the lead time is significant relative to the length of the contract horizon, we show that the improvement in the inventory system can be over 5%. Further, we extend our basic setting to incorporate the penalty for failing to meet a target, and show how one can solve large-scale problems via stochastic approximation. The primary managerial implication of our study is that ignoring the performance review horizon in an SLA will cause overstocking, especially when the lead time is large.

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Cutter edge temperature in milling is an important factor to cutter life. With high cutting speed and feedrate, the cutting efficiency is high; however, the cutter edge temperature is high, shortening the cutter life. Therefore, it is necessary to know the cutter edge temperature in milling. Unfortunately, the cutter edge temperature is difficult to measure and predict in milling. To address the technical challenge, an analytical approach was proposed to predict cutter edge temperature in milling. First, the heat flux into the cutter edge was calculated. Second, by using the Green function, the cutter edge temperature was figured out, and the results obtained from this approach agreed well with that of a recognized test. Then, based on the engagement between the cutter and workpiece in trochoidal milling, the cutter edge temperature was obtained in trochoidal milling. Finally, a temperature comparison was made between trochoidal and side milling based on this analytical approach, and the reasons that trochoidal machining could extend the cutter life were found. This approach is first proposed to calculate the cutter edge temperature in trochoidal milling and can be applied to machining parameters optimization in trochoidal milling and cutter design optimization.