In this paper we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey and Ruszczyński [13] and the Parallel Coordinate Descent Method (PCDM) of Richtárik and Takáč [23]. We show that the two methods are equivalent for feasibility problems up to the selection of a single step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8(L /L)(ω − 1) 2 times better than the best known bound for DQAM, where ω is the degree of partial separability and L andL are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian.
In this work, we introduce a novel queueing model with two classes of users, where users wait in the system to match with a candidate from the other class, instead of accessing a resource. This new model is useful for analyzing the traffic in web portals that match people who provide a service with people who demand the service, e.g. employment portals, matrimonial and dating sites and rental portals. We provide a Markov chain model for these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. We prove that if no control mechanism is employed these systems are unstable for any set of parameters, and suggest four different classes of control policies to assure stability. Contrary to the intuition that the rejection rate should decrease as the users get more likely to match, we show that for certain control policies the rejection rate is insensitive to the matching probability. Even more surprisingly, we show that for reasonable policies the rejection rate may be an increasing function of the matching probability. We also prove insensitivity results related to the average queue lengths and waiting times.
This paper focuses on probabilistic matching systems where two classes of users arrive at the system to match with users from the other class. The users are selective and the matchings occur probabilistically. Recently, Markov chain models were proposed to analyze these systems; however, an exact analysis of these models to completely characterize the performance is not possible due to the probabilistic matching structure. In this work, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyze the basic properties of these approximations and show that some performance measures are insensitive to the matching probability, agreeing with the existing results. We also perform numerical experiments with our approximations to gain insight into probabilistic matching systems.
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