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This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.

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This paper studies the capacity allocation game between duopolistic airlines which could offer callable products. Previous literature has shown that callable products provide a riskless source of additional rev- enue for a monopolistic airline. We examine the impact of the introduction of callable products on the revenues and the booking limits of duopolistic airlines. The analytical results demonstrate that, when there is no spill of low-fare customers, offering callable products is a dominant strategy of both airlines and provides Pareto gains to both airlines. When customers of both fare classes spill, offering callable products is no longer a dominant strategy and may harm the revenues of the airlines. Numerical ex- amples demonstrate that whether the two airlines offer callable products and whether offering callable products is beneficial to the two airlines mainly depends on their loads and capacities. Specifically, when the difference between the loads of the airlines is large, the loads of the airlines play the most important role. When the difference between the loads of the airlines is small, the capacities of the airlines play the most important role. Moreover, numerical examples show that the booking limits of the two airlines in the case with callable products are always higher than those in the case without callable products.

We develop a generic game platform that can be used to model various real-world systems with multiple intelligent cloud-computing pools and parallel-queues for resources-competing users. Inside the platform, the software structure is modelled as Blockchain. All the users are associated with Big Data arrival streams whose random dynamics is modelled by triply stochastic renewal reward processes (TSRRPs). Each user may be served simultaneously by multiple pools while each pool with parallel- servers may also serve multi-users at the same time via smart policies in the Blockchain, e.g. a Nash equilibrium point myopically at each fixed time to a game-theoretic scheduling problem. To illustrate the effectiveness of our game platform, we model the performance measures of its internal data flow dynamics (queue length and workload processes) as reflecting diffusion with regime-switchings (RDRSs) under our scheduling policies. By RDRS models, we can prove our myopic game-theoretic policy to be an asymptotic Pareto minimal-dual-cost Nash equilibrium one globally over the whole time horizon to a randomly evolving dynamic game problem. Iterative schemes for simulating our multi-dimensional RDRS models are also developed with the support of numerical comparisons.