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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.

. Motivated by block partitioned problems arising from group sparsity representation and generalized noncooperative potential games, this paper presents a basic decomposition method for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an optimization problem is given by the sum of two nonseparable functions minus a sum of separable, pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1 , i.e., differentiable with a Lipschitz gradient, while the other summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By taking advantage of the (negative) pointwise maximum structure in the objective, the developed algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth problems. This aim is accomplished by combining the alternating direction method of multipliers (ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall problem into subproblems each involving the individual blocks. To arrive at a stationary solution of the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex function in each pointwise maximum. In order to lessen the potential computational burden in each iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is established.

This work combines a level-set approach and the optimal transport-based Wasserstein distance in a data assimilation framework. The primary motivation of this work is to reduce assimilation artifacts resulting from the position and observation error in the tracking and forecast of pollutants present on the surface of oceans or lakes. Both errors lead to spurious effect on the forecast that need to be corrected. In general, the geometric contour of such pollution can be retrieved from observation while more detailed characteristics such as concentration remain unknown. Herein, level sets are tools of choice to model such contours and the dynamical evolution of their topology structures. They are compared with contours extracted from observation using the Wasserstein distance. This allows to better capture position mismatches between both sources compared with the more classical Euclidean distance. Finally, the viability of this approach is demonstrated through academic test cases and its numerical performance is discussed.

Many optimization algorithms converge to stationary points. When the underlying problem is nonconvex, they may get trapped at local minimizers and occasionally stagnate near saddle points. We propose the Run-and-Inspect Method, which adds an “inspect” phase to existing algorithms that helps escape from non-global stationary points. The inspection samples a set of points in a radius R around the current point. When a sample point yields a sufficient decrease in the objective, we resume an existing algorithm from that point. If no sufficient decrease is found, the current point is called an approximate R-local minimizer. We show that an R-local minimizer is globally optimal, up to a specific error depending on R, if the objective function can be implicitly decomposed into a smooth convex function plus a restricted function that is possibly nonconvex, nonsmooth. Therefore, for such nonconvex objective functions, verifying global optimality is fundamentally easier. For high-dimensional problems, we introduce blockwise inspections to overcome the curse of dimensionality while still maintaining optimality bounds up to a factor equal to the number of blocks. Our method performs well on a set of artificial and realistic nonconvex problems by coupling with gradient descent, coordinate descent, EM, and prox-linear algorithms.

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No abstract uploaded!

No abstract uploaded!

Many iterative methods in optimization are fixed-point iterations with averaged operators. As such methods converge at an O(1/k) rate with the constant determined by the averagedness coefficient, establishing small averagedness coefficients for operators is of broad interest. In this paper, we show that the averagedness coefficients of the composition of averaged operators by Ogura and Yamada (Numer Func Anal Opt 32(1–2):113–137, 2002) and the threeoperator splitting by Davis and Yin (Set Valued Var Anal 25(4):829–858, 2017) are tight. The analysis relies on the scaled relative graph, a geometric tool recently proposed by Ryu, Hannah, and Yin (arXiv:1902.09788, 2019).

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