We study the smoothed online quadratic optimization (SOQO) problem where, at each round $t$, a player plays an action $x_t$ in response to a quadratic hitting cost and an additional squared $\ell_2$-norm cost for switching actions. This problem class has strong connections to a wide range of application domains including smart grid management, adaptive control, and data center management, where switching-efficient algorithms are highly sought after. We study the SOQO problem in both adversarial and stochastic settings, and in this process, perform the first stochastic analysis of this class of problems. We provide the online optimal algorithm when the minimizers of the hitting cost function evolve as a general stochastic process, which, for the case of martingale process, takes the form of a distribution-agnostic dynamic interpolation algorithm that we call Lazy Adaptive Interpolation (LAI). Next, we present the stochastic-adversarial trade-off by proving an $\Omega(T)$ expected regret for the adversarial optimal algorithm in the literature (ROBD) with respect to LAI and, a sub-optimal competitive ratio for LAI in the adversarial setting. Finally, we present a best-of-both-worlds algorithm that obtains a robust adversarial performance while simultaneously achieving a near-optimal stochastic performance.

We consider a large-scale service system where incoming tasks have to be instantaneously dispatched to one out of many parallel server pools. The user-perceived performance degrades with the number of concurrent tasks and the dispatcher aims at maximizing the overall quality of service by balancing the load through a simple threshold policy. We demonstrate that such a policy is optimal on the fluid and diffusion scales, while only involving a small communication overhead, which is crucial for large-scale deployments. In order to set the threshold optimally, it is important, however, to learn the load of the system, which may be unknown. For that purpose, we design a control rule for tuning the threshold in an online manner. We derive conditions that guarantee that this adaptive threshold settles at the optimal value, along with estimates for the time until this happens. In addition, we provide numerical experiments that support the theoretical results and further indicate that our policy copes effectively with time-varying demand patterns. Summary of Contribution: Data centers and cloud computing platforms are the digital factories of the world, and managing resources and workloads in these systems involves operations research challenges of an unprecedented scale. Due to the massive size, complex dynamics, and wide range of time scales, the design and implementation of optimal resource-allocation strategies is prohibitively demanding from a computation and communication perspective. These resource-allocation strategies are essential for certain interactive applications, for which the available computing resources need to be distributed optimally among users in order to provide the best overall experienced performance. This is the subject of the present article, which considers the problem of distributing tasks among the various server pools of a large-scale service system, with the objective of optimizing the overall quality of service provided to users. A solution to this load-balancing problem cannot rely on maintaining complete state information at the gateway of the system, since this is computationally unfeasible, due to the magnitude and complexity of modern data centers and cloud computing platforms. Therefore, we examine a computationally light load-balancing algorithm that is yet asymptotically optimal in a regime where the size of the system approaches infinity. The analysis is based on a Markovian stochastic model, which is studied through fluid and diffusion limits in the aforementioned large-scale regime. The article analyzes the load-balancing algorithm theoretically and provides numerical experiments that support and extend the theoretical results.

Traditionally, reconfiguration problems ask the question whether a given solution of an optimization problem can be transformed to a target solution in a sequence of small steps that preserve feasibility of the intermediate solutions. In this paper, rather than asking this question from an algorithmic perspective, we analyze the combinatorial structure behind it. We consider the problem of reconfiguring one independent set into another, using two different processes: (1) exchanging exactly k vertices in each step, or (2) removing or adding one vertex in each step while ensuring the intermediate sets contain at most k fewer vertices than the initial solution. We are interested in determining the minimum value of k for which this reconfiguration is possible, and bound these threshold values in terms of several structural graph parameters. For hereditary graph classes we identify structures that cause the reconfiguration threshold to be large.