Mathematics

FISTA is a popular convex optimisation algorithm which is known to converge at an optimal rate whenever the optimisation domain is contained in a suitable Hilbert space. We propose a modified algorithm where each iteration is performed in a subspace, and that subspace is allowed to change at every iteration. Analytically, this allows us to guarantee convergence in a Banach space setting, although at a reduced rate depending on the conditioning of the specific problem. Numerically we show that a greedy adaptive choice of discretisation can greatly increase the time and memory efficiency in infinite dimensional Lasso optimisation problems.

Given a real function f , the rate function for the large deviations of the diffusion process of drift ?�f given by the Freidlin-Wentzell theorem coincides with the time integral of the energy dissipation for the gradient flow associated with f . This paper is concerned with the stability in the hilbertian framework of this common action functional when f varies. More precisely, we show that if ( f h ) h is uniformly λ -convex for some λ?�R and converges towards f in the sense of Mosco convergence, then the related functionals ? -converge in the strong topology of curves.

The cyber risk insurance market is at a nascent stage of its development, even as the magnitude of cyber losses is significant and the rate of cyber risk events is increasing. Existing cyber risk insurance products as well as academic studies have been focusing on classifying cyber risk events and developing models of these events, but little attention has been paid to proposing insurance risk transfer strategies that incentivize mitigation of cyber loss through adjusting the premium of the risk transfer product. To address this important gap, we develop a Bonus-Malus model for cyber risk insurance. Specifically, we propose a mathematical model of cyber risk insurance and cybersecurity provisioning supported with an efficient numerical algorithm based on dynamic programming. Through a numerical experiment, we demonstrate how a properly designed cyber risk insurance contract with a Bonus-Malus system can resolve the issue of moral hazard and benefit the insurer.

The Merton problem is the well-known stochastic control problem of choosing consumption over time, as well as an investment mix, to maximize expected constant relative risk aversion (CRRA) utility of consumption. Merton formulated the problem and provided an analytical solution in 1970; since then a number of extensions of the original formulation have been solved. In this note we identify a certainty equivalent problem, i.e., a deterministic optimal control problem with the same optimal value function and optimal policy, for the base Merton problem, as well as a number of extensions. When time is discretized, the certainty equivalent problem becomes a second-order cone program (SOCP), readily formulated and solved using domain specific languages for convex optimization. This makes it a good starting point for model predictive control, a policy that can handle extensions that are either too cumbersome or impossible to handle exactly using standard dynamic programming methods.

What type of delegation contract should be offered when facing a risk of the magnitude of the pandemic we are currently experiencing and how does the likelihood of an exogenous early termination of the relationship modify the terms of a full-commitment contract? We study these questions by considering a dynamic principal-agent model that naturally extends the classical Holmstr{ö}m-Milgrom setting to include a risk of default whose origin is independent of the inherent agency problem. We obtain an explicit characterization of the optimal wage along with the optimal action provided by the agent. The optimal contract is linear by offering both a fixed share of the output which is similar to the standard shutdown-free Holmstr{ö}m-Milgrom model and a linear prevention mechanism that is proportional to the random lifetime of the contract. We then tweak the model to add a possibility for risk mitigation through investment and study its optimality.

The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.

The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.

It is challenging to recover the large-scale internet traffic data purely from the link measurements. With the rapid growth of the problem scale, it will be extremely difficult to sustain the recovery accuracy and the computational cost. In this work, we propose a new Sparsity Low-Rank Recovery (SLRR) model and its Schur Complement Based semi-proximal Alternating Direction Method of Multipliers (SCB-spADMM) solver. Our approach distinguishes itself mainly for the following two aspects. First, we fully exploit the spatial low-rank property and the sparsity of traffic data, which are barely considered in the literature. Our model can be divided into a series of subproblems, which only relate to the traffics in a certain individual time interval. Thus, the model scale is significantly reduced. Second, we establish a globally convergent ADMM-type algorithm inspired by [Li et al., Math. Program., 155(2016)] to solve the SLRR model. In each iteration, all the intermediate variables' optimums can be calculated analytically, which makes the algorithm fast and accurate. Besides, due to the separability of the SLRR model, it is possible to design a parallel algorithm to further reduce computational time. According to the numerical results on the classic datasets Abilene and GEANT, our method achieves the best accuracy with a low computational cost. Moreover, in our newly released large-scale Huawei Origin-Destination (HOD) network traffics, our method perfectly reaches the seconds-level feedback, which meets the essential requirement for practical scenarios.

Data Envelopment Analysis (DEA) is a multi-criteria technique based on linear programming to deal with many real-life problems, mostly in nonprofit organizations. The slacks-based measure (SBM) model is one of the DEA model used to assess the relative efficiencies of decision-making units (DMUs). The SBM DEA model directly used input slacks and output slacks to determine the relative efficiency of DMUs. In order to deal with qualitative or uncertain data, a fuzzy SBM DEA model is used to assess the performance of DMUs in this study. The credibility measure approach, transform the fuzzy SBM DEA model into a crisp linear programming model at different credibility levels is used. The results came from the fuzzy DEA model are more rational to the real-world situation than the conventional DEA model. In the end, the data of Indian oil refineries is collected, and the efficiency behavior of the companies obtained by applying the proposed model for its numerical illustration.

This paper studies finite-horizon robust tracking control for discrete-time linear systems, based on input-output data. We leverage behavioral theory to represent system trajectories through a set of noiseless historical data, instead of using an explicit system model. By assuming that recent output data available to the controller are affected by noise terms verifying a quadratic bound, we formulate an optimization problem with a linear cost and LMI constraints for solving the robust tracking problem without any approximations. Our approach hinges on a parameterization of noise trajectories compatible with the data-dependent system representation and on a reformulation of the tracking cost, which enables the application of the S-lemma. In addition, we propose a method for reducing the computational complexity and demonstrate that the size of the resulting LMIs does not scale with the number of historical data. Finally, we show that the proposed formulation can easily incorporate actuator disturbances as well as constraints on inputs and outputs. The performance of the new controllers is discussed through simulations.