An Optimal Control Experiment for an SEIRS Epidemiological Model

This work studies an optimal control model for a discrete-time Susceptible/Exposed/Infective/Removed/Susceptible (SEIRS) deterministic epidemiological model with a finite time horizon and changing population. The model presented converts a continuous SEIRS model that would typically be solved using differential equations into a discrete model that can be solved using dynamic programming. The discrete approach more closely resembles real life situations, as the number of individuals in a population, the rate of vaccination to be applied, and the time steps are all discrete values. The model utilizes a previously developed algorithm and applies it to the presented SEIRS model. To demonstrate the applicability of the algorithm, a series of numerical results are presented for various parameter values. KEYWORDS: Control; Cost; Discrete; Disease; Epidemiology; Minimization; Modeling; Optimality; SEIRS; Vaccination

Controlling Mixed Connected and Non-Connected Vehicle Traffic Through a Diamond Interchange

Diamond interchanges (DIs) allow movement of vehicles between surface streets and freeways for all types of vehicles, including normal non-connected human-driven vehicle (NHDV) traffic and the connected vehicles (CVs). Unlike simple intersections, DIs consist of a pair of closely spaced intersections that are controlled together with complicated traffic movements and heavy demand fluctuations. This paper reviews the movements being controlled at DIs and presents a dynamic programming (DP)-based real-time proactive traffic control algorithm called MIDAS, to control both NHDVs and CVs. Like seminal cycle-free adaptive control methods such as OPAC and RHODES, MIDAS uses a forward recursion DP approach with efficient data structures for any large set of phase movements being controlled at DIs, over a finite-time horizon that rolls forward, and then uses a backward recursion to retrieve the optimal phase sequence and duration of phases. MIDAS captures Eulerian measurements from fixed loop detectors for all vehicles, and also captures Lagrangian measurements like in-vehicle GPS from CVs to estimate link travel times, arrival times, turning movements, etc. For every time horizon MIDAS predicts future arrivals, estimates queues at the interchange, and then minimizes a user-defined metric like delays, stops, or queues at an interchange. The paper compares performances of MIDAS with those of an optimal fixed cycle time signal control (OFTC) scheme and RHODES control on a simulated DI. The simulation is of Phoenix, AZ, DI (on I-17/19th Ave.) that uses the VISSIM micro-simulation platform. Performance is evaluated for various traffic loads and various CV market penetrations. Results show that MIDAS control outperforms RHODES and OFTC.

ARPA: An autonomous renderer placement algorithm in distributed multimedia fog networks with delay guarantees

Multimedia cloud computing has emerged as a popular paradigm for the support of delay-intolerable immersive multimedia applications with high-end three-dimensional rendering. To that end, fog computing offers distributed computational offloading solutions, by positioning rendering servers in close proximity to end users promising in this way continuous service provision, that is otherwise not easily attainable under the strictly centralized cloud-only model. Yet, in order to alleviate the multimedia providers from unnecessary capital expenditure, a strategic placement approach of the servers at the fog layer must be implemented, that can effectively cope both with the network dynamics and the overall imposed deployment cost, and still adhere to the delay bounds set forth by the multimedia application. In this paper, we formally formulate the problem as a facility location problem using constrained optimization over a finite time horizon. We then theoretically analyze the minimum acceptable conditions necessary for a decentralized location of the servers, utilizing solely local information around their immediate neighborhood, that iteratively leads to better solutions. Based on the analysis, we propose a distributed algorithm, namely the Autonomous Renderer Placement Algorithm (ARPA), to address it. ARPA employs localized service relocation to shift the placement according to simple rules that designate elastic migration, replication, and complementary consolidation of the underlying renderers. Simulation results under diversified deployment scenarios, as well as trace-driven comparisons against other approaches, testify to ARPA's accountability in obeying the delay limits and fast converge in finite time slots to a placement solution that both outperforms the baseline alternatives and is close to the optimal one, rendering it suitable for scaling up and down to meet the current demands of the offered multimedia applications.

Discrete-time inverse linear quadratic optimal control over finite time-horizon under noisy output measurements

In this paper, the problem of inverse quadratic optimal control over finite time-horizon for discrete-time linear systems is considered. Our goal is to recover the corresponding quadratic objective function using noisy observations. First, the identifiability of the model structure for the inverse optimal control problem is analyzed under relative degree assumption and we show the model structure is strictly globally identifiable. Next, we study the inverse optimal control problem whose initial state distribution and the observation noise distribution are unknown, yet the exact observations on the initial states are available. We formulate the problem as a risk minimization problem and approximate the problem using empirical average. It is further shown that the solution to the approximated problem is statistically consistent under the assumption of relative degrees. We then study the case where the exact observations on the initial states are not available, yet the observation noises are known to be white Gaussian distributed and the distribution of the initial state is also Gaussian (with unknown mean and covariance). EM-algorithm is used to estimate the parameters in the objective function. The effectiveness of our results are demonstrated by numerical examples.

Optimal Selling Strategies under Regime-Switching Market Environment with Finite Expiry

This paper addresses an optimal stock liquidation problem over a finite-time horizon; to that end, we model it as an optimal stopping problem in a regime-switching market. The optimal stopping time is written as a solution to a system of Volterra type integral equations. Moreover, it reveals that when the risk-free interest rate is always lower than the return rate of the stock, it is never optimal to sell the stock early; otherwise, one should sell the stock in bear market if the stock price reaches a critical value and hold the stock in bull market until the maturity date. Finally, we present a trinomial tree method for numerical implementation. The numerical results are consistent with the theoretical findings.

The effect of consolidated periods in heterogeneous lot-sizing games

We consider a cooperative game defined by an economic lot-sizing problem with heterogeneous costs over a finite time horizon, in which each firm faces demand for a single product in each period and coalitions can pool orders. The model of cooperation works as follows: ordering channels and holding and backlogging technologies are shared among the members of the coalitions. This implies that each firm uses the best ordering channel and holding technology provided by the participants in the consortium. That is, they produce, hold inventory, pay backlogged demand and make orders at the minimum cost of the coalition members. Thus, firms aim at satisfying their demand over the planing horizon with minimal operation cost. Our contribution is to show that there exist fair allocations of the overall operation cost among the firms so that no group of agents profit from leaving the consortium. Then we propose a parametric family of cost allocations and provide sufficient conditions for this to be a stable family against coalitional defections of firms. Finally, we focus on those periods of the time horizon that are consolidated and we analyze their effect on the stability of cost allocations.

Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.

Combinatorial Heuristics for Inventory Routing Problems

We consider the deterministic inventory routing problem over a discrete finite time horizon. Given clients on a metric, each with daily demands that must be delivered from a depot and holding costs over the planning horizon, an optimal solution selects a set of daily tours through a subset of clients to deliver all demands before they are due and minimizes the total holding and tour routing costs over the horizon. In the capacitated case, a limited number of vehicles are available, where each vehicle makes at most one trip per day. Each trip from the depot is allowed to carry a limited amount of supply to deliver. We develop fast heuristics for both cases by solving a family of prize-collecting Steiner tree instances. Computational experiments show our heuristics can find near-optimal solutions for both cases and substantially reduce the runtime compared with a pure mixed integer programming formulation approach.

Sequential Shortest Path Interdiction with Incomplete Information and Limited Feedback

We study sequential shortest path interdiction, where in each period an interdictor with incomplete knowledge of the arc costs blocks at most [Formula: see text] arcs, and an evader with complete knowledge about the costs traverses a shortest path between two fixed nodes in the interdicted network. In each period, the interdictor, who aims at maximizing the evader’s cumulative cost over a finite time horizon, and whose initial knowledge is limited to valid lower and upper bounds on the costs, observes only the total cost of the path traversed by the evader, but not the path itself. This limited information feedback is then used by the interdictor to refine knowledge of the network’s costs, which should lead to better decisions. Different interdiction decisions lead to different responses by the evader and thus to different feedback. Focusing on minimizing the number of periods it takes a policy to recover a full information interdiction decision (that taken by an interdictor with complete knowledge about costs), we show that a class of greedy interdiction policies requires, in the worst case, an exponential number of periods to converge. Nonetheless, we show that under less stringent modes of feedback, convergence in polynomial time is possible. In particular, we consider different versions of imperfect randomized feedback that allow establishing polynomial expected convergence bounds. Finally, we also discuss a generalization of our approach for the case of a strategic evader, who does not necessarily follow a shortest path in each period.