Discrete Stochastic Optimization Problem Research Articles

Optimal Multi Zones Search Technique to Detect a Lost Target by Using K Sensors

This paper presents the discrete search technique on multi zones to detect a lost target by using sensors. The search region is divided into zones. These zones contain an equal number of states (cells) not necessarily identical. Each zone has a one sensor to detect the target. The target moves over the cells according to a random process. We consider the searching effort as a random variable with a known probability distribution. The detection function with the discounted reward function in a certain state and time interval are given. The optimal effort distribution that minimizes the probability of undetection is obtained after solving a discrete stochastic optimization problem. An algorithm is constructed to obtain the optimal solution as in the numerical application.

Detection of an appropriate pharmaceutical company to get a suitable vaccine against COVID-19 with minimum cost under the quality control process.

Some international pharmaceutical companies have succeeded in producing vaccines against COVID‐19. Countries all over the world have aimed to obtain these vaccines with minimum cost. We consider a set of K‐independent Markovian waiting lists. Each list contains a set of countries, where each one of them has an exponential service time and a Poisson arrival process. These companies differ in some characteristics such as the vaccine production cost and the speed of the required quantity delivery. We present a new detection model that helps in providing an appropriate decision to choose a suitable company. Moreover, the concept of balking and the retention of reneged countries is taken into consideration under the quality control process of each waiting list. Under steady state, we face an interesting and difficult discrete stochastic optimization problem. Its solution gives an optimal distribution of the searching effort, which is bounded by a known probability distribution. A simulation study has been derived to get the minimum value of the paid cost random values. The highest service rate, the total expected profit of each queuing system, and the optimum performance measures, which depend on this cost, have been obtained to show the effectiveness of this model.

Hybrid Niche Immune Genetic Algorithm for Fault Detection Coverage in Industry Wireless Sensor Network

The industry wireless sensor network (IWSN) technology, which is used to monitor industrial equipment, has attracted more and more attention in recent years. Sensor nodes in IWSN can spontaneously complete distributed networking and carry out monitoring tasks under random deployment conditions. Therefore, a self‐organized IWSN is particularly suitable for the fault detection and diagnosis of industrial equipment in complex environments. However, due to the detection, ability of a single sensor node is limited, and the monitoring distribution problem is a typical multidimensional discrete NP‐hard combinatorial stochastic optimization problem, which is challenging to solve for the traditional mathematical methods. With the purpose of improving the target monitoring capability and prolonging lifetime of IWSN, a novel hybrid niche immune genetic algorithm (HNIGA) for optimizing the target coverage model of fault detection is proposed. It uses the genetic operation to evolve antibody groups and applies niche technology to maintain the diversity of antibody groups. As a result, HNIGA can effectively reduce the failure rate of detection targets. To verify the performance of HNIGA, a series of simulations under different simulation conditions are carried out. Specifically, HNIGA is compared with genetic algorithm (GA) and simulated annealing (SA). Simulation results show that HNIGA has a faster convergence speed and more robust global search capability than the other two algorithms.

Augmented simulation methods for discrete stochastic optimization with recourse

We develop an augmented simulation approach to solve discrete stochastic optimization problems by converting them into a grand simulation problem in the joint space of random and decision variables. The optimal decision is obtained via the mode of the augmented probability model, using a new multivariate extension of the classic Barker’s algorithm. Illustrations on different versions of univariate and multivariate discrete news-vendor problems with exogenous and endogenous uncertainties are detailed. We contrast our method with the Metropolis–Hastings algorithm, the nested sampling-based augmented simulation method, and traditional Monte Carlo simulation-based optimization schemes. The proposed method is shown to be computationally efficient and could serve as another tool to solve discrete stochastic optimization problems with recourse.

Quality control for a detected an appropriate queue from K‐independent M/M/C/N queueing models

AbstractIn this paper, we consider a system of K‐independent Markovian queues such that each one of them has a Poisson arrival process and exponential service time. We assume that every server has some characteristics such as the speed of the service performance or the service cost. To find an appropriate queue, which meets customer needs for the service performance, we present a new approach that gives a suitable decision to choose an appropriate queue from our system. This allows the customer to deal with minimum cost and faster server under steady state. We solve an interesting discrete stochastic optimization problem where the paid cost by the customer is bounded by a Gaussian distribution. Using these hypotheses, we perform a simulation study by generating the paid cost random values and choosing the minimum value between them. This minimum cost gives the highest service rate, which is used to obtain the optimum values of the system effectiveness measures.

Integer Programming on the Junction Tree Polytope for Influence Diagrams

Influence diagrams (ID) and limited memory influence diagrams (LIMID) are flexible tools to represent discrete stochastic optimization problems, with the Markov decision process (MDP) and partially observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In an ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types: chance, decision, and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the “single policy update,” the default algorithm for addressing IDs.

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. Our framework and SAA results cover a large variety of resource allocation problems for which at each stage after the first one, new information becomes available and the allocation can be readjusted, under constraints that involve expectations estimated by Monte Carlo. As an illustration, we apply this SAA method to a staffing problem in a call center, in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service (QoS). The staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage, and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages.

Rounding heuristics for multiple product dynamic lot-sizing in the presence of queueing behavior

We present heuristics for solving a difficult nonlinear integer programming (NIP) model arising from a multi-item single machine dynamic lot-sizing problem. The heuristic obtains a local optimum for the continuous relaxation of the NIP model and rounds the resulting fractional solution to a feasible integer solution by solving a series of shortest path problems. We also implement two benchmarks: a version of the well-known Feasibility Pump heuristic and the Surrogate Method developed for stochastic discrete optimization problems. Computational experiments reveal that our shortest path based rounding procedure finds better production plans than the previously developed myopic heuristic and the benchmarks.

Network-Based Approximate Linear Programming for Discrete Optimization

Several prescriptive tasks in business and engineering as well as prediction in machine learning entail the solution of challenging discrete optimization problems. We recast the typical optimization formulation of these problems as high-dimensional dynamic programs and approach their approximation via linear programming. We develop tractable approximate linear programs with supporting theory by bringing together tools from state-space aggregations, networks, and perfect graphs (i.e., graph completions). We embed these models in a simple branch-and-bound scheme to solve applications in marketing analytics and the maintenance of energy or city-owned assets. We find that the resulting technique substantially outperforms a state-of-the-art commercial solver as well as aggregation-heuristics in terms of both solution quality and time. Our results motivate further consideration of networks and graph theory in approximate linear programming for solving deterministic and stochastic discrete optimization problems.

Virtual Resource Allocation in Software-Defined Information-Centric Cellular Networks With Device-to-Device Communications and Imperfect CSI

In this paper, we propose an architecture of software-defined information-centric network virtualization with device-to-device (D2D) communications, which facilitates dynamic virtual resource allocation and content caching via a software-defined networking (SDN) controller with a global view of the system. In our proposed framework, substrate physical resources can be virtualized and shared among multiple mobile virtual network operators (MVNOs). Meanwhile, by means of integrating D2D communications into information-centric wireless networks, content caching is enabled not only in the air but in mobile devices as well. In addition, taking into consideration inaccurate channel estimation and measurement, we formulate the virtual resource allocation and caching optimization as a discrete stochastic optimization problem in which imperfect channel state information is incorporated. Because the formulated virtual resource allocation problem is a large-scale combinational optimization problem, we exploit discrete stochastic approximation approaches to cope with it. Finally, extensive simulations are conducted to demonstrate the effectiveness of the proposed scheme with different system parameters. Simulation results show that MVNOs can benefit from not only the sharing of physical infrastructure but from the caching functionality that exists in both the air and mobile devices as well.

Software-Defined Device-to-Device (D2D) Communications in Virtual Wireless Networks With Imperfect Network State Information (NSI)

Software-defined networking (SDN) and network function virtualization (NFV) are a promising system architecture and control mechanism for future networks. Although some works have been done on wireless SDN and NFV, recent advancements in device-to-device (D2D) communications are largely ignored in this novel framework. In this paper, we study the integration of D2D communication in the framework of SDN and NFV. An inherent challenge in supporting software-defined D2D is the imperfectness of network state information, including channel state information (CSI) and queuing state information, in virtual wireless (QSI) networks. To address this challenge, we formulate the resource sharing problem in this framework as a discrete stochastic optimization problem and develop discrete stochastic approximation algorithms to solve this problem. Such algorithms can reduce the computational complexity compared with exhaustive search while achieving satisfactory performance. Both the static wireless channel and time-varying channels are considered. Extensive simulations show that users can benefit from both wireless network virtualization and software-defined D2D communications, and our proposed scheme can achieve considerable performance gains in both system throughput and user utility under practical network settings.

Medium Access Control for Unmanned Aerial Vehicle (UAV) Ad-Hoc Networks With Full-Duplex Radios and Multipacket Reception Capability

Recent advances in interference cancellation and signal-processing techniques can enable full-duplex radios and multipacket reception (MPR), which will have significant impact on medium access control (MAC) schemes. In this paper, we propose a MAC scheme in unmanned aerial vehicle (UAV) ad-hoc networks with full-duplex radios and MPR. To efficiently handle the highly mobile environment of a UAV ad-hoc network, a token-based technique is used to update information in the network as well. The MAC schemes in the presence of perfect and imperfect channel state information (CSI) are formulated as a combinatorial optimization problem and a discrete stochastic optimization problem, respectively. Simulation results show the effectiveness of the proposed MAC.

Sample Average Approximation Method for Compound Stochastic Optimization Problems

The paper studies stochastic optimization (programming) problems with compound functions containing expectations and extreme values of other random functions as arguments. Compound functions arise in various applications. A typical example is a variance function of nonlinear outcomes. Other examples include stochastic minimax problems, econometric models with latent variables, and multilevel and multicriteria stochastic optimization problems. As a solution technique a sample average approximation (SAA) method (also known as statistical or empirical (sample) mean method) is used. The method consists in approximation of all expectation functions by their empirical means and solving the resulting approximate deterministic optimization problems. In stochastic optimization, this method is widely used for optimization of standard expectation functions under constraints. In this paper, SAA method is extended to general compound stochastic optimization problems. The conditions for convergence in mean, almost surely, and rate of convergence are established. The study of the convergence rate is based on properties of Rademacher averages of functional sets, concentration inequalities for bounded random functions, and the concept of uniform normalized convergence of random variables. The convergence results are applicable both for discrete and continuous stochastic optimization problems.

A Global Optimization Approach to Solve the Traffic Signal Synchronization Problem

The Traffic Signal Synchronization is a traffic engineering technique of matching the green light times for a series of intersections to enable the maximum number of vehicles to pass through, thereby reducing stops and delays experienced by motorists. Synchronizing traffic signals ensures a better flow of traffic and minimizes gas consumption and pollutant emissions. The objective function used in this work is a weighted sum of the delays caused by the signalized intersections. In this paper, we apply generalized’surrogate problem’ methodology that is based on an on-line control scheme which transforms the problem into a’surrogate’ continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches while simultaneously updating both actual and surrogate system states. We extend a ‘surrogate problem’ approach that is developed for a class of stochastic discrete optimization problems so as to tackle the traffic signal synchronization problem to minimize the total delay (DTSS). Numerical experiments conducted on a test and a real networks show that the surrogate method converges in a very small area.

On Solving Some Stochastic Discrete Optimization Problems Under General Regret Function

In this paper we consider stochastic discrete optimization problems (DOP) in which feasible solutions remain feasible irrespective of the randomness of the problem parameters. We introduce the concept of the risk associated with a solution and define optimal solution in terms of having least possible risk. We show that a least risk solution can be obtained by solving a non-stochastic discrete optimization problem similar to the stochastic problem in certain problems and present results regarding the generation of the non-stochastic problem in terms of finding the parameter of the distribution which may act as surrogate for the random element in its no-stochastic counterpart. While this surrogate in the mean for a linear regret function, the situation is complex under general regret. Our results show that the above result continues to hold (in general) if the DOP has only one random element having symmetric distribution. We obtain some bounds for this parameter for certain group of asymmetric distributions and study its limiting behavior under two asymptotic setup. We establish through various examples that the results from uni-dimensional case cannot be extended to stochastic DOP with multiple random element with any reasonable generality. However, we characterize a finite number of solutions which will include the optimal solution in this case. An heuristic based on local search type algorithm is also devised when the number of random elements is too high, and we study the performance of this algorithm through simulation.

Joint Topology Control and Authentication Design in Mobile Ad Hoc Networks With Cooperative Communications

Security is the main concern and bottleneck for widely deployed wireless applications due to the fact that wireless channels are vulnerable to attacks and that wireless bandwidth is a constrained resource. In this sense, it is desirable to adaptively achieve security according to the available resource. In particular, mobile ad hoc networks (MANETs) based on cooperative communication (CC) present significant challenges to security issues, as well as issues of network performance and management. In this paper, we focus on authentication and topology control issues. Although authentication and topology control are separately studied in most existing works, they are, in fact, closely correlated in MANETs. For example, both authentication and topology control schemes have significant impacts on throughput. In this paper, we jointly consider authentication and topology control. Specifically, we analyze the effective throughput with upper layer authentication schemes and physical-layer schemes related to channel conditions and relay selections for CCs. A joint authentication and topology control (JATC) scheme is proposed to improve the throughput. JATC is formulated as a discrete stochastic optimization problem, which does not require prior perfect channel status but only channel estimate. We also mathematically prove the tracking convergence property and the convergence rate of the discrete stochastic optimization approach in this paper. Simulation results show that our scheme can substantially improve throughput in MANETs with CC.

Joint power allocation and beamforming with users selection for cognitive radio networks via discrete stochastic optimization

In this paper, we study the problem of mutual interference cancellation among secondary users (SUs) and interference control to primary users (PUs) in spectrum sharing underlay cognitive radio networks (CRNs). Multiple antennas are used at the secondary base station (SBS) to form multiple beams towards to individual SUs, and a set of SUs are selected to adapt to the beams. For the interference control to PUs, we study power allocation among SUs to guarantee the interference to PUs below a tolerable level while maximizing SUs' QoS. Based on these conditions, the problem of joint power allocation and beamforming with SUs selection is studied. Specifically, we emphasize on the condition of imperfect channel sensing. And we formulate the problem as a discrete stochastic optimization problem, then an efficient algorithm based on a discrete stochastic optimization method is proposed to solve the joint power allocation and beamforming with SUs selection problem. The proposed algorithm has fast convergence rate. Finally, simulation results are presented to demonstrate the performance of the proposed scheme.

Capacity-Optimized Topology Control for MANETs with Cooperative Communications

Cooperative communications can significantly enhance transmission reliability and bandwidth efficiency in wireless networks. However, many upper layer aspects of cooperative communications merit further research. In this paper, we investigate its impacts on network topology and network capacity, which is determined by considerable aspects, such as physical layer capacity, interference, path length, etc. Since cooperative communications enhance physical layer capacity and relay selection impacts network topology directly, we present a Capacity-Optimized COoperative (COCO) topology control scheme for mobile ad hoc networks (MANETs) with cooperative communications. We consider both upper layer network capacity and physical layer relay selections in the proposed scheme. In addition, only the channel estimate, not the perfect channel status, is assumed to be known in our scheme. The topology control problem in MANETs is then formulated as a discrete stochastic optimization problem, which can be solved using a stochastic approximation approach. Further, an improved COCO is presented to reconfigure network topology to track the changing mobile environment dynamically. Simulation results are presented to show the effectiveness of the proposed scheme.

Evolutionary Policy Iteration Under a Sampling Regime for Stochastic Combinatorial Optimization

This article modifies the evolutionary policy selection algorithm of Chang et al., which was designed for use in infinite horizon Markov decision processes (MDPs) with a large action space to a discrete stochastic optimization problem, in an algorithm called Evolutionary Policy Iteration-Monte Carlo (EPI-MC). EPI-MC allows EPI to be used in a stochastic combinatorial optimization setting with a finite action space and a noisy cost (value) function by introducing a sampling schedule. Convergence of EPI-MC to the optimal action is proven and experimental results are given.

Persistency Model and Its Applications in Choice Modeling

Given a discrete maximization problem with a linear objective function where the coefficients are chosen randomly from a distribution, we would like to evaluate the expected optimal value and the marginal distribution of the optimal solution. We call this the persistency problem for a discrete optimization problem under uncertain objective, and the marginal probability mass function of the optimal solution is named the persistence value. In general, this is a difficult problem to solve, even if the distribution of the objective coefficient is well specified. In this paper, we solve a subclass of this problem when the distribution is assumed to belong to the class of distributions defined by given marginal distributions, or given marginal moment conditions. Under this model, we show that the persistency problem maximizing the expected objective value over the set of distributions can be solved via a concave maximization model. The persistency model solved using this formulation can be used to obtain important qualitative insights to the behavior of stochastic discrete optimization problems. We demonstrate how the approach can be used to obtain insights to problems in discrete choice modeling. Using a set of survey data from a transport choice modeling study, we calibrate the random utility model with choice probabilities obtained from the persistency model. Numerical results suggest that our persistency model is capable of obtaining estimates that perform as well, if not better, than classical methods, such as logit and cross-nested logit models. We can also use the persistency model to obtain choice probability estimates for more complex choice problems. We illustrate this on a stochastic knapsack problem, which is essentially a discrete choice problem under budget constraint.