Discrete Stochastic Optimization Research Articles

On asymptotic convergence rate of random search

This paper presents general theoretical studies on asymptotic convergence rate (ACR) for finite dimensional optimization. Given the continuous problem function and discrete time stochastic optimization process, the ACR is the optimal constant for control of the asymptotic behaviour of the expected approximation errors. Under general assumptions, condition ACR<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<1$$\\end{document} implies the linear behaviour of the expected time of hitting the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}- optimal sublevel set with ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+ $$\\end{document} and determines the upper bound for the convergence rate of the trajectories of the process. This paper provides general characterization of ACR and, in particular, shows that some algorithms cannot converge linearly fast for any nontrivial continuous optimization problem. The relation between asymptotic convergence rate in the objective space and asymptotic convergence rate in the search space is provided. Examples and numerical simulations with use of a (1+1) self-adaptive evolution strategy and other algorithms are presented.

Восстановление геополей на основе комбинирования методов ядерного сглаживания и нейросетевых моделей

The purpose of the study, the results of which are described in the article, was to improve solving the matter of the geo-fields’ values restoring based on processing high-intensity spatial-temporal data streams received from a highly mobile geo-sensors network. Previously, the authors proposed an original approach to solving this task, which means applying the kernel smoothing methods, the nuclear function for which is determined automatically, using discrete stochastic optimization, in particular, the annealing simulation method. The idea of a new approach proposed by the authors is as follows

Discrete Stochastic Optimization for Public Health Interventions with Constraints

Many public health threats exist, motivating the need to find optimal intervention strategies. Given the stochastic nature of the threats (e.g., the spread of pandemic influenza, the occurrence of drug overdoses, and the prevalence of alcohol-related threats), deterministic optimization approaches may be inappropriate. In this paper, we implement a stochastic optimization method to address aspects of the 2009 H1N1 and the COVID-19 pandemics, with the spread of disease modeled by the open-source Monte Carlo simulations, FluTE, and Covasim, respectively. Without testing every possible option, the objective of the optimization is to determine the best combination of intervention strategies so as to result in minimal economic loss to society. To reach our objective, this application-oriented paper uses the discrete simultaneous perturbation stochastic approximation method (DSPSA), a recursive simulation-based optimization algorithm, to update the input parameters in the disease simulation software so that the output iteratively approaches minimal economic loss. Assuming that the simulation models for the spread of disease (FluTE for H1N1 and Covasim for COVID-19 in our case) are accurate representations for the population being studied, the simulation-based strategy we present provides decision makers a powerful tool to mitigate potential human and economic losses from any epidemic. The basic approach is also applicable in other public health problems, such as opioid abuse and drunk driving.

Analyzing the economic water loss level with a discrete stochastic optimization algorithm by considering budget constraints

Abstract Water losses in water distribution systems (WDSs) cause inefficiency of water resources and increase operating costs. Water loss management (WLM) methods generally have high initial investment and operating costs. However, the budget planned within the scope of WLM in administrations is generally limited. Therefore, the most appropriate method should be determined by considering the budget conditions and cost–benefit analysis. The aim of this study is to propose a new economic water loss level (EWLL) model in WDSs with the different budget constraints of 5, 15 and 20% of the annual income. This EWLL model was developed by the discrete stochastic optimization algorithm. The EWLL and economically recoverable leakage volumes were determined by considering the budget constraints. Moreover, the most appropriate methods were determined to reach the EWLL values defined according to the budget constraints. The EWLL was calculated as 8.62% by the optimization model without budget constraints. Moreover, the EWLL values with budget constraints of 5, 15 and 20% of the annual income were determined as 56.82, 21.14 and 18.02%, respectively. This EWLL model will make a significant contribution to the annual planning in WLM depending on the budget constraints of the administrations.

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.

Determination of Economic Leakage Level with Optimization Algorithm in Water Distribution Systems

It is necessary to set the most appropriate targets in water loss management (WLM) and to choose the most economically and technically suitable method to achieve these targets. The aim of this work is to define the level of economic leakage with the optimization algorithm in sustainable WLM, considering the system components, requirements and costs of the methods. For this purpose, a new model based on discrete stochastic optimization has been developed to determine the most appropriate values of the pressure level and the number of teams. After the system components were entered into the model, the optimal values of parameters were determined according to the problem-specific defined objective function. It was calculated that the pressure should be reduced from 40[Formula: see text]m to 25.30[Formula: see text]m, and the number of teams was 4 for the sample system. The Economic Leakage Level for the system was calculated as 145.85[Formula: see text]m3/day in volume and 11.9% in percentage.

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.

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.

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.

Single observation adaptive search for discrete and continuous stochastic optimization

Solving a stochastic optimization problem often involves performing repeated noisy function evaluations at points encountered during the algorithm. Recently, a continuous optimization framework for executing a single observation per search point was shown to exhibit a martingale property so that associated estimation errors are guaranteed to converge to zero. We generalize this martingale single observation approach to problems with mixed discrete–continuous variables. We establish mild regularity conditions for this class of algorithms to converge to a global optimum.

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.

Fine-Tuning of Feedback Gain Control for Hover Quad Copter Rotors by Stochastic Optimization Methods

Three degree of freedom (3 DOF) Hover Quad Copter (HQC) platforms are implemented for various missions in diverse scales from the micro to macro platforms. As HQC platforms scale down, micro platform requires rather robust and effective control techniques. This study investigates applicability of some stochastic optimization methods for tuning feedback gain control of HQC rotors and compares optimization results with results of linear quadratic regulator (LQR) method that has been widely used analytical method for optimal feedback gain control of HQCs. This study considers the utilization of two stochastic methods for tuning of HQCs. These methods are stochastic multi-parameter divergence optimization method (SMDO) and discrete stochastic optimization method (DSO). These methods are employed to optimize feedback gain coefficients of an experimental HQC test platform. Simulation and experimental results of SMDO and DSO methods are reported and compared with results of LQR method.

Algorithms to Approximate Column-sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms , to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k -column-sparse packing integer programs (Bansal et al., Theory of Computing , 2012) and stochastic k -set packing (Bansal et al., Algorithmica , 2012), and go “half the remaining distance” to optimal for a major integrality-gap conjecture of Füredi, Kahn, and Seymour on hypergraph matching ( Combinatorica , 1993).

5G Software-Defined Heterogeneous Networks With Cooperation and Partial Connectivity

The heterogeneous network composed of small cell base stations (SCBSs) and macro base stations (MBSs) has been widely deployed to alleviate the load of conventional MBSs, which may also cause severe interference and large energy consumption. The existing literature generally focus on the heterogeneous networks either without cooperation or with cross-tier cooperation. In this paper, we introduce a central controller and mobile edge cloud (MEC) servers and propose a heterogeneous network with SCBSs cooperation, i.e., the intra-tier cooperation. A novel load balancing metric instead of the simple access probability ratio is introduced. Based on the utility theory, we comprehensively evaluate several important performance metrics such as connectivity probability, load balancing, and energy efficiency to optimize the overall performance of the network. Moreover, in the special case of allowing partial connectivity, the discrete stochastic optimization algorithm is proposed so that the software-defined networks (SDNs) controller can adjust network parameters with the help of MEC servers to maximize the overall energy efficiency. The simulation results demonstrate that our proposed approach is valid to optimize the network in accordance with the user requirements. This paper provides a useful reference for the practical deployment of software-defined heterogeneous networks where the energy efficiency is increasingly becoming a key concern.

Optimal procurement and production planning for multi-product multi-stage production under yield uncertainty

Many products with several types are produced in more than one production stage subject to various potential complications, yield uncertainty being a very costly one. This study investigates the input procurement and semi-processed item allocation decisions of a producer in a multi-product-type and two-stage production system subject to yield uncertainty. A make-to-order production environment with deterministic production leadtimes is considered. For the problem, a discrete stochastic optimization model with binomial yield at each production stage is proposed. Optimal solution is determined with a novel solution algorithm developed by showing the convexity of the problem. The model is applicable not only to production in industries such as chemical, electronic, and mechanical manufacturing, but also to process types such as transportation and remanufacturing. As result of a comprehensive sensitivity analysis, it is found that improving yield in a single stage may not be effective in ensuring cost savings, improvements in different stages should be considered together. More interestingly, higher demand does not always lead to higher production, in contrast it may even reduce production amounts under yield uncertainty.

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.

A robust algorithm to solve the signal setting problem considering different traffic assignment approaches

AbstractIn this paper we extend a stochastic discrete optimization algorithm so as to tackle the signal setting problem. Signalized junctions represent critical points of an urban transportation network, and the efficiency of their traffic signal setting influences the overall network performance. Since road congestion usually takes place at or close to junction areas, an improvement in signal settings contributes to improving travel times, drivers’ comfort, fuel consumption efficiency, pollution and safety. In a traffic network, the signal control strategy affects the travel time on the roads and influences drivers’ route choice behavior. The paper presents an algorithm for signal setting optimization of signalized junctions in a congested road network. The objective function used in this work is a weighted sum of delays caused by the signalized intersections. We propose an iterative procedure to solve the problem by alternately updating signal settings based on fixed flows and traffic assignment based on fixed signal settings. To show the robustness of our method, we consider two different assignment methods: one based on user equilibrium assignment, well established in the literature as well as in practice, and the other based on a platoon simulation model with vehicular flow propagation and spill-back. Our optimization algorithm is also compared with others well known in the literature for this problem. The surrogate method (SM), particle swarm optimization (PSO) and the genetic algorithm (GA) are compared for a combined problem of global optimization of signal settings and traffic assignment (GOSSTA). Numerical experiments on a real test network are reported.

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.