Stochastic programming based multi-arm bandit offloading strategy for internet of things

In order to solve the high latency of traditional cloud computing and the processing capacity limitation of Internet of Things(IoT)users,Multi-access Edge Computing(MEC)migrates computing and storage capabilities from the remote data center to the edge of network,providing users with computation services quickly and directly.In this paper,we investigate the impact of the randomness caused by the movement of the IoT user on decision-making for offloading,where the connection between the IoT user and the MEC servers is uncertain.This uncertainty would be the main obstacle to assign the task accurately.Consequently,if the assigned task cannot match well with the real connection time,a migration(connection time is not enough to process)would be caused.In order to address the impact of this uncertainty,we formulate the offloading decision as an optimization problem considering the transmission,computation and migration.With the help of Stochastic Programming(SP),we use the posteriori recourse to compensate for inaccurate predictions.Meanwhile,in heterogeneous networks,considering multiple candidate MEC servers could be selected simultaneously due to overlapping,we also introduce the Multi-Arm Bandit(MAB)theory for MEC selection.The extensive simulations validate the improvement and effectiveness of the proposed SP-based Multi-arm bandit Method(SMM)for offloading in terms of reward,cost,energy consumption and delay.The results showthat SMMcan achieve about 20%improvement compared with the traditional offloading method that does not consider the randomness,and it also outperforms the existing SP/MAB based method for offloading.

Multistage Stochastic Programming Model for the Portfolio Problem of a Property-Liability Insurance Company

The current portfolio model for property-liability insurance company is only single period that can not meet the practical demands of portfolio management, and the purpose of this paper is to develop a multiperiod model for its portfolio problem. The model is a multistage stochastic programming which considers transaction costs, cash flow between time periods, and the matching of asset and liability; it does not depend on the assumption for normality of return distribution. Additionally, an investment constraint is added. The numerical example manifests that the multiperiod model can more effectively assist the property-liability insurer to determine the optimal composition of insurance and investment portfolio and outperforms the single period one.

A Stochastic Programming Strategy in Microgrid Cyber Physical Energy System for Energy Optimal Operation

This paper focuses on the energy optimal operation problem of microgrids (MGs) under stochastic environment. The deterministic method of MGs operation is often uneconomical because it fails to consider the high randomness of unconventional energy resources. Therefore, it is necessary to develop a novel operation approach combining the uncertainty in the physical world with modeling strategy in the cyber system. This paper proposes an energy scheduling optimization strategy based on stochastic programming model by considering the uncertainty in MGs. The goal is to minimize the expected operation cost of MGs. The uncertainties are modeled based on autoregressive moving average (ARMA) model to expose the effects of physical world on cyber world. Through the comparison of the simulation results with deterministic method, it is shown that the effectiveness and robustness of proposed stochastic energy scheduling optimization strategy for MGs are valid. © 2014 Chinese Association of Automation.

A simulation-based two-stage interval-stochastic programming model for water resources management in Kaidu-Konqi watershed,China

This study presented a simulation-based two-stage interval-stochastic programming （STIP） model to support water resources management in the Kaidu-Konqi watershed in Northwest China. The modeling system coupled a distributed hydrological model with an interval two-stage stochastic programing （ITSP）. The distributed hydrological model was used for establishing a rainfall-runoff forecast system, while random parameters were pro- vided by the statistical analysis of simulation outcomes water resources management planning in Kaidu-Konqi The developed STIP model was applied to a real case of watershed, where three scenarios with different water re- sources management policies were analyzed. The results indicated that water shortage mainly occurred in agri- culture, ecology and forestry sectors. In comparison, the water demand from municipality, industry and stock- breeding sectors can be satisfied due to their lower consumptions and higher economic values. Different policies for ecological water allocation can result in varied system benefits, and can help to identify desired water allocation plans with a maximum economic benefit and a minimum risk of system disruption under uncertainty.

Approximation-Exact Penalty Function Method for Solving a Class of Stochastic Programming

We present an approximation-exact penalty function method for solving the single stage stochastic programming problem with continuous random variable. The original problem is transformed into a determinate nonlinear programming problem with a discrete random variable sequence, which is obtained by some discrete method. We construct an exact penalty function and obtain an unconstrained optimization. It avoids the difficulty in solution by the rapid growing of the number of constraints for discrete precision. Under lenient conditions, we prove the equivalence of the minimum solution of penalty function and the solution of the determinate programming, and prove that the solution sequences of the discrete problem converge to a solution to the original problem.

Gaining-Sharing Knowledge Based Algorithm for Solving Stochastic Programming Problems

This paper presents a novel application of metaheuristic algorithmsfor solving stochastic programming problems using a recently developed gaining sharing knowledge based optimization (GSK) algorithm. The algorithmis based on human behavior in which people gain and share their knowledgewith others. Different types of stochastic fractional programming problemsare considered in this study. The augmented Lagrangian method (ALM)is used to handle these constrained optimization problems by convertingthem into unconstrained optimization problems. Three examples from theliterature are considered and transformed into their deterministic form usingthe chance-constrained technique. The transformed problems are solved usingGSK algorithm and the results are compared with eight other state-of-the-artmetaheuristic algorithms. The obtained results are also compared with theoptimal global solution and the results quoted in the literature. To investigatethe performance of the GSK algorithm on a real-world problem, a solidstochastic fixed charge transportation problem is examined, in which theparameters of the problem are considered as random variables. The obtainedresults show that the GSK algorithm outperforms other algorithms in termsof convergence, robustness, computational time, and quality of obtainedsolutions.

Stochastic Programming For Order Allocation And Production Planning

Stochastic demand is an important factor that heavily affects production planning.It influences activities such as purchasing,manufacturing,and selling,and quick adaption is required.In production planning,for reasons such as reducing costs and obtaining supplier discounts,many decisions must be made in the initial stage when demand has not been realized.The effects of non-optimal decisions will propagate to later stages,which can lead to losses due to overstocks or out-of-stocks.To find the optimal solutions for the initial and later stage regarding demand realization,this study proposes a stochastic two-stage linear program-ming model for a multi-supplier,multi-material,and multi-product purchasing and production planning process.The objective function is the expected total cost after two stages,and the results include detailed plans for purchasing and production in each demand scenario.Small-scale problems are solved through a deterministic equivalent transformation technique.To solve the problems in the large scale,an algorithm combining metaheuristic and sample average approximation is suggested.This algorithm can be implemented in parallel to utilize the power of the solver.The algorithm based on the observation that if the remaining quantity of materials and number of units of products at the end of the initial stage are given,then the problems of the first and second stages can be decomposed.

A Complex Algorithm for Solving a Kind of Stochastic Programming

Considering that the probability distribution of random variables in stochastic programming usually has incomplete information due to a perfect sample data in many real applications, this paper discusses a class of two-stage stochastic programming problems modeling with maximum minimum expectation compensation criterion (MaxEMin) under the probability distribution having linear partial information (LPI). In view of the nondifferentiability of this kind of stochastic programming modeling, an improved complex algorithm is designed and analyzed. This algorithm can effectively solve the nondifferentiable stochastic programming problem under LPI through the variable polyhedron iteration. The calculation and discussion of numerical examples show the effectiveness of the proposed algorithm.

Two-stage Stochastic Programming for Coordinated Operation of Distributed Energy Resources in Unbalanced Active Distribution Networks with Diverse Correlated Uncertainties

This paper proposes a stochastic programming(SP)method for coordinated operation of distributed energy resources(DERs)in the unbalanced active distribution network(ADN)with diverse correlated uncertainties.First,the threephase branch flow is modeled to characterize the unbalanced nature of the ADN,schedule DER for three phases,and derive a realistic DER allocation.Then,both active and reactive power resources are co-optimized for voltage regulation and power loss reduction.Second,the battery degradation is considered to model the aging cost for each charging or discharging event,leading to a more realistic cost estimation.Further,copulabased uncertainty modeling is applied to capture the correlations between renewable generation and power loads,and the twostage SP method is then used to get final solutions.Finally,numerical case studies are conducted on an IEEE 34-bus three-phase ADN,verifying that the proposed method can effectively reduce the system cost and co-optimize the active and reactive power.

Two-stage stochastic programming with robust constraints for the logistics network post-disruption response strategy optimization

Logistics networks (LNs) are essential for the transportation and distribution of goods or services from suppliers to consumers. However, LNs with complex structures are more vulnerable to disruptions due to natural disasters and accidents. To address the LN post-disruption response strategy optimization problem, this study proposes a novel two-stage stochastic programming model with robust delivery time constraints. The proposed model jointly optimizes the new-line-opening and rerouting decisions in the face of uncertain transport demands and transportation times. To enhance the robustness of the response strategy obtained, the conditional value at risk (CVaR) criterion is utilized to reduce the operational risk, and robust constraints based on the scenario-based uncertainty sets are proposed to guarantee the delivery time requirement. An equivalent tractable mixed-integer linear programming reformulation is further derived by linearizing the CVaR objective function and dualizing the infinite number of robust constraints into finite ones. A case study based on the practical operations of the JD LN is conducted to validate the practical significance of the proposed model. A comparison with the rerouting strategy and two benchmark models demonstrates the superiority of the proposed model in terms of operational cost, delivery time, and loading rate.

Foreign exchange trading and management with the stochastic dual dynamic programming method

We present a novel tool for generating speculative and hedging foreign exchange(FX)trading policies.Our solution provides a schedule that determines trades in each rebalancing period based on future currency prices,net foreign account positions,and incoming(outgoing)flows from business operations.To obtain such policies,we construct a multistage stochastic programming(MSP)model and solve it using the stochastic dual dynamic programming(SDDP)numerical method,which specializes in solving high-dimensional MSP models.We construct our methodology within an open-source SDDP package,avoiding implementing the method from scratch.To measure the performance of our policies,we model FX prices as a mean-reverting stochastic process with random events that incorporate stochastic trends.We calibrate this price model on seven currency pairs,demonstrating that our trading policies not only outperform the benchmarks for each currency,but may also be close to ex-post optimal solutions.We also show how the tool can be used to generate more or less conservative strategies by adjusting the risk tolerance,and how it can be used in a vari-ety of contexts and time scales,ranging from intraday speculative trading to monthly hedging for business operations.Finally,we examine the impact of increasing trade policy uncertainty(TPU)levels on our findings.Our findings show that the volatility of currencies from emerging economies rises in comparison to currencies from devel-oped markets.We discover that an increase in the TPU level has no effect on the aver-age profit obtained by our method.However,the risk exposure of the policies increases(decreases)for the group of currencies from emerging(developed)markets.

Improved Unit Commitment with Accurate Dynamic Scenarios Clustering Based on Multi-Parametric Programming and Benders Decomposition

Stochastic unit commitment is one of the most powerful methods to address uncertainty. However, the existingscenario clustering technique for stochastic unit commitment cannot accurately select representative scenarios,which threatens the robustness of stochastic unit commitment and hinders its application. This paper providesa stochastic unit commitment with dynamic scenario clustering based on multi-parametric programming andBenders decomposition. The stochastic unit commitment is solved via the Benders decomposition, which decouplesthe primal problem into the master problem and two types of subproblems. In the master problem, the committedgenerator is determined, while the feasibility and optimality of generator output are checked in these twosubproblems. Scenarios are dynamically clustered during the subproblem solution process through the multiparametric programming with respect to the solution of the master problem. In other words, multiple scenariosare clustered into several representative scenarios after the subproblem is solved, and the Benders cut obtainedby the representative scenario is generated for the master problem. Different from the conventional stochasticunit commitment, the proposed approach integrates scenario clustering into the Benders decomposition solutionprocess. Such a clustering approach could accurately cluster representative scenarios that have impacts on theunit commitment. The proposed method is tested on a 6-bus system and the modified IEEE 118-bus system.Numerical results illustrate the effectiveness of the proposed method in clustering scenarios. Compared withthe conventional clustering method, the proposed method can accurately select representative scenarios whilemitigating computational burden, thus guaranteeing the robustness of unit commitment.

Risk Management for International Portfolios with Basket Options:A Multi-Stage Stochastic Programming Approach

The authors consider the problem of active international portfolio management with basket options to achieve optimal asset allocation and combined market risk and currency risk management via multi-stage stochastic programming(MSSP). The authors note particularly the novel consideration and signi?cant bene?t of basket options in the context of portfolio optimization and risk management.Extensive empirical tests strongly demonstrate that basket options consistently have more clearly improvement on portfolio performances than a portfolio of vanilla options written on the same underlying assets. The authors further show that the MSSP model provides as a supportive tool for asset allocation,and a suitable test bed to empirically investigate the performance of alternative strategies.

STOCHASTIC PROGRAMMING METHOD FOR MULTIPERIOD CONSUMPTION AND INVESTMENT PROBLEMS WITH TRANSACTIONS COSTS

Using the GARCH model to describe the risky asset's return process so thatits time-varying moments and conditional heteroskedasticity can be properly reflected,general multiperiod optimal investment and consumption problems with both fixed andproportional transactions costs are investigated in this paper. We model this kind ofdifficult problems as a dynamic stochastic optimization problem, which can cope withdifferent utility functions and any number of time periods. The procedure to solve theresulting complex nonlinear stochastic optimization problem is discussed in detail and abranch-decomposition algorithm is devised.

STOCHASTIC GENERALIZED GOAL PROGRAMMING

This thesis presents the combination of the stochastic programming and generalized goal programming. We puts forward several generalized goal programming models with stochastic parameter--stochastic generalized goal programming. Furthermore, we probe into the theory. and algorithm of these models. At last, this method was applied to an example of an industrial problem.

APPROXIMATE LAGRANGE MULTIPLIER ALGORITHM FOR STOCHASTIC PROGRAMS WITH COMPLETE RECOURSE:NONLINEAR DETERMINISTIC CONSTRAINTS

In this paper we design an approximation method for solving stochastic programs with com-plete recourse and nonlinear deterministic constraints. This method is obtained by combiningapproximation method and Lagrange multiplier algorithm of Bertsekas type. Thus this methodhas the advantages of both the two.

AN INEXACT LAGRANGE-NEWTON METHOD FOR STOCHASTIC QUADRATIC PROGRAMS WITH RECOURSE

In this paper,two-stage stochastic quadratic programming problems with equality constraints are considered.By Monte Carlo simulation-based approximations of the objective function and its first(second)derivative,an inexact Lagrange-Newton type method is proposed.It is showed that this method is globally convergent with probability one.In particular,the convergence is local superlinear under an integral approximation error bound condition.Moreover,this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.

Experimental Study of Methods of Scenario Lattice Construction for Stochastic Dual Dynamic Programming

The stochastic dual dynamic programming (SDDP) algorithm is becoming increasingly used. In this paper we present analysis of different methods of lattice construction for SDDP exemplifying a realistic variant of the newsvendor problem, incorporating storage of production. We model several days of work and compare the profits realized using different methods of the lattice construction and the corresponding computer time spent in lattice construction. Our case differs from the known one because we consider not only a multidimensional but also a multistage case with stage dependence. We construct scenario lattice for different Markov processes which play a crucial role in stochastic modeling. The novelty of our work is comparing different methods of scenario lattice construction. We considered a realistic variant of the newsvendor problem. The results presented in this article show that the Voronoi method slightly outperforms others, but the k-means method is much faster overall.

Some Explicit Results for the Distribution Problem of Stochastic Linear Programming

A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. This is the “wait and see” problem of stochastic linear programming. Explicit results for the distribution problem are extremely difficult to obtain;indeed, previous results are known only if the right hand side coefficients have an exponential distribution [1]. To date, no explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s. In this paper, we obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. A transformation is utilized that greatly reduces computational time.

A Hybrid Programming Model for Optimal Production Planning under Demand Uncertainty in Refinery

Abstract Production planning under uncertainty is considered as one of the most important problems in plant-wide optimization. In this article, first, a stochastic programming model with uniform distribution assumption is developed for refinery production planning under demand uncertainty, and then a hybrid programming model incorporating the linear programming model with the stochastic programming one by a weight factor is proposed. Subsequently, piecewise linear approximation functions are derived and applied to solve the hybrid programming model-under uniform distribution assumption. Case studies show that the linear approximation algorithm is effective to solve.the hybrid programming model, along with an error≤0.5% when the deviatiorgmean≤20%. The simulation results indicate that the hybrid programming model with an appropriate weight factor （0.1-0.2） can effectively improve the optimal operational strategies under demand uncertainty, achieving higher profit than the linear programming model and the stochastic programming one with about 1.3% and 0.4% enhancement, respectavely.