Stochastic Constraints Research Articles

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACTBy utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.

Joint Economic Lot-sizing in Multi-product Multi-level Integrated Supply Chains: Generalized Benders Decomposition

Joint Economic Lot-sizing Problems (JELPs) have received a great deal of attention in recent years because they are the bedrock of pushing forward the frontiers of knowledge in supply chain (SC). JELP can manage efficiently SC through promoting cooperation among the levels of the SC. In this paper, we deal with JELP in the context of the integrated four-level SC in order to minimise the total inventory cost. The model is multi-product and considers raw material replenishment. Real stochastic constraints, including space limitation, procurement cost, ordering, and joint economic lot-size limitations can make the model more applicable for real-world SCs. The objectives are both to determine the joint economic lot-sizing policy and the optimum period length such that the total inventory cost of the SC is minimised, while the stochastic constraints are satisfied. Mathematical formulation of the problem is stochastic, MINLP, large scale and hard to solve. In this regard, we utilised Generalised Benders Decomposition (GBD) in order to minimise the MINLP model of research. The results of numerical examples and sensitivity analysis illustrate that proposed method has satisfactory performance in optimum solution, number of taken iterations, the dual infeasibility, constraint violation, and the complementarity.

Robust Beamforming and Power Allocation in CR MISO Networks with SWIPT to Maximize the Minimum Achievable Rate

In this paper, a new method is presented to maximize the minimum achievable rate of the users in a cognitive radio multiple input single output network. The secondary system provides simultaneous wireless information and power transfer (SWIPT) for energy harvesting receivers. Considering random Gaussian vector for the estimation error of the channel state information (CSI), we impose some outage constraints to guarantee the quality of service of the network. Outage constraints on the received power and the information leakage at the energy harvesting users are imposed to assure SWIPT. We introduce new convex inequalities to replace the original non-convex constraints and write the objective function as the minimization of the maximum of some fractional functions. This new objective function and convex inequalities result in a new quasi-convex general fractional programming problem. The new quasi-convex optimization problem is written in such a way that the computational costs to solve this problem are as low as possible. We develop an algorithm to obtain the optimal beamforming weights and artificial noise covariance matrix. All stochastic constraints are rewritten for the special case that the covariance matrices of error in the estimation of CSI are a factor of identity matrix. This reformulation results in less computation costs to obtain optimal beamforming weights and AN covariance matrix. Simulation results confirm that, all stochastic constraints are satisfied with certain probability. In comparison with the previous robust related work, the proposed method achieves higher rates which confirms superiority of our method.

Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

A Decomposition Approach for Stochastic Shortest-Path Network Interdiction with Goal Threshold

Shortest-path network interdiction, where a defender strategically allocates interdiction resource on the arcs or nodes in a network and an attacker traverses the capacitated network along a shortest s-t path from a source to a terminus, is an important research problem with potential real-world impact. In this paper, based on game-theoretic methodologies, we consider a novel stochastic extension of the shortest-path network interdiction problem with goal threshold, abbreviated as SSPIT. The attacker attempts to minimize the length of the shortest path, while the defender attempts to force it to exceed a specific threshold with the least resource consumption. In our model, threshold constraint is introduced as a trade-off between utility maximization and resource consumption, and stochastic cases with some known probability p of successful interdiction are considered. Existing algorithms do not perform well when dealing with threshold and stochastic constraints. To address the NP-hard problem, SSPIT-D, a decomposition approach based on Benders decomposition, was adopted. To optimize the master problem and subproblem iteration, an efficient dual subgraph interdiction algorithm SSPIT-S and a local research based better-response algorithm SSPIT-DL were designed, adding to the SSPIT-D. Numerical experiments on networks of different sizes and attributes were used to illustrate and validate the decomposition approach. The results showed that the dual subgraph and better-response procedure can significantly improve the efficiency and scalability of the decomposition algorithm. In addition, the improved enhancement algorithms are less sensitive and robust to parameters. Furthermore, the application in a real-world road network demonstrates the scalability of our decomposition approach.

An efficient simulation procedure for ranking the top simulated designs in the presence of stochastic constraints

This research considers the problem of ranking the top simulated designs in the presence of stochastic constraints. The objective and constraint measures of each design must be estimated via simulation. Given a fixed simulation budget, the ranking of the top feasible designs cannot be determined with certainty. The objective of this research is to derive an efficient simulation budget allocation strategy such that the probability of correct ranking (PCR) can be maximized. To deal with this problem, we propose a lower bound on the PCR and develop an asymptotically optimal allocation rule based on the lower bound. Useful insights on characterizing the allocation rule are provided, and numerical experiments are carried out to demonstrate the efficiency of the suggested simulation procedure.

Stochastic Successive Convex Approximation for Non-Convex Constrained Stochastic Optimization

This paper proposes a constrained stochastic successive convex approximation (CSSCA) algorithm to find a stationary point for a general non-convex stochastic optimization problem, whose objective and constraint functions are non-convex and involve expectations over random states. Most existing methods for non-convex stochastic optimization, such as the stochastic (average) gradient and stochastic majorization-minimization, only consider minimizing a stochastic non-convex objective over a deterministic convex set. The proposed CSSCA algorithm can also handle stochastic non-convex constraints in optimization problems, and it opens the way to solving more challenging optimization problems that occur in many applications. The algorithm is based on solving a sequence of convex objective/feasibility optimization problems obtained by replacing the objective/constraint functions in the original problems with some convex surrogate functions. The CSSCA algorithm allows a wide class of surrogate functions and thus provides many freedoms to design good surrogate functions for specific applications. Moreover, it also facilitates parallel implementation for solving large scale stochastic optimization problems, which arise naturally in today's signal processing such as machine learning and big data analysis. We establish the convergence of CSSCA algorithm with a feasible initial point, and customize the algorithmic framework to solve several important application problems. Simulations show that the CSSCA algorithm can achieve superior performance over existing solutions.

Modelling And optimal lot-sizing of the replenishments in constrained, multi-product and bi-objective EPQ models with defective products: Generalised Cross Decomposition

ABSTRACT The optimal lot-sizing of the replenishments has a cumulative effect on practical Economic Production Quantity (EPQ) models with the aim of inventory system management. In this paper, an EPQ model of the replenishment is proposed by taking into account the real-world conditions. A bi-objective inventory model is designed in order to minimise the total inventory cost and to maximise the profit, simultaneously. Realistic and practical stochastic constraints are imposed on the procurement cost, the screening cost, the disposal cost, and the space cost. The objective of the present study is to optimise the lot-sizing of the replenishment, while the stochastic constraints are satisfied and the optimum number of lots and optimum volume of each lot are calculated. Responding to the inconsistency of objective functions, an Lp-metric method is utilised to integrate and gain a single objective function. The mathematical formulation of the model is stochastic, Mix Integer Nonlinear Programming (MINLP), large-scale and hard to solve. In this regards, Generalised Cross Decomposition (GCD) is utilised to optimise the MINLP model of this research. The results of numerical examples and sensitivity analyses give us a complete insight into the applicability and accuracy of the suggested model and the solution method.

Reliability-based optimum design of hydraulic water retaining structure constructed on heterogeneous porous media: utilizing stochastic ensemble surrogate model-based linked simulation optimization model

Seepage characteristics under hydraulic water retaining structures (HWRSs) significantly affect the hydraulic serviceability and stability of such structures. The expected hydraulic conductivity value and its spatial and directional variation substantially influence seepage characteristics. Furthermore, the uniform and homogenous hydraulic conductive is rarely seen in the real field. To study the effects of uncertainty and variation in hydraulic conductivity, a random field concept was used to generate different realizations of heterogeneous hydraulic conductivity with constant mean and varied standard deviation. Consequently, the seepage characteristics stochastically varied, creating uncertainty in seepage characteristics which influenced the HWRS design. Therefore, the objective of this paper was to integrate the reliability concept in the linked simulation optimization (S-O) model to address the uncertainty of hydraulic conductivity. Hence, the safest and most cost-effective HWRS design could be attained considering the effects of the uncertainty of hydraulic conductivity. The reliability-based optimum design (RBOD) concept was implemented utilizing the multiple realization optimization technique based on stochastic ensemble surrogate models. The reliability degree of each candidate design was measured based on the number of stochastic constraints satisfying the design requirements to the total number of constraints. The S-O model-based RBOD was formulated to find the most cost-effective HWRS design satisfying a beforehand desired degree of reliability. Each surrogate model was trained utilizing several (input–output) data sets simulated using numerical seepage modeling code (SEEPW). The Gaussian process regression machine learning technique was used to train several surrogate models to imitate the responses of the numerical seepage modeling. Each single input data set was solved (4 × 5) times to simulate four different realizations of spatial variation resulting from one of the five different standard deviation values (0.85, 1.55, 2.25, 2.95, 3.65 m/day) with constant mean (2 m/day). Each group of data for single standard deviation was utilized to train a single surrogate model. The genetic algorithm-based S-O model was utilized as an efficient optimization solver for such complex optimization task. The results of this study demonstrated that the reliability significantly influenced the design of HWRS. Further, the deterministic optimum design of HWRS was insufficient to be considered a reliable design, especially when a high degree of uncertainty due to the hydraulic conductivity estimation is included. Hence, RBOD for such problem is essential and helps the designer make a decision.

Optimization of cutting conditions using an evolutive online procedure

This paper proposes an online evolutive procedure to optimize the Material Removal Rate in a turning process considering a stochastic constraint. The usual industrial approach in finishing operations is to change the tool insert at the end of each machining feature to avoid defective parts. Consequently, all parts are produced at highly conservative conditions (low levels of feed and speed), and therefore, at low productivity. In this work, a framework to estimate the stochastic constraint of tool wear during the production of a batch is proposed. A simulation campaign was carried out to evaluate the performances of the proposed procedure. The results showed that it was possible to improve the Material Removal Rate during the production of the batch and keeping the probability of defective parts under a desired level.

Robust multi-period portfolio selection based on downside risk with asymmetrically distributed uncertainty set

Motivated by the asymmetrical attitudes of investors towards downside losses and upside gains, this paper proposes a robust multi-period portfolio selection model based on downside risk with asymmetrically distributed uncertainty set, in which the downside losses of a portfolio are controlled by the lower partial moment (LPM). A computationally tractable approximation approach based on second-order cone optimization is used for solving the proposed model. We show in theory that the optimal solution of the robust model can generate a given probability guarantee for individual and joint stochastic constraints. The effect of the asymmetrically distributed uncertainty set on performance of the optimal solution is analyzed by the usual comparative static method. Comprehensive numerical comparisons with real market data are reported and indicate that the proposed model can obtain the smaller standard deviation and turnover ratios which reduce the Sharpe ratios of optimal portfolio, compared with some well-known models in the literature.

A second-order cone programming formulation for two player zero-sum games with chance constraints

We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium in mixed strategies if the row vectors of the random matrices defining the stochastic constraints are elliptically symmetric distributed random vectors. We further show that a saddle point equilibrium can be obtained from the optimal solutions of a primal-dual pair of second-order cone programs.

An asset -- liability management stochastic program of a leasing company

We build a multi-stage stochastic program of an asset-liability management problem of a leasing company, analyse model results and present a stress-testing methodology suited for financial applications. At the beginning, the business model of such a company is formulated. We introduce three various risk constraints, namely the chance constraint, the Value-at-Risk constraint and the conditional Value-at-Risk constraint along with the second-order stochastic dominance constraint, which are applied to the model to control risk of the optimal strategy. We also present the structure and the generation process of our scenarios. To capture the evolution of interest rates the Hull-White model is used. Thereafter, results of the model and the effect of the risk constraints on the optimal decisions are thoroughly investigated. In the final part, the performance of the optimal solutions of the problems for unconsidered and unfavourable crisis scenarios is inspected. The methodology of a stress test we used was proposed in such a way that it answers typical questions asked by asset-liability managers.

A CVaR-robust-based multi-objective optimization model and three-stage solution algorithm for a virtual power plant considering uncertainties and carbon emission allowances

In order to make full use of distribute energy resources and decrease the abandoned energy of clean energy, the paper aggregates wind power plant (WPP), photovoltaic power generation (PV), biomass power generation (BPG), energy storage system (ESS), conventional gas turbines (CGT) and flexible load into a virtual power plant (VPP). Firstly, the basic structure and uncertainty factors of VPP operation are analyzed, and the output model of power sources is proposed. Then, a multi-objective optimization model is proposed with three objective functions, namely, the maximum operation revenue, the minimum operation risk and the minimum carbon emissions. Furthermore, the robust optimization theory is applied to construct a risk aversion model by converting the constraint conditions into stochastic constraint conditions with uncertainty factors. Thirdly, a three-stage solution algorithm is put forward for the proposed multi-objective model including payoff table establishment, fuzzy linearization and objective weight calculation. Finally, the modified IEEE30 node system is chosen as simulation system. The results show: (1) VPP could utilize the complementary nature of different distributed energy sources to balance the revenue, risk and carbon emission. (2) Robust optimization theory and conditional value at risk could describe the uncertainty risk, and, the prediction accuracy needs to be improved for controlling the operation risk. (3) The risk attitude of decision makers would affect VPP scheduling scheme, the less uncertainty lead to greater risk when confidence degree ≤0.85 and robust coefficient ≥0.95 (risk extreme aversion). (3) Price-based demand response (PBDR) could smooth load demand curve and the maximum total emission allowances (MTEA) can highlight the environmentally friendly characteristics, which could improve more grid-connected space of clean energy and achieve the optimal VPP operation. Therefore, the proposed multi-objective model could obtain higher economic benefit and achieve lower carbon emissions while rationally controlling risks, which could be taken as an reliable decision support for decision makers.

Stochastic Optimization of AISI 52100 Hard Turning With Six Sigma Capability Constraint

Hard turning optimization problems are usually approached using response surface methodology. By running designed experiments, researchers build analytical models to represent the outputs under interest. However, most studies focus on the expected values of the outputs, and only a few consider the variances of the models, even though there are several stochastic programming (SP) techniques available in the literature. Such variances may have a significant impact on the problem solution. This paper aims to optimize the AISI 52100 hardened steel turning process using SP. The decision variables are cutting speed, feed rate, and depth of cut, outputs have cost per part and material removal rate, and average surface roughness six sigma capability is modeled as a stochastic constraint. The SP method is also compared with the conventional one, which did not include the variances into the problem. The results show that taking the variability of the models into account is necessary to obtain a satisfactory process capability and to analyze different scenarios. Finally, this paper shows that competitive results can be achieved by simplifying the problem formulation.

Investigating carbon emissions in a production-inventory model under imperfect production, inspection errors and service-level constraint

This paper discusses an integrated inventory model for vendor-buyer system with stochastic demand and service-level constraint. The buyer's demand follows a normal distribution and the lead time depends upon lot size. The buyer uses a periodic review policy to manage the inventory level. An imperfect production processes and inspection errors are considered in the model. Carbon emissions and energy impacts which are generated from transportation activity and production activity are also included in the model. An incentive and penalty policy are adopted to reduce the carbon emissions generated from supply chain system. The model contributes to the current literature by allowing the inclusion of imperfect production, inspection errors, carbon emissions and setup cost reduction. The objective of the model is to determine the review period, setup cost and the number of deliveries so that the total cost incurred has the minimum value. An iterative procedure is suggested to solve the model. A numerical example and sensitivity analysis are performed to illustrate the application of the model and to show the impact of the changes of key parameters on model behaviour.

A Simple Proof of Strong Duality in the Linear Persuasion Problem

We provide a simple proof of strong duality for the linear persuasion problem. The duality is established in Dworczak and Martini (2019), under slightly stronger assumptions, using techniques from the literature on optimization with stochastic dominance constraints and several approximation arguments. We provide a short, alternative proof that is based on a direct argument to show the existence of optimal price functions, and on switching the roles of the primal and the dual to show that there is no duality gap.

Advancing Constrained Ranking and Selection With Regression in Partitioned Domains

Ranking and selection (R&S) procedures are powerful tools to enhance the efficiency of simulation-based optimization. In this paper, we consider the R&S problem subject to stochastic constraints and seek to improve the selection efficiency by incorporating the information from across the domain into quadratic regression metamodels. To better fulfill the quadratic assumption of the regression metamodel used in this paper, we divide the solution space into adjacent partitions such that the underlying functions of both the objective and constraint measures in each partition are approximately quadratic with homogeneous noise. Using the large deviations theory, we characterize the asymptotically optimal allocation rule by maximizing the rate at which the probability of false selection tends to zero. Numerical experiments demonstrate that our approach dramatically improves the selection efficiency by 50%-90% on some typical selection examples compared with the existing approaches.

Multi-item multi-choice integrated optimisation in inventory transportation problem with stochastic supply

This paper explores the study of multi-item multi-choice transportation problem (TP) in the ground of inventory optimisation. Using the concept of basic inventory optimisation, we develop a methodology for integrated optimisation in inventory transportation (IOIT) to reduce the logistic cost of a system. To accommodate the present situations of real-life TP, the stochastic supply is taken into consideration in the article. We describe a technique to reduce stochastic constraint to deterministic constraint with the help of stochastic programming. An algorithm is presented to solve the proposed problem using MATLAB. Then the proposed problem is solved by well-known optimisation technique; and the obtained solution is compared with the solution of basic inventory optimisation method. An example is presented to verify the effectiveness of the paper.

Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric

Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric