Two-stage Optimization Problem Research Articles

Submodular reassignment problem for reallocating agents to tasks with synergy effects

We propose a new combinatorial optimization problem that we call the submodular reassignment problem. We are given k submodular functions over the same ground set, and we want to find a set that minimizes the sum of the distances to the sets of minimizers of all functions. The problem is motivated by a two-stage stochastic optimization problem with recourse summarized as follows. We are given two tasks to be processed and want to assign a set of workers to maximize the sum of profits. However, we do not know the value functions exactly, but only know a finite number of possible scenarios. Our goal is to determine the first-stage allocation of workers to minimize the expected number of reallocated workers after a scenario is realized at the second stage. This problem can be modeled by the submodular reassignment problem. We prove that the submodular reassignment problem can be solved in strongly polynomial time via submodular function minimization. We further provide a maximum-flow formulation of the problem that enables us to solve the problem without using a general submodular function minimization algorithm, and more efficiently both in theory and in practice. In our algorithm, we make use of Birkhoff’s representation theorem for distributive lattices.

A multicountry, multicommodity stochastic game theory network model of competition for medical supplies inspired by the Covid-19 pandemic

In this paper, we construct the first stochastic Generalized Nash Equilibrium model for the study of competition among countries for limited supplies of medical items (PPEs, ventilators, etc.) in the disaster preparedness and response phases in the Covid-19 pandemic. The government of each country is faced with a two-stage stochastic optimization problem in which the first stage is prior to the pandemic declaration and the second stage is post the pandemic declaration. We provide the theoretical constructs, a qualitative analysis, and an algorithm, accompanied by convergence results. Both illustrative examples are presented as well as algorithmically solved numerical examples, inspired by the need for N95 masks and ventilators. The results reveal that, in addition to the preparedness of countries before the pandemic declaration, their ability to adapt to the conditions in different scenarios has a significant impact on their overall success in the management of the pandemic crisis. The framework can capture competition for other medical supplies, including Covid-19 vaccines and possible treatments, with modifications to handle perishability.

A Two-Stage Deep Autoencoder-Based Missing Data Imputation Method for Wind Farm SCADA Data

This paper proposes a novel two-stage method for imputing missing SCADA data of wind turbines with high accuracy based on deep nonparametric models, sparse autoencoders (SAE), and a gradient-based optimization algorithm, coordinate descent (CD). A complex pattern of missing data, namely, data loss in correlated attributes (DLCA) that occurs simultaneously, is focused on and studied. In this paper, the missing data imputation is formulated as a two-stage optimization problem. In the first stage, the reconstruction error (RE) of SAE is regarded as the loss for training nonparametric attribute reconstruction models via a complete dataset to learn a low-dimensional manifold, in which data are densely distributed. At the second stage, RE serves as an objective function for optimizing the missing data imputation of a similar but incomplete dataset based on the developed SAE. According to the potential convexity of REs with respect to the imputation of missing attributes, which is empirically discovered through preliminary experiments, the CD algorithm is applied to efficiently solve the optimization problem. The efficacy of the proposed method is validated by using a large real wind turbine dataset. The results of the computational experiments demonstrate that the proposed method performs well on considered benchmarks that are well known for imputing missing data.

On the Optimality of Affine Policies for Budgeted Uncertainty Sets

In this paper, we study the performance of affine policies for a two-stage, adjustable, robust optimization problem with a fixed recourse and an uncertain right-hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets, widely used in practice, in which we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than [Formula: see text] even for budget of uncertainty sets in which [Formula: see text] is the number of decision variables. Affine policies, in which the second-stage decisions are constrained to be an affine function of the uncertain parameters provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an [Formula: see text]-approximation for the two-stage, adjustable, robust problem with fixed nonnegative recourse for budgeted uncertainty sets. This matches the hardness of approximation, and therefore, surprisingly, affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for a significantly more general class of intersection of budgeted sets, including disjoint constrained budgeted sets, permutation invariant sets, and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal, feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute a near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

Trading-oriented battery energy storage planning for distribution market

In this paper, we present a trading-oriented battery energy storage system (BESS) planning model for a distribution market. The proposed planning model is formulated as a mutual-iteration and multi-objective two-stage optimization problem. The first stage is designed to optimize the internal resources allocation including PV system, wind turbine, gas turbine and BESS. Then a double-side auction model named as averaging pricing market (APM) mechanism is presented in the second stage, which aims at solving the issue of energy trading benefit loss caused by the relatively high difference between time-of-use (ToU) electricity price and feed-in-tariff. The contribution of this paper lies in combining the planning of BESS with the optimization of the distribution electricity market. Numerical results on a modified IEEE 24-bus system are presented to demonstrate the superiority of the two-stage optimization model and the sensitivity of the model is also investigated based on regret analysis.

Sequencing and scheduling appointments for multi-stage service systems with stochastic service durations and no-shows

ABSTRACT In this work, we consider a joint sequencing and scheduling appointments problem with stochastic service times and no-shows in multi-stage service systems. The objective is to minimise the total expected weighted costs of customers' waiting times and service providers' idle times over multiple stages. For the problem, we first formulate it as a stochastic program and exploit the sample average approximation approach to reformulate it as a mixed-integer program in further. Then we transform the stochastic program into a two-stage optimisation problem and develop a standard Benders decomposition algorithm. To overcome the long running time of Benders decomposition, we simplify the master problem in the algorithm and propose a Benders decomposition-based algorithm to find a near-optimal solution. Finally, we conduct a series of numerical experiments to illustrate the efficiency of our proposed algorithm, examine the impact of the number of stages, stochastic service times and no-shows on the optimal job allowances and performance indicators (i.e. waiting times and idle times) and investigate two easy-to-implement sequence rules. The computational results show that both our proposed Benders decomposition-based algorithm and easy-to-implement sequence rules perform well.

A Study on Efficient Computing Budget Allocation for a Two-Stage Problem

We consider how to allocate simulation budget to estimate the risk measure of a system in a two-stage simulation optimization problem. In this problem, the first stage simulation generates scenarios that serve as inputs to the second stage simulation. For each sampled first stage scenario, the second stage procedure solves a simulation optimization problem by evaluating a number of decisions and selecting the optimal decision for the scenario. It also provides the estimated performance of the system over all sampled first stage scenarios to estimate the system’s reliability or risk measure, which is defined as the probability of the system’s performance exceeding a given threshold under various scenarios. Usually, such a two-stage procedure is very computationally expensive. To address this challenge, we propose a simulation budget allocation procedure to improve the computational efficiency for two-stage simulation optimization. After generating first stage scenarios, a sequential allocation procedure selects the scenario to simulate, followed by an optimal computing budget allocation scheme that determines the decision to simulate in the second stage simulation. Numerical experiments show that the proposed procedure significantly improves the efficiency of the two-stage simulation optimization for estimating system’s reliability.

Content Promotion for Online Content Platforms with Network Diffusion Effect

Online content platform aims to maximize content adoption which is simultaneously driven by the platform promotion and network diffusion. We study the candidate generation and promotion (CGP) problem for online content with the diffusion effect in this paper. Motivated by real-world datasets from the industry partner, we propose a novel diffusion model for online content that captures time variant diffusion effect and the existence of promoting probability. Based on the diffusion model, we formulate the CGP problem as a two-stage optimization problem. We show that the problem is NP-hard and can be seen as a submodular maximization problem. By utilizing the two-stage structure and combining the greedy idea from submodularity, we propose an $(1-\frac{1}{e})(1-\epsilon)$-approximation algorithm. We also consider the online version of this problem where the platform is simultaneously estimating the diffusion parameters and deploying the content promotion decisions. We propose a state-dependent non-anticipatory online policy that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ of $\alpha$-Cumulative Myopic Regret ($\alpha$-CMRegret). Numerical results from a real-world dataset show that the diffusion process for online content deviates drastically from the traditional diffusion models and our proposed model provides much better fit. We also demonstrate the efficiency of our offline and online algorithms with the dataset.

A Bi-Level Model for the Bidding Strategy of an Inter-Regional Electricity Trading Aggregator Considering Reliability Requirements and Transmission Losses

In countries with large-scale power systems and nascent electricity markets, the inter-regional electricity trading aggregators (IRETAs) are in charge of participating in inter-regional electricity trading (IRET) on behalf of the generators and loads within the region except for operating the regional power systems (RPSs). As price makers in IRET and system operators of RPSs, IRETAs should make their bidding strategies considering both the RPSs’ reliability requirements and the IRET’s transmission losses to ensure both the safe operation of RPS and the profits in the IRET, which cannot be handled by the current methods. Therefore, this paper investigates an approach to help IRETAs make comprehensive bidding strategies in the IRET. Firstly, a hierarchical framework for an IRETA to participate in IRET is introduced. Then a bi-level model is proposed, in which the optimal operation of RPS and the market-clearing process of IRET are regarded as the upper and lower-level problems. Specifically, the upper-level problem is modeled as a two-stage stochastic optimization problem considering both the fault states and the reliability requirements of RPS. Moreover, the non-negligible losses from the inter-regional electricity transmission are internalized in the lower-level problem to consider their impacts on the IRETA’s bidding strategy-making. The proposed bi-level model is nonlinear and transformed into a linear single-level one through the Karush-Kuhn-Tucker (KKT) conditions and the dual theory. Finally, simulations demonstrate that the bidding strategy formed by the proposed method reduces the number of buses exceeding the limit of reliability requirement by over 50% and cuts down the transmission loss by over 40% compared with the former works.

Efficient Algorithms for Distributionally Robust Stochastic Optimization with Discrete Scenario Support

Recently, there has been a growing interest in distributionally robust optimization (DRO) as a principled approach to data-driven decision making. In this paper, we consider a distributionally robust two-stage stochastic optimization problem with discrete scenario support. While much research effort has been devoted to tractable reformulations for DRO problems, especially those with continuous scenario support, few efficient numerical algorithms were developed, and most of them can neither handle the non-smooth second-stage cost function nor the large number of scenarios $K$ effectively. We fill the gap by reformulating the DRO problem as a trilinear min-max-max saddle point problem and developing novel algorithms that can achieve an $\mathcal{O}(1/\epsilon)$ iteration complexity which only mildly depends on $K$. The major computations involved in each iteration of these algorithms can be conducted in parallel if necessary. Besides, for solving an important class of DRO problems with the Kantorovich ball ambiguity set, we propose a slight modification of our algorithms to avoid the expensive computation of the probability vector projection at the price of an $\mathcal{O}(\sqrt{K})$ times more iterations. Finally, preliminary numerical experiments are conducted to demonstrate the empirical advantages of the proposed algorithms.

Two-Stage Stochastic Optimization Via Primal-Dual Decomposition and Deep Unrolling

We consider a two-stage stochastic optimization problem, in which a long-term optimization variable is coupled with a set of short-term optimization variables in both objective and constraint functions. Despite that two-stage stochastic optimization plays a critical role in various engineering and scientific applications, there still lack efficient algorithms, especially when the long-term and short-term variables are coupled in the constraints. To overcome the challenge caused by tightly coupled stochastic constraints, we first establish a two-stage primal-dual decomposition (PDD) method to decompose the two-stage problem into a long-term problem and a family of short-term subproblems. Then we propose a PDD-based stochastic successive convex approximation (PDD-SSCA) algorithmic framework to find KKT solutions for two-stage stochastic optimization problems. At each iteration, PDD-SSCA first runs a short-term sub-algorithm to find stationary points of the short-term subproblems associated with a mini-batch of the state samples. Then it constructs a convex surrogate for the long-term problem based on the deep unrolling of the short-term sub-algorithm and the back propagation method. Finally, the optimal solution of the convex surrogate problem is solved to generate the next iterate. We establish the almost sure convergence of PDD-SSCA and customize the algorithmic framework to solve two important application problems. Simulations show that PDD-SSCA can achieve superior performance over existing solutions.

Efficient Sparse Coding Using Hierarchical Riemannian Pursuit

Sparse coding is a class of unsupervised methods for learning a sparse representation of the input data in the form of a linear combination of a dictionary and a sparse code. This learning framework has led to state-of-the-art results in various image and video processing tasks. However, classical methods learn the dictionary and the sparse code based on alternating optimizations, usually without theoretical guarantees for either optimality or convergence due to non-convexity of the problem. Recent works on sparse coding with a complete dictionary provide strong theoretical guarantees thanks to the development of the non-convex optimization. However, initial non-convex approaches learn the dictionary in the sparse coding problem sequentially in an atom-by-atom manner, which leads to a long execution time. More recent works seek to directly learn the entire dictionary at once, which substantially reduces the execution time. However, the associated recovery performance is degraded with a finite number of data samples. In this paper, we propose an efficient sparse coding scheme with a two-stage optimization. The proposed scheme leverages the global and local Riemannian geometry of the two-stage optimization problem and facilitates fast implementation for superb dictionary recovery performance by a finite number of samples without atom-by-atom calculation. We further prove that, with high probability, the proposed scheme can exactly recover any atom in the target dictionary with a finite number of samples if it is adopted to recover one atom of the dictionary. An application on wireless sensor data compression is also proposed. Experiments on both synthetic and real-world data verify the efficiency and effectiveness of the proposed scheme.

Optimal demand response resource exploitation for efficient accommodation of renewable energy sources in multi-energy systems considering correlated uncertainties

Abstract This paper proposes a comprehensive planning model for enhancing accommodation of renewable energy sources into a combined heat and power based multi-energy system by utilizing the flexibility of a demand response program. As distinct from existing works, demand response has been implemented through a price-based program not only in terms of its effects on system operation but also for the planning decisions, and the potential correlations among uncertainties (i.e., renewable energy sources availability, load demand, and demand-side responsiveness) have been explicitly considered in our study. The concerned problem is interpreted into a two-stage optimization problem, where the placement of advanced metering infrastructures, the installation of renewable generation units along with the relevant pricing strategy for the demand side are jointly optimized to minimize the overall economic costs of the system. The flexibility of customers’ energy demands in response to real-time price changes is represented by using a generalized elasticity model, through which the demand response in terms of both load shifting in the temporal dimension, and the switching of energy source on the demand side can be properly captured under a unified framework. An efficient scenario generation method leveraging a series of correlation-handling techniques is employed to address the uncertainties in the proposed problem and a comprehensive scenario reduction method based on a novel optimal clustering technique is further introduced to alleviate the computational burden of the resultant model with the correlated uncertainties. Compared with the existing literatures, the novelty of this paper is in three-fold: 1) This study investigates the potential value of demand response for promoting renewable energy exploitation from a long-term planning perspective, instead of the conventional operation aspect. 2) The impacts from both system uncertainties and their potential correlations have been explicitly considered. 3) A comprehensive scenario reduction method based on optimal clustering analysis is introduced and employed to alleviate the solving complexity and improve the computational efficiency of the proposed model. The proposed planning framework is demonstrated on an illustrative test case and the simulation results verified the effectiveness of the developed model.

Linearized Robust Counterparts of Two-Stage Robust Optimization Problems with Applications in Operations Management

In this article, we discuss an alternative method for deriving conservative approximation models for two-stage robust optimization problems. The method mainly relies on a linearization scheme employed in bilinear programming; therefore, we will say that it gives rise to the linearized robust counterpart models. We identify a close relation between this linearized robust counterpart model and the popular affinely adjustable robust counterpart model. We also describe methods of modifying both types of models to make these approximations less conservative. These methods are heavily inspired by the use of valid linear and conic inequalities in the linearization process for bilinear models. We finally demonstrate how to employ this new scheme in location-transportation and multi-item newsvendor problems to improve the numerical efficiency and performance guarantees of robust optimization.

Electric vehicles reactive power management and reconfiguration of distribution system to minimise losses

This study investigates the optimal operation of distribution feeder reconfiguration (DFR) strategy in smart distribution system integrated with electric vehicles (EVs) and wind power generation. A two-stage framework is proposed for real and reactive power management and DFR. In the first stage, demand dispatch of EVs is coordinated to utilise available wind power and minimise the total load variations. In the second stage, a comprehensive problem formulation is developed to minimise the losses in the system with simultaneous EVs reactive power management and DFR. A genetic algorithm is used to solve this two-stage optimisation problem. A case study of 33-bus radial distribution system is presented to highlight the efficacy of proposed approach. The obtained results are quantitatively assessed in terms of losses and node voltages. A comparative benchmarking analysis is presented to substantiate the role of DFR and reactive power capability of EVs in improving the system performance.

Adaptively Meshed Video Stabilization

Video stabilization is essential for improving the visual quality of shaky videos. Current video stabilization methods usually take feature trajectories in the background to estimate one global transformation matrix or several transformation matrices based on a fixed mesh, and warp shaky frames into their stabilized views. However, these methods may not model the shaky camera motion well in complicated scenes, such as scenes containing large foreground objects or strong parallax, and may result in notable visual artifacts in the stabilized videos. To resolve the above issues, this paper proposes an adaptively meshed method to stabilize a shaky video based on all of its feature trajectories and an adaptive blocking strategy. More specifically, we first extract the feature trajectories of the shaky video and then generate a triangle mesh according to the distribution of the feature trajectories in each frame. Then, the transformations between shaky frames and their stabilized views over all triangular grids of the mesh are calculated to stabilize the shaky video. Since more feature trajectories can usually be extracted from all of the regions, including both the background and foreground regions, a finer mesh will be obtained and provided for camera motion estimation and frame warping. We estimate the mesh-based transformations of each frame by solving a two-stage optimization problem. Moreover, foreground and background feature trajectories are no longer distinguished and both contribute to the estimation of the camera motion in the proposed optimization problem, yielding better estimation performance than previous works, particularly in challenging videos with large foreground objects or strong parallax. To further enhance the robustness of our method, we propose two adaptive weighting mechanisms to improve its spatial and temporal adaptability. Experimental results demonstrate the effectiveness of our method in producing visually pleasing stabilization effects in various challenging videos.

Multi-task processing oriented production layout based on evolutionary programming mechanism

Traditional job scheduling is an independent optimization phase without considering material flow among machines. However, in actual production, the machine layout will affect material flow time, therefore affecting job completion time. In this paper, a novel multi-task-oriented production layout problem is proposed, with the goal to optimize material flow by adjusting machine placement. To solve this problem, a production layout evolution (PLEV) framework is established, which models it as a two-stage optimization problem. In the first stage, a concept of ‘relevance between machines (RBM)’ is proposed. The RBM measures the relationship between two machines according to the processing data. A machine placement scheme is designed based on RBM. In the second stage, a job scheduling rule based on the earliest finish time (EFT) is adopted to evaluate production layouts. Then, through an evolutionary programming mechanism, the production layout can evolve to optimize makespan. Experiments are carried out to verify the feasibility and performance of the proposed PLEV framework. The results demonstrate that promising product layouts for multi-task processing can be obtained by using this framework.

Decision making under wind power generation and load demand uncertainties: a two-stage stochastic optimal reactive power dispatch problem

ABSTRACT For the modern power systems with intensive renewable energy sources integration, the uncertainty analysis of its performance become necessary for desired decision making. An appropriate choice of modelling and handling methodology, for uncertain parameters associated with the renewable energy, remains one of major concerns for the power system engineers. This paper presents a stochastic optimization framework for some critical decisions making on reactive power dispatch in two consecutive stages by incorporating both generation outputs of wind farms along with load demand uncertainties. The proposed method is generic for optimal reactive power dispatch in any modern wind-integrated power system. A methodology based on scenario-based analysis is adopted for handling these uncertainties. Moreover, a hybrid fuzzy evolutionary algorithm (HFEA) is applied to obtain high-quality solutions for this complex two-stage stochastic optimization problem. Furthermore, the proposed HFEA-based stochastic optimization approach is tested on IEEE 14-bus power system with two distinct wind farms. The simulation results prove the robustness of HFEA for generating feasible and optimal solutions for all the considered scenarios.

Resource allocation for contingency planning: An inexact proximal bundle method for stochastic optimization

Resource contingency planning aims to mitigate the effects of unexpected disruptions in supply chains. While these failures occur infrequently, they often have disastrous consequences. This paper formulates the resource allocation problem in contingency planning as a two-stage stochastic optimization problem with a risk-averse recourse function. The solution method proposed relies on an inexact proximal bundle method with subgradient approximations through a scenario reduction mechanism. The paper extends the inexact oracle to a more general risk-averse setting, and proves that it meets the requirements of the oracle in the inexact bundle method, ensuring convergence to an optimal solution. The practical performance of the developed inexact bundle method under risk aversion is investigated for our resource allocation problem. We create a library of test problems and obtain their optimal values by applying the exact bundle method. The computed solutions from the developed inexact bundle method are compared against these optimal values, under different coherent risk measures. Our analyses indicate that our inexact bundle method significantly reduces the computational time of solving the resource allocation problem in comparison to the exact bundle method, and is capable of achieving a high percentage of optimality within a much shorter time.

Two-stage inventory management with financing under demand updates

Considered is a retailer (she) facing non-stationary stochastic demand. Demand can be fully observed and backlogged, consequently the retailer can update the initial demand information using a Bayesian approach. To alleviate the demand risk, the retailer may use a secondary opportunity to replenish through an option contract. In addition, the retailer also has access to an immediate loan if she faces capital constraints and to a risk-free investment if she has surplus funds. The paper presents a recourse approach to solve the two-stage optimization problem and derive the optimal inventory/financing policies. The results show that the option procurement policy has a two-threshold base-stock structure depending on the first procurement, demand update and also the retailer’s financial state. The initial procurement can be computed subsequently. A sufficiently large initial demand will induce the retailer to seize the secondary procurement opportunity. Finally, a series of numerical examples demonstrate the resulting policy under various inventory/financial situations. This research incorporates the financial and operational decisions into demand updates, and brings new managerial results and insights.