Hierarchical Optimization Problem Research Articles

Equilibrium Interaction Strategies for Integrated Energy System Incorporating Demand-Side Management Based on Stackelberg Game Approach

This paper analyzes the balanced interaction strategy of an integrated energy system (IES) operator and an industrial user in the operation process of the IES under the demand-side management (DSM) based on game theory. Firstly, we establish an electric–thermal IES, which includes a power grid, a heat grid and a natural gas grid. Secondly, a two-stage Stackelberg dynamic game model is proposed to describe the game behavior of IES operators and industrial users in the process of participating in DSM. The interactions between the IES operator (leader) and the user (follower) are formulated into a one-leader–one-follower Stackelberg game, where optimization problems are formed for each player to help select the optimal strategy. A pricing function is adopted for regulating time-of-use (TOU), which acts as a coordinator, inducing users to join the game. Then, for the complex two-stage dynamic game model established, the lower user-side constraint optimization problem is replaced by its KKT condition, so that the two-stage hierarchical optimization problem is transformed into a single-stage mixed-integer nonlinear optimization model, and the branch-and-bound method is introduced to solve it. Finally, the equilibrium strategies and income values of both sides of the game are obtained through a case simulation, and the dynamic equilibrium strategy curves under different capacity configurations are obtained through the sensitivity analysis of key parameters. The equilibrium income of the IES is USD 93.859, while the equilibrium income of industrial users in the park is USD 92.720. The simulation results show that the proposed method and model are effective.

Enhancing dynamic operation optimization feasibility for constrained economic model predictive control systems

AbstractBased on the hierarchical control structure, optimization and control problems of large‐scale and multivariate plants are solved sequentially. The economic performance of the plant plays an essential role in the plant‐wide modern industry. The optimal operating conditions will change as the economic criteria changes throughout the operation of the plant as the result of variations in raw material prices, product prices, production demand, market fluctuations, disturbances, and so forth. In reality, soft constraints are frequently used to denote the production requirements of various operating conditions. In order to improve economic performance and guarantee feasibility for the entire plant operation, a novel economic model predictive control (EMPC) strategy is proposed to control the constrained multi‐variable process system with varying economic performance criteria under soft constraints. By incorporating the transient steady‐state and two categories of slack variables for soft constraints, a modified economic performance index is optimized to cope with the changing criteria. In addition, a contractive constraint is added to the closed‐loop system to guarantee stability for non‐dissipative stage costs. This approach ensures recursive feasibility and asymptotic stability. The effectiveness of the proposed method is demonstrated by numerical examples and the fluid catalytic cracking unit (FCCU) process.

Hierarchical MPC for a dynamic process system employing parametric global optimization strategy

The hierarchical decision making in process industries has been traditionally viewed as having a common objective, such as the overall cost, which needs to be optimized. However, a more appropriate approach is to formulate and solve hierarchical optimization and control problems. The solution algorithms for hierarchical optimization problems have been reported in the literature. The idea is to recast each optimization sub-problem in the hierarchy into a multiparametric programming problem, considering the variables of upper-level problems as unknown parameters. In this paper, explicit Model Predictive Control (MPC) and hierarchical optimization techniques, employing multiparametric programming, are combined for hierarchical MPC. The solution algorithm for hierarchical MPC is described in detail. Note that the solution to a hierarchical MPC problem is challenging, even for the simplest case of linear-quadratic objectives. Closed-loop simulations of a thermal mixing process, under two different hierarchical MPC formulations, are performed and the control performance is studied.

Learning-Based Hierarchical Edge Caching for Cloud-Aided Heterogeneous Networks

Edge caching has emerged as a promising technique against latency caused by explosive growth of mobile data traffic through caching popular contents at the edge networks. However, the dynamically changing content popularity nature and limited caching capacity make it challenging to design an effective caching scheme to reduce latency. To solve this, a learning-based hierarchical edge caching (LHEC) scheme is proposed in this work. We first propose a novel deep learning architecture, namely Stacked Autoencoder-Long Short Term Memory Network (SAE-LSTMNet) to capture both the correlation of the request patterns among different content and the periodicity in time domain to improve the prediction accuracy of the content popularity. Then, to predict the popularity of these newly-added contents, a dynamic content catalog is introduced and a similarity-based content popularity prediction (SCPP) approach is proposed. Based on the content popularity prediction, a hierarchical edge caching optimization problem is formulated to minimize the average content downloading latency. Since the formulated problem is NP-hard and difficult to be solved, a low-complexity algorithm is proposed to obtain the near-optimal solutions. Simulation results show that the proposed content popularity prediction approach outperforms up to 6.36% in terms of the mean absolute error compared with the state-of-the-art methods and the proposed LHEC scheme reduces the average downloading latency at about 5.3%~7.9% compared with those existing caching schemes.

A set‐based approach for hierarchical optimization problem using Bayesian active learning

AbstractIn the development of complex engineering systems, many engineers from different disciplines collaborate to identify feasible designs that satisfy all system requirements. Analytical target cascading (ATC) is a method for the design optimization of hierarchical, multilevel systems and has been successfully employed in the design of complex engineering systems. In this paper, we propose a novel data‐driven set‐based ATC (SBATC) method for hierarchical design optimization problems using machine learning techniques. The proposed SBATC offers two advantages in engineering processes. First, it decomposes hierarchical design optimization problems into sets of suboptimization problems. The feasible regions of the suboptimization problems are explored and cascaded to lower‐level optimization problems. Using the set‐based approach, couplings between two adjacent levels in the optimization process are not required. Second, an efficient strategy is employed to determine feasible regions based on Bayesian active learning (BAL). In BAL, the Gaussian process (GP) of all cost functions is trained. An acquisition function that combines the probability of feasibility and entropy search is evaluated using posterior distributions of the trained GP. The acquisition function is maximized to generate new sampling points around the feasible regions by balancing the exploitation and exploration of the design space. To verify the effectiveness of the proposed method, numerical examples of hierarchical optimization problems are evaluated.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

Supervised Learning of Analysis-Sparsity Priors With Automatic Differentiation

Sparsity priors are commonly used in denoising and image reconstruction. For analysis-type priors, a dictionary defines a representation of signals that is likely to be sparse. In most situations, this dictionary is unknown, and is to be recovered from pairs of ground-truth signals and measurements, by minimizing the reconstruction error. This defines a hierarchical optimization problem, which can be cast as a bi-level optimization. Yet, this problem is unsolvable, as reconstructions and their derivative <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w.r.t.</i> the dictionary have no closed-form expression. However, reconstructions can be iteratively computed using the Forward-Backward splitting (FB) algorithm. In this paper, we approximate reconstructions by the output of the FB algorithm, up to a linear transformation. Then, we leverage automatic differentiation to evaluate the gradient of this output <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w.r.t.</i> the dictionary, which we learn with projected gradient descent. Experiments show that our algorithm successfully learns the 1D Total Variation (TV) dictionary from piecewise constant signals. For the same case study, we propose to constrain our search to dictionaries of 0-centered columns, which removes undesired local minima and improves numerical stability.

Hierarchical optimization of policy and design for standalone hybrid power systems considering lifecycle carbon reduction subsidy

To achieve the target of carbon neutrality, the present work proposes a lifecycle carbon reduction subsidy policy to promote the exploitation of renewable energy technologies in remote areas. The hierarchical optimization of policy and design for a standalone hybrid renewable energy system is further conducted based on the proposed subsidy policy. The upper-level policy optimization model aims to maximize the renewable energy fraction and minimize the subsidy expenses borne by the government simultaneously, and the lower-level design optimization model aims to minimize the net present cost borne by the system operator. Four meta-heuristic algorithms coupled with a standard iterative method are employed to solve the hierarchical optimization problem, and their performances are comprehensively assessed in terms of convergence, robustness, and computational efficiency. Besides, an improved K-means clustering algorithm along with a tailored optimal K-value selection strategy is employed for scenario reduction. The results of the case study show that: (ⅰ) teaching-learning-based optimization performs the best convergence and robustness as it achieves the balance of exploration and exploitation. (ⅱ) the nexus between policy and system design presents a piecewise pattern, implying that the system configuration will change significantly only when the subsidy policy exceeds a certain threshold. (ⅲ) the proposed subsidy policy is expected to increase the renewable energy fraction to 99.23% and reduce the lifecycle carbon emissions by 80.7%. Overall, the proposed subsidy policy and its impact mechanism on system design can provide theoretical support for the government to formulate tailored policies for different preferences, and the proposed optimization framework and methodology can be directly extended to practical applications.

Optimal Scheduling of Demand Response Aggregators in Industrial Parks Based on Load Disaggregation Algorithm

Nowadays, industrial parks play a significant role in the development of electricity market plans, and can thus provide excellent opportunities for market players to actively participate in various electricity markets. The demand response aggregator (DRA) is a major market player that can take advantage of these opportunities. In restructured electricity markets, identifying the consumption patterns of different classes of consumers can be effective in furthering the goals of the DRA. In previous studies on the self-scheduling of the DRA, consumer behavior has not been considered. Such an approach leads to numerous technical problems in the restructured electricity markets. For this purpose, herein, a practical mechanism is presented for executing the self-scheduling process of the DRA by considering the load disaggregation algorithm. The integration of self-scheduling and load disaggregation processes creates a hierarchical optimization problem. The main aim of the constructed hierarchical structure is to find the optimal self-scheduling of the DRA to consciously participate in the electricity markets by identifying the behavior of different consumers. The proposed structure is implemented and evaluated on the industrial park in Saveh, Iran. The time-of-use (TOU) and reward-based demand response (DR) programs are considered as the tools available for the DRA to trade the DR volumes in the day-ahead and balancing electricity markets.

Efficient learning rate adaptation based on hierarchical optimization approach

This paper proposes a new hierarchical approach to learning rate adaptation in gradient methods, called learning rate optimization (LRO). LRO formulates the learning rate adaption problem as a hierarchical optimization problem that minimizes the loss function with respect to the learning rate for current model parameters and gradients. Then, LRO optimizes the learning rate based on the alternating direction method of multipliers (ADMM). In the process of this learning rate optimization, LRO does not require any second-order information and probabilistic model, so it is highly efficient. Furthermore, LRO does not require any additional hyperparameters when compared to the vanilla gradient method with the simple exponential learning rate decay. In the experiments, we integrated LRO with vanilla SGD and Adam. Then, we compared their optimization performance with the state-of-the-art learning rate adaptation methods and also the most commonly-used adaptive gradient methods. The SGD and Adam with LRO outperformed all the competitors on the benchmark datasets in image classification tasks.

A hierarchical scheme for remaining useful life prediction with long short-term memory networks

Remaining useful life (RUL) prediction is essential in prognostics and health management (PHM) applications, where data-driven approaches employ the tendency of the degradation process using operating data of complex systems, and have attracted more and more attention. With the idea that forecasting the time period before the equipment reaches the critical degradation stage (e.g., failure, fault, etc.), RUL prediction is usually formed as an optimization problem (in particular, a regression problem between the inputs–real-time measurements and the outputs–the RUL predictions). This work formulates the RUL prediction as a bi-level optimization problem, (i) the lower level is intended to forecast the time-series in the near future, and (ii) the upper level is to predict the RULs by integrating the available measurements up-to-date and the predicted ones by the lower-level prediction. To tackle the hierarchical optimization problem, a bi-level deep learning scheme is proposed for the machine RUL prediction, where long short-term memory (LSTM) networks are applied as of the unique characteristics in processing time-series and extracting recursive and non-recursive features among them. Case studies using PHM08 data challenge data set, 4 data sets in C-MAPSS package and 1 data set in the new CMAPSS dataset are implemented, to validate the proposed framework. The results show that the presented method outperforms the state-of-the-art approaches.

Riemannian stochastic fixed point optimization algorithm

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad.

Hierarchical Task-Space Optimal Covariance Control With Chance Constraints

This letter presents a new control paradigm applicable to nonlinear systems such as robots subject to chance and covariance assignment constraints which we refer to as hierarchical optimal covariance control. To the best of our knowledge, this is the first study to formulate the hierarchical optimal covariance control problem involving multiple operational tasks. The framework is defined as a multi-stage optimization problem considering multiple hierarchical tasks specified in lexicographic order. Towards this goal, we first approximate the nonlinear dynamic model of a robot into multiple linear stochastic systems by linearizing the model along given trajectories. We then project these stochastic models onto the null-space of the previous task models for efficiently solving lexicographical optimization. In addition, we specify probability functions to account for chance constraints using the Whittacker M function. We formulate the chance constraints as a positive semi-definite matrix constraint and solve the hierarchical optimal covariance control problem using sequential semi-definite programming. We demonstrate that this procedure yields higher accuracy for multiple hierarchical tasks than employing deterministic operational space control models.

Solving a Multi-Objective Chance Constrained Hierarchical Optimization Problem Under Intuitionistic Fuzzy Environment with its Application

Solving a Multi-Objective Chance Constrained Hierarchical Optimization Problem Under Intuitionistic Fuzzy Environment with its Application

A two‐stage multi‐period distribution network expansion planning considering the integration of private investors

In recent years, the emerging of renewable energy resources (RERs) in distribution networks has been rapidly raised by network planners. Although the integration of RERs brings a lot of advantages to distribution networks, they impose some disadvantages such as high initial capital investment on distribution companies (DISCO). This paper proposes a new two-stage multi-period distribution network expansion planning (DNEP), in which the potential of private investor's (PIs) participation is considered in the problem. In the proposed model, the planner presents an incentive price to buy energy from PIs so that it attracts them to participate in the planning projects. This incorporation is based on a long-term contract that guarantees the benefits of both contract correspondents. The proposed model is presented as a hierarchical two-stage optimization problem where the upper stage determines the general structure of the system and the lower level considers the operational conditions. Moreover, the photovoltaic plants (PVs) and energy storage systems (ESSs) are investigated as clean and new technologies in the planning, and DISCO determines the optimal sizing and sitting of these resources regarding technical and economical assessments. The fuzzy clustering method (FCM) is used to capture the intermittency of the system loads and solar irradiance by creating a day-ahead scheme. The effectiveness of the proposed DNEP problem is evaluated through a 54-bus distribution test system and a real 104-bus test system as well. Simulation and economic results show that how PIs can play an important role in long-term planning by investing RERs.

Off-policy algorithm based Hierarchical optimal control for completely unknown dynamic systems

This paper proposes an online reinforcement learning(RL) for solving Stackelberg games with completely unknown dynamic systems. To deal with the hierarchical optimal control problem, the key is to find the solution of a two-level optimal control with the leader’s optimal problem being constrained. Firstly, the leader–follower coupled Hamiltonian-Jacobi(HJ) equations subject to the follower’s costate equation is derived. Secondly, an off-policy scheme is designed following the policy iteration(PI) algorithm based on model. The improved off-policy algorithm is given for obtaining the solution to the leader–follower coupled HJ equations. The algorithm is built without requiring any knowledge of the dynamic system, and the leader–follower optimal control policies are directly constructed by the extracted data. Meanwhile, the existence of the Stackelberg equilibrium is demonstrated. Lastly, NNs for each player are built, and NN learning is accomplished online with Kronecker product technique and vectorization, which simples the NN form and could decrease computation burden. Simulation examples are presented to demonstrate the proposed learning algorithm.

Hierarchical nonlinear model predictive control with multi-time-scale for wastewater treatment process

Different time scales are inevitable for the controlled variables of wastewater treatment process (WWTP), which may degrade the operation performance or even destroy the stability of the closed-loop system. In this paper, a hierarchical nonlinear model predictive control (HNMPC) strategy is developed to deal with the different time scales of controlled variables to improve the operation performance of WWTP. The main merits of HNMPC are three-fold. First, a hierarchical control structure is developed to obtain reasonable control laws. Then, the controlled variables with different time scales can be tracked by different level controllers. Second, a gradient method is developed to solve the hierarchical optimization problem of HNMPC to reduce the computational cost. Then, the fast response of HNMPC can be achieved to improve the operation performance. Third, the stability of HNMPC is proved in theory. Then, the corresponding stability conditions are given to guide the practical application. Finally, the testing results on the benchmark simulation model verify that the proposed HNMPC can achieve suitable operation performance in terms of control accuracy.

Fuzzy multiobjective hierarchical optimization with application to identify antienzymes of colon cancer cells

BackgroundDeveloping a new drug from conception to launch is costly and time consuming. Computer-aided methods can reduce research cost and accelerate the development process during the early drug discovery and development stages. MethodsThe present study proposed a fuzzy hierarchical optimization framework for identifying potential anticancer targets in genome-scale metabolic models. In this framework used for evaluating treatment side effects, the mortality of treated cancer cells, viability and metabolic deviation of perturbed normal cells were assigned fuzzy objective functions. Fuzzy set theory and the nested hybrid differential evolution (NHDE) algorithm were applied to solve a maximizing decision-making problem transformed from the hierarchical optimization problem. We verified the existence and limitation of this transformation. Significant FindingsThe NHDE algorithm in the optimization framework identified 12 one-target genes with high hierarchical fitness scores by using a small population size. The experimental data accessed from the DepMap database indicated that most of the identified targets can cause most cell lines to die (except EBP, LSS, and NSDHL). Two-target combinations resulted in higher cell viability and lower side effects than did one-target treatments. However, three- and four-target combinations did not significantly improve cell viability and reduce side effects relative to the two-target combinations.

Modeling Visual Containment for Web Page Layout Optimization

AbstractWeb pages have become fundamental in conveying information for companies and individuals, yet designing web page layouts remains a challenging task for inexperienced individuals despite web builders and templates. Visual containment, in which elements are grouped together and placed inside container elements, is an efficient design strategy for organizing elements in a limited display, and is widely implemented in most web page designs. Yet, visual containment has not been explicitly addressed in the research on generating layouts from scratch, which may be due to the lack of hierarchical structure. In this work, we represent such visual containment as a layout tree, and formulate the layout design task as a hierarchical optimization problem. We first estimate the layout tree from a given a set of elements, which is then used to compute tree‐aware energies corresponding to various desirable design properties such as alignment or spacing. Using an optimization approach also allows our method to naturally incorporate user intentions and create an interactive web design application. We obtain a dataset of diverse and popular real‐world web designs to optimize and evaluate various aspects of our method. Experimental results show that our method generates better quality layouts compared to the baseline method.

RETRACTED ARTICLE: Cooperative Virtual Connectivity Control in Uplink Small Cell Network: Towards Optimal Resource Allocation

Nowadays, the application of soft computational techniques in power systems has been increasing with good acceptance under the machine learning umbrella. In this paper, first we presented a deep reinforcement learning approach to find the optimal configuration for HetNet systems. We used a huge quantity of radial configurations from a test system for training purposes. We also studied the joint carrier/power allocation problem in a multi-layer hierarchical networks to achieve the optimal power efficiency in addition to ensure the quality-of-experience for all subscribers. The proposed approach utilizes an adaptive Load Balancing model that aims to obtain “almost optimal” fairness among all servers from the key performance indicator viewpoints. Contrary to current model-based energy efficiency methods, we proposed a Joint Resource Allocation, Energy Efficiency and Flow Control algorithm to solve conventional non-convex and hierarchical optimization problems. Also, referred to SLA-based continuous resource allocation, we generalized the proposed algorithm to Joint Flow/Power Control, and Throughput Optimization algorithm to achieve the optimal energy efficiency beside of guaranteeing user throughput constraints. The simulation results exhibit that the proposed flow-controlled power optimization approach using the capability of network topology adjustment, is able to considerably enhance the energy efficiency compared to the conventional schemes.