Proposed Decomposition Algorithm Research Articles

Enhanced Dynamic Surface EMG Decomposition Using the Non-Negative Matrix Factorization and Three-Dimensional Motor Unit Localization.

Surface electromyography (sEMG) signal decomposition is of great importance in examining neuromuscular diseases and neuromuscular research, especially dynamic sEMG decomposition is even more technically challenging. A novel two-step sEMG decomposition approach was developed. The linear minimum mean square error estimation was first employed to extract estimated firing trains (EFTs) from the eigenvector matrices constructed using the non-negative matrix factorization (NMF). The firing instants of each EFT were then classified into motor units (MUs) according to their specific three-dimensional (3D) space position. The performance of the proposed approach was evaluated using simulated and experimentally recorded sEMG. The simulation results demonstrated that the proposed approach can reconstruct MUAPTs with true positive rates of 89.12 ± 2.71%, 94.34 ± 1.85% and 95.45 ± 2.11% at signal-to-noise ratios of 10, 20, and 30 dB, respectively. The experimental results also demonstrated a high decomposition accuracy of 90.13 ± 1.31% in the two-source evaluation, and a high accuracy of 91.86 ± 1.14% in decompose-synthesize-decompose- compare evaluation. The adoption of NMF reduces the dimension of random pattern under the restriction of non-negativity, as well as keeps the information unchanged as much as possible. The 3D space information of MUs enhances the classification accuracy by tackling the issue of relative movements between MUs and electrodes during dynamic contractions. The accuracy achieved in this study demonstrates the good performance and reliability of the proposed decomposition algorithm in dynamic surface EMG decomposition. The spatiotemporal information is applied to the dynamic surface EMG decomposition.

Decomposing Two-Stage Stochastic Scheduling Problems on a continuous-time basis via Slot Similarity

Incorporating uncertainty into process scheduling by explicitly considering future forecasts is crucial for achieving optimal and feasible solutions in practice. However, the complexity of two-stage stochastic scheduling formulations increases exponentially with the number of scenarios, making it difficult to solve large-scale problems in real-time using mathematical programming, which is often necessary for production-maintenance scheduling. This is where decomposition strategies on a scenario basis can be advantageous. Recently, the authors developed a method for decomposing two-stage scheduling problems (TSSPs) formulated on a discrete-time basis using the Similarity Index. This paper extends this concept to TSSPs formulated on a continuous-time basis by fuzzifying discrete decisions among slots instead of time periods, and incorporating the progressive-hedging algorithm to manage non-anticipativity in continuous decisions. The proposed algorithm is tested in a case study of a multiproduct plant consisting of a single processing unit from the literature. Results show that the proposed decomposition algorithm solves the problem faster than the monolithic formulation.

Capacity planning with uncertain endogenous technology learning

Optimal capacity expansion requires complex decision-making, often influenced by technology learning, which represents the reduction in expansion cost due to factors such as cumulative installed capacity. However, having perfect foresight over the technology cost reduction is highly unlikely. In this work, we develop a multistage stochastic programming framework to model capacity planning problems with endogenous uncertainty in technology learning. To assess the benefit of the proposed framework over deterministic optimization, we apply a shrinking-horizon approach to compute the value of stochastic solution. Further, a decomposition scheme based on column generation is developed to solve large instances. Results from our computational experiments indicate substantial potential cost savings and the effectiveness of the proposed decomposition algorithm in solving instances with large numbers of scenarios. Lastly, a power capacity planning case study is presented, highlighting the stochastic optimization’s ability to anticipate significantly different expansion and production decisions in low- and high-learning scenarios.

MmWave relay systems with robust hybrid transceiver designs under correlated channel estimation errors

In this paper, we incorporate mmWave systems with relay techniques to propose a hybrid transceiver design approach. The existing researches of hybrid transceiver designs mostly rely on accurate channel state information acquisitions. However, it is hard to avoid estimation errors in practice, which motivates us to propose a robust design based on imperfect channel estimations. We first design the fully digital processors and then decompose them into hybrid ones. An alternating algorithm is proposed to solve the fully digital processors by utilizing the relationship between the maximization of the upper bound of the averaged mutual information and the weighted minimum mean squared error problem. A decomposition algorithm with closed-form solutions is also proposed to obtain hybrid processors with analog-digital-analog structure at the relay terminal. Compared with related works, the proposed decomposition algorithm outperforms existing decomposition methods under imperfect CSI condition and the proposed hybrid design sacrifices certain complexity to achieve obvious performance improvements, which are illustrated by numerical simulations.

A two-stage decision dependent stochastic approach for airline flight network expansion

Airlines need to expand their flight networks with developing new routes and introducing more flights to increase their market share. In this work, we propose a two-stage stochastic mixed integer nonlinear program (MINLP), which expands an existing flight schedule by operating new flights either with existing fleet resources or a leased aircraft while considering the impact of departure time decisions on the probability distribution of random demand. Moreover, our study helps an airline to link a strategic decision of leasing an aircraft to the tactical aircraft assignment decisions by considering fuel efficiency and seat capacity of the aircraft alternatives in response to new passenger demand. However, the large number of scenarios, nonlinear fuel burn function and nonlinearities due to the decision dependent probabilities become main challenges of solving the problem. In order to deal with the computational requirements of a two-stage stochastic MINLP with decision dependent probabilities, we propose strong conic quadratic and McCormick inequalities, and an exact scenario group wise decomposition algorithm along with a new bounding method. In our computational results, we clearly demonstrate the effectiveness of proposed decomposition algorithm and the strength of the reformulations.

Lane-changing decision modelling in congested traffic with a game theory-based decomposition algorithm

This study presents a cellular lane-changing model where the traffic lanes are discretized into cells and formulates the lane-changing process as a multi-player non-zero-sum non-cooperative game in a connected environment where the real-time surrounding traffic data is shared. In a congested traffic scenario where a large quantity of mandatory lane-changing maneuvers are required simultaneously, the size of the game is extended dynamically (i.e., 2–5 players in this case, and it can be extended more if necessary) based on the traffic situation. Discretionary lane-changing maneuvers are also considered in the decision-making process to maximize the use of the capacity of each traffic lane, i.e., distributing the vehicles to each traffic lane as uniform as possible. Moreover, the competing vehicle on the target lane has more flexible actions (i.e., changing to other lanes) except for accelerating and decelerating actions. Thus, its temporary benefit is sometimes sacrificed by taking these actions to pursue the global benefit so that other vehicles could complete mandatory lane-changing maneuvers. It is a known fact that the space complexity expands exponentially as the game size increases, so a novel decomposition algorithm based on game theory is proposed to reduce the complexity and improve computational efficiency. Finally, a rule-based approach, a classic Nash equilibrium approach and the proposed decomposition algorithm are compared by the critical indicators such as the number of lane-changing vehicles, the maximum incoming queues during the process, the mean of computational time per iteration, etc. The performance shows no significant difference in the efficacy of lane-changing maneuvers among these approaches under the uncongested traffic condition. At the same time, the decomposition algorithm is more efficient in computing time than the classic Nash equilibrium approach. As the traffic gets congested, the game theory-based approaches prove more effective in lane-changing behaviours than the rule-based approach. Meanwhile, the decomposition algorithm outperforms the classic Nash equilibrium approach more significantly in terms of computational time.

A two-stage decomposition method for integrated optimization of islanded AC grid operation scheduling and network reconfiguration

A restructuring of electricity systems promises energy resilience, prosperity, and sustainability for communities. In particular, Reconfigurable networks, such as distribution systems or microgrids, as smart electricity systems are more controllable and flexible compared to traditional systems. In this study, we integrate network reconfiguration decisions into a multi-period grid operation scheduling problem to minimize the operational costs while taking into account both technical and inter-temporal constraints for distributed energy resources, physical constraints on alternating current (AC) network, and system reliability constraints. We develop a comprehensive optimization model for the grid operation planning problem by taking into account the operation scheduling and economic dispatch decisions from earlier cycles and grid operational constraints while determining the optimal network topology. Due to the NP-hard nature of the problem, we introduce a two-stage decomposition algorithm where the solution of the operation scheduling problem (without switching variables) feeds the network reconfiguration problem embedding the dispatch problem for each time period. The performance of the proposed two-stage decomposition algorithm is demonstrated on three different available IEEE testbeds against CPLEX and COUENNE solvers for benchmarking purposes. The selected IEEE testbeds have been widely adopted by previous literature for the aforementioned grid operation planning problem due to their capability in reproducibility. The proposed decomposition algorithm outperforms CPLEX in terms of both solution quality and computational times, particularly our algorithm provides 0.5% and 3.17% better solutions in mid-size and large-size networks, respectively.

An extended formulation of moldable task scheduling problem and its application to quay crane assignments

In this paper, we study an extended formulation of moldable task scheduling problem (MTSP) motivated by the assignments of quay cranes to vessels. In container terminals, handling time of a vessel depends on the number of quay cranes assigned to that vessel. This characteristic allows us to model quay crane assignment problem (QCAP) as a variant of MTSP. By considering the modeling requirements of various properties of QCAP, we develop an extended formulation of MTSP with specific task to machine assignments. Even though this formulation brings modeling flexibility, it can only be solved for small instances because of its size. For this reason, we provide a generic solution algorithm based on a logic based Benders decomposition by utilizing the extended formulation. There are various characteristics of QCAP observed in different terminals. Accordingly, we implement the proposed decomposition algorithm for contiguous assignments of QCs, uniform QCs as well as the availability of QCs. Computational experiments show that the proposed algorithm is able to solve instances of considerable sizes to optimality and provides a modeling flexibility that allows implementation to different terminal settings.

A Distributionally Robust AC Network-Constrained Unit Commitment

This paper presents a distributionally robust network-constrained unit commitment (DR-NCUC) model considering AC network modeling and uncertainties of demands and renewable productions. The proposed model characterizes uncertain parameters using a data-driven ambiguity set constructed by training samples. The non-convex AC power flow equations are approximated by convex quadratic and McCormick relaxations. Since the proposed min-max-min DR-NCUC problem cannot be solved directly by available solvers, a new decomposition algorithm with proof of convergence is reported in this paper. The master problem of this algorithm is solved using both primal and dual cuts, while the max-min sub-problem is solved using the primal-dual hybrid gradient method, obviating the need for using duality theory. Also, an active set strategy is proposed to enhance the tractability of the decomposition algorithm by ignoring the subset of inactive constraints. The proposed model is applied to a 6-bus test system and the IEEE 118-bus test system under different conditions. These case studies illustrate the performance of the proposed DR-NCUC model to characterize uncertainties and the superiority of the proposed decomposition algorithm over other decomposition approaches using either primal or dual cuts.

An exact robust approach for the integrated berth allocation and quay crane scheduling problem under uncertain arrival times

We consider an integrated berth allocation and quay crane assignment and scheduling problem where the arrival times of the vessels may be affected by uncertainty. The problem is modelled as a two-stage robust mixed integer program where the berth allocation decisions are taken before the exact arrival times are known, and the crane assignment and scheduling operations are adjusted to the arrival times. To solve the robust two-stage model, we follow a decomposition algorithm that decomposes the problem into a master problem and a separation problem. A new scenario reduction procedure for solving the separation problem is proposed as well as a warm start technique for reducing the number of iterations performed by the decomposition algorithm. To scale the proposed decomposition algorithm for large size instances, it is combined with a rolling horizon heuristic.The efficiency and effectiveness of the proposed algorithms are demonstrated through extensive computational experiments carried out on randomly generated instances with both homogeneous and heterogeneous cranes as well as on instances from the literature.

Core decomposition and maintenance in weighted graph

Coreness is an important index to reflect the cohesiveness of a graph. The problems of core computation in static graphs and core update in dynamic graphs, known as the core decomposition and core maintenance problems respectively, have been extensively studied in previous work. However, most of these work focus on unweighted graphs. Considering that graphs are weighted in a lot of realistic applications, it is indispensable to extend the coreness to weighted graphs and devise efficient algorithms for weighted core decomposition and weighted core maintenance. In this work, we present a new definition of weighted coreness for vertices in a weighted graph, by taking into account the weights of vertices, which makes the coreness in unweighted graph be a special case. We propose efficient algorithms for both weighted core decomposition and weighted core maintenance problems. The coreness of vertices can be computed in linear time by the proposed decomposition algorithm, while the proposed core maintenance algorithm can process multiple-edge insertions/deletions simultaneously, which greatly reduces the core update time. Comprehensive experiments on both realistic networks and temporal graphs exhibit our algorithms are efficient and scalable.

Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems

Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems, see Rockafellar and Wets (Math. Oper. Res. 16:119–147, 1991), Ruszczynski and Shapiro (Stochastic Programming, Handbook in OR & MS, North-Holland Publishing Company, Amsterdam, 2003) and Ruszczynski (Math. Program. 79:333–353, 1997). In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. This is achieved by introducing nonanticipativity constraints for the first stage decision variables, rearranging the minimax problem through Lagrange decomposition and applying the well-known primal-dual hybrid gradient (PDHG) method to the new minimax problem. The algorithmic framework does not depend on specific structure of the ambiguity set. To extend the algorithm to the case that the underlying random variables are continuously distributed, we propose a discretization scheme and quantify the error arising from the discretization in terms of the optimal value and the optimal solutions when the ambiguity set is constructed through generalized prior moment conditions, the Kantorovich ball and $$\phi$$ -divergence centred at an empirical probability distribution. Some preliminary numerical tests show the proposed decomposition algorithm featured with parallel computing performs well.

Cooperative coevolution for large-scale global optimization based on fuzzy decomposition

Cooperative coevolution (CC) is an effective evolutionary divide-and-conquer strategy that solves large-scale global optimization (LSGO) by decomposing the problem into a set of lower-dimensional subproblems. The main challenge of CC is to find an optimal decomposition. Differential Grouping (DG) is a competitive decomposition method to identify the variable interaction with several improved versions like GDG and DG2. Although DG-based decomposition methods have shown superior performance compared to the other decomposition methods, they still have difficulty to deal with the overlapping problems since their optimal decomposition is unknown. To address this issue, instead of pursuing the high accuracy of decomposition, we propose a novel fuzzy decomposition algorithm that groups the variables according to their interaction degree. In the proposed fuzzy decomposition algorithm, the interaction structure matrix and the interactive degree for a LSGO problem are calculated at first according to the interaction among all the decision variables. Then the number of subgroups is determined based on the interactive degree. Based on the interaction structure matrix, a spectral clustering algorithm is proposed to decompose the decision variables with regard to the number of subgroups in order to achieve a better balance between high grouping accuracy and suitable group size. The proposed decomposition algorithm with DECC has been proven to outperform several state-of-the-art algorithms on the latest LSGO benchmark functions.

Transmission Network Investment Using Incentive Regulation: A Disjunctive Programming Approach

A well-planned electric transmission infrastructure is the foundation of a reliable and efficient power system, especially in the presence of large scale renewable generation. However, the current electricity market designs lack incentive mechanisms which can guarantee optimal transmission investments and ensure reliable integration of renewable generation such as wind. This paper first proposes a stochastic bilevel disjunctive program for optimal transmission investment based on the newly proposed theoretical H-R-G-V incentive mechanism. The upper level is a profit-maximization problem of an independent transmission company (Transco), while the lower level is a welfare maximization problem. The revenue of the Transco is bounded by a regulatory constraint set by the regulator in order to induce socially optimal investments. The application of the H-R-G-V mechanism allows the regulator to ensure social maximum transmission investments and helps to reduce transmission congestion and wind power spillage. The transmission investment under the H-R-G-V mechanism is modeled as a stochastic bilevel disjunctive program. To solve the developed mathematical model we first propose a series of linearization and reformulation techniques to recast the original model as a stochastic mixed integer linear problem (MILP). We exploit the disjunctive nature of the reformulated stochastic MILP model and further propose a Bean decomposition algorithm to efficiently solve the stochastic MILP model. The proposed decomposition algorithm is also modified and accelerated to improve the computational performance. The computational performance of our MILP modeling approach and modified and accelerated Bean decomposition algorithm is studied through several examples in detail. The simulation results confirm a promising performance of both the modeling approach and its solution algorithm.

Multistage Stochastic Optimization for Microgrid Operation Under Islanding Uncertainty

Microgrids can be distinguished from traditional power systems based on their ability to perform the islanded operation. This study explores the problem of optimizing the microgrid operation in the context of uncertain islanding events. This occurs when the microgrid is isolated from the main grid and it operates as an independent system. A multistage stochastic optimization model that considers multi-occurrence and multi-period islanding events is proposed to optimize the proactive policy. This is achieved by creating parameters for the maximum number of time periods of the islanded operation in the planning horizon. To solve the resulting large-scale mixed-integer program, an efficient algorithm based on the two-stage Benders' decomposition method is proposed. Numerical experiments show that the proposed policy significantly reduces the expected operation costs of a microgrid. This is more cost-effective in comparison to the other reactive policies that prepare a certain reserve level and reschedule the resources after an islanding event occurs. This study demonstrates that the proposed decomposition algorithm can efficiently solve large-scale problems that have longer planning horizons or a larger number of time periods for the islanded operation.

A decomposition algorithm to solve dynamic city-scale ridesharing problem

This paper proposes a computational efficient, rapid, dynamic ridesharing algorithm. Firstly, a designed pre-processing procedure reduces size of the input date of potential nodes for vehicle to pass through. The proposed decomposition algorithm decomposes the large-scale problem with pre-processing date into independent sub-problems, and the method of parallel computing is adopted to solve the sub-problems. We call the dynamic programming algorithm to deal with the vehicle routing problem involved in sub-problems. Greedy random adaptive search algorithm is compared with the decomposition algorithm to verify the algorithm. Finally, a mathematical example shows that the algorithm can effectively solve large-scale dynamic ridesharing problems.

An Incoherent Decomposition Algorithm Based on Polarimetric Symmetry for Multilook Polarimetric SAR Data

Polarimetric symmetry, including reflection symmetry, rotation symmetry, and azimuthal symmetry, are useful concepts for describing the scattering characteristics of polarimetric synthetic aperture radar (SAR) data. In this study, for the coherency matrices of multilook polarimetric SAR data, three new scattering models were first proposed based on the polarimetric symmetry. Then, a four-component incoherent decomposition algorithm was introduced, which was based on the three aforementioned new models, along with a classic volume scattering model. The proposed algorithm was considered to be a complete incoherent decomposition algorithm. No polarimetric information was lost during the decomposition. Also, there were no negative powers in its output. Then, by utilizing three polarimetric SAR images which had been derived by ESAR, RADARSAT-2, and GF-3, respectively, the experimental results had revealed that the proposed decomposition algorithm was effective in the analysis of the scattering mechanisms of terrain targets. It was observed that more than 90% of the coherency matrices of the real polarimetric SAR data had been completely decomposed by the proposed algorithm into four components. These components were found to be exactly consistent with the scattering models for surface scattering, double-bounce scattering, volume scattering, and helix scattering. The proposed decomposition algorithm had provided insight into the limitations of incoherent decomposition.

Universal Perceptron Theory on Decomposition Structures of n-Bit Boolean Functions

By using the universal perceptron theory, a decomposition algorithm called improved distance and DNA-like decomposition algorithm (ID-DNA-like DA) is proposed to investigate the linear decomposition structures of [Formula: see text]-bit Boolean functions. The specific linear decomposition structure of an [Formula: see text]-bit Boolean function is generated by the proposed decomposition algorithm ([Formula: see text]). Especially, the linear decomposition structure of 4-bit Boolean function is shown to have the following properties: the total number of 4-bit Boolean functions is 65[Formula: see text]536 and the number of linearly separable Boolean functions is 1882; the number of 2-length decomposition Boolean functions is not less than 41[Formula: see text]366; the number of 3-length decomposition Boolean functions does not exceed 22[Formula: see text]286; and the number of 4-length decomposition Boolean functions is only 2. At the same time, it is shown that the decomposition length of [Formula: see text]-bit Boolean function does not exceed [Formula: see text] [Formula: see text]. In addition, a concept of the [Formula: see text]-bit Boolean function decomposition structure figure is proposed. By decomposition calculation, the related results of decomposition structure figures show that there is some symmetry with a strong self-similar fractal feature.

A generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variables

In this paper, we propose a generalized Benders decomposition-based branch and cut algorithm for solving two stage stochastic mixed-integer nonlinear programs (SMINLPs) with mixed binary first and second stage variables. At a high level, the proposed decomposition algorithm performs spatial branch and bound search on the first stage variables. Each node in the branch and bound search is solved with a Benders-like decomposition algorithm where both Lagrangean cuts and Benders cuts are included in the Benders master problem. The Lagrangean cuts are derived from Lagrangean decomposition. The Benders cuts are derived from the Benders subproblems, which are convexified by cutting planes, such as rank-one lift-and-project cuts. We prove that the proposed algorithm converges in the limit. We apply the proposed algorithm to a stochastic pooling problem, a crude selection problem, and a storage design problem. The performance of the proposed algorithm is compared with a Lagrangean decomposition-based branch and bound algorithm and solving the corresponding deterministic equivalent with the solvers including BARON, ANTIGONE, and SCIP.

Emission-Intensity-Based Carbon Tax and Its Impact on Generation Self-Scheduling

We propose an emission-intensity-based carbon-tax policy for the electric-power industry and investigate the impact of the policy on thermal generation self-scheduling in a deregulated electricity market. The carbon-tax policy is designed to take a variable tax rate that increases stepwise with the increase of generation emission intensity. By introducing a step function to express the variable tax rate, we formulate the generation self-scheduling problem under the proposed carbon-tax policy as a mixed integer nonlinear programming model. The objective function is to maximize total generation profits, which are determined by generation revenue and the levied carbon tax over the scheduling horizon. To solve the problem, a decomposition algorithm is developed where the variable tax rate is transformed into a pure integer linear formulation and the resulting problem is decomposed into multiple generation self-scheduling problems with a constant tax rate and emission-intensity constraints. Numerical results demonstrate that the proposed decomposition algorithm can solve the considered problem in a reasonable time and indicate that the proposed carbon-tax policy can enhance the incentive for generation companies to invest in low-carbon generation capacity.