Set Of Constraints Research Articles

DC-DistADMM: ADMM Algorithm for Constrained Optimization Over Directed Graphs

This article reports an algorithm for multi-agent distributed optimization problems with a common decision variable, local linear equality and inequality constraints and set constraints with convergence rate guarantees. \textcolor{black}{The algorithm accrues all the benefits of the Alternating Direction Method of Multipliers (ADMM) approach}. It also overcomes the limitations of existing methods on convex optimization problems with linear inequality, equality and set constraints by allowing directed communication topologies. Moreover, the algorithm can be synthesized distributively. The developed algorithm has: (i) a $O(1/k)$ rate of convergence, where $k$ is the iteration counter, when individual functions are convex but not-necessarily differentiable, and (ii) a geometric rate of convergence to any arbitrary small neighborhood of the optimal solution, when the objective functions are smooth and restricted strongly convex at the optimal solution. The efficacy of the algorithm is evaluated by a comparison with state-of-the-art constrained optimization algorithms in solving a constrained distributed $\ell_1$-regularized logistic regression problem, and unconstrained optimization algorithms in solving a $\ell_1$-regularized Huber loss minimization problem. Additionally, a comparison of the algorithm's performance with other algorithms in the literature that utilize multiple communication steps is provided.

Distributed Optimization for Second-Order Discrete-Time Multiagent Systems With Set Constraints.

The optimization problem of second-order discrete-time multiagent systems with set constraints is studied in this article. In particular, the involved agents cooperatively search an optimal solution of a global objective function summed by multiple local ones within the intersection of multiple constrained sets. We also consider that each pair of local objective function and constrained set is exclusively accessible to the respective agent, and each agent just interacts with its local neighbors. By borrowing from the consensus idea, a projection-based distributed optimization algorithm resorting to an auxiliary dynamics is first proposed without interacting the gradient information of local objective functions. Next, by considering the local objective functions being strongly convex, selection criteria of step size and algorithm parameter are built such that the unique solution to the concerned optimization problem is obtained. Moreover, by fixing a unit step size, it is also shown that the optimization result can be relaxed to the case with just convex local objective functions given a properly chosen algorithm parameter. Finally, practical and numerical examples are taken to verify the proposed optimization results.

A QUBO Formulation for Minimum Loss Network Reconfiguration

We introduce a novel quadratic unconstrained binary optimization (QUBO) formulation for a classical problem in electrical engineering – the optimal reconfiguration of distribution grids. For a given graph representing the grid infrastructure and known nodal loads, the problem consists in finding the spanning tree that minimizes the total link ohmic losses. A set of constraints is initially defined to impose topologically valid solutions. These constraints are then converted to a QUBO model as penalty terms. The electrical losses terms are formulated as QUBO, taking advantage of the relationship between several model variables in order to reduce the overall losses expression. Finally, these terms are added to the model as the objective function to minimize. In order to achieve the best possible performance of solution searching with classical solvers, with hybrid quantum-classical solvers and with quantum annealers, our QUBO formulation has the goal of being very efficient in terms of variables usage. A standard 33-node test network is used as an illustrative example of our general formulation. Model metrics for this example are presented and discussed. Finally, the optimal solution for this example was obtained and validated through comparison with the optimal solution from an independent method.

Optimal Placement of PV Smart Inverters With Volt-VAr Control in Electric Distribution Systems

The high R/X ratio of typical distribution systems makes the system voltage vulnerable to the active power injection from distributed energy resources (DERs). Moreover, the intermittent and uncertain nature of the DER generation brings new challenges to the voltage control. This article proposes a two-stage stochastic optimization strategy to optimally place the photovoltaic (PV) smart inverters with Volt-VAr capability for distribution systems with high PV penetration to mitigate voltage violation issues. The proposed optimization strategy enables a planning-stage guide for upgrading the existing PV inverters while considering the operation-stage characteristics of the Volt-VAr control. One advantage of this planning strategy is that it utilizes the local control capability of the smart inverter that requires no communication, thus avoiding issues related to communication delays and failures. Another advantage is that the Volt-VAr control characteristic is internally integrated into the optimization model as a set of constraints, making placement decisions more accurate. The objective of the optimization is to minimize the upgrading cost and the number of the smart inverters required while maintaining the voltage profile within the acceptable range. Case studies on an actual 12.47 kV, 9-km-long Arizona utility feeder have been conducted using OpenDSS to validate the effectiveness of the proposed placement strategy in both static and dynamic simulations.

Homogeneous search for helium in the atmosphere of 11 gas giant exoplanets with SPIRou

The metastable helium triplet in the near-infrared (10 833 Å) is among the most important probes of exoplanet atmospheres. It can trace their extended outer layers and constrain mass loss. We used the near-infrared high-resolution spectropolarimeter SPIRou on the CFHT to search for the spectrally resolved helium triplet in the atmospheres of eleven exoplanets, ranging from warm mini-Neptunes to hot Jupiters and orbiting G, K, and M dwarfs. Observations were obtained as part of the SPIRou Legacy Survey and complementary open-time programs. We applied a homogeneous data reduction to all datasets and set constraints on the presence of metastable helium, despite the presence of systematics in the data. We confirm published detections for HAT-P-11 b, HD 189733 b, and WASP-69 b and set upper limits for the other planets. We applied the p–winds open source code to set upper limits on the mass-loss rate for the nondetections and to constrain the thermosphere temperature, mass-loss rate, line-of-sight velocity, and the altitude of the thermosphere for the detections. We confirm that the presence of metastable helium correlates with the stellar mass and the extreme-ultraviolet flux received by the planets. We investigated the correlation between the mass-loss rate and the presence of metastable helium, but it remains difficult to draw definitive conclusions. Finally, some of our results are in contradiction with previous results in the literature, and therefore we stress the importance of repeatable, homogeneous, and larger-scale analyses of the helium triplet to obtain robust statistics, study temporal variability, and better understand how the helium triplet can be used to explore the evolution of exoplanets.

Artificial neural networks for the prediction of mechanical properties of CGNP/PLGA nanocomposites

The mechanical properties of composite-based components have an impact on both their quality and efficacy. As a result, a scientific framework is needed to discover the practical set of progression constraints that will result in exceptional mechanical qualities. The possibilities of an artificial neural network (ANN) for predicting mechanical properties—specifically, the density and hardness of a carboxyl decorated graphene Nanocomposites (CGNP)/Poly Lactic-co-Glycolic Acid (PLGA) nanocomposite created under various spark plasma sintering (SPS) operating conditions—are explored in this work. A back-propagation with a 2–10-2 structural design and Levenberg-Marquardt algorithm was advanced to anticipate the automated performance of CGNP/PLGA nanocomposites in expressions of density and hardness parameters. The hypothesized outcomes of the models were compared to the actual experimental value. Density (0.9842) and hardness (0.9737) were found to be near to one; the archetypal accomplished well, with a good correlation coefficient (R) for both outputs and a low root-mean-squared error. The anticipated data findings were found to be quite compatible with the values acquired from the investigational assessment results. Our findings demonstrate the utility of a well-trained ANN method in assessing the density and hardness properties of SPSed CGNP/PLGA nanocomposites. As a result, the ANN system provides an unswerving tool for making decisions that can reduce the high costs involved with the experimental characterization of newly manufactured polymer composites.

Sequential Load Restoration With Soft Open Points and Time-Dependent Cold Load Pickup for Resilient Distribution Systems

Global changes in climate conditions and increasing number of weather-related disasters in the world have elevated the quest for better load restorations in power distribution systems. This paper proposes a distribution system restoration (DSR) model where non-exponential time-dependent cold load pickup (CLPU) characteristics and soft open point operations are simultaneously addressed. We develop a time-dependent CLPU model for attaining a more realistic and secure DSR in which we demonstrate that the traditional delayed exponential model (DEM) for a long outage duration is a special case of the adopted CLPU model. The current soft open point (SOP) models would typically relax the operation constraints without devising any tightening procedures, which might lead to inaccurate or even infeasible SOP states. In this paper, we analyze the reasons traditional SOP constraints would lead to unsecure operations and present a set of novel SOP operation constraints for recovering bindings. Also, we adopt the sequential second order cone programming (SSOCP) algorithm to establish the sequential framework of DSR and solve the proposed model effectively. Our presented case studies discuss the rationality of the proposed DSR model and algorithm.

Safe and Stable Secondary Voltage Control of Microgrids Based on Explicit Neural Networks

This paper proposes a novel safety-critical secondary voltage control method based on explicit neural networks (NNs) for islanded microgrids (MGs) that can guarantee any state inside the desired safety bound even during the transient. Firstly, an integrator is introduced in the feedback loop to fully eliminate the steady-state error caused by primary control. Then, considering the impact of secondary control on the stability of the whole system, a set of transient stability and safety constraints is developed. In order to achieve online implementation that requires fast computation, an explicit NN-based secondary voltage controller is designed to cast the time-consuming constrained optimization in the offline NN training phase, by leveraging the local Lipschitzness of activation functions. Specially, instead of using the NN as a black box, the explicit representation of NN is substituted into the closed-loop MG for transferring the stability and safety constraints. Finally, the NN is trained by safe imitation learning, where an optimization problem is formulated by maximizing the imitation accuracy and volume of the stable region while satisfying the stability and safety constraints. Thus, the safe and stable region is approximated that any trajectory initiates within will converge to the equilibrium while bounded by safety conditions. The effectiveness of the proposed method is verified on a prototype MG with detailed dynamics.

Efficacious Models of Discrete Control Devices for Convex Optimization in Distribution Systems

Distribution system optimizations often lead to NP-hard programming problems. Although convex relaxations have proved effective for treating most of these problems, solution time increases drastically when discrete control devices (DCDs), e.g., tap-changers, and capacitor banks, are involved. The models of DCDs are categorized in two categories. The models of the first category rely on auxiliary binary variables to achieve a set of convex constraints. The first proposed model in this paper expedites the models of the first category by upgrading the coding of variables. The second category includes the models that elude the computational burden of handling the auxiliary binary variables by relaxing the equations of DCDs and restricting the relaxation region through a sequential bound tightening. Though this approach expedites solving large problems, the risk of encountering a suboptimal/infeasible solution is considerable. The second propounded model resolves such issues while being kept computationally competitive, by designing a parameter-free stopping criterion, a systematic mechanism to reduce the step-size in bound tightening, and an error-cognizant convex model selection function. A wide range of studies demonstrates that the proposed models resolve the issues of those available. The paper concludes with broad recommendations on the suitability of these models.

Condition-based Design of Variable Impedance Controllers from User Demonstrations

This paper presents an approach to ensure conditions on Variable Impedance Controllers through the off-line tuning of the parameters involved in its description. In particular, we prove its application to term modulations defined by a Learning from Demonstration technique. This is performed through the assessment of conditions regarding safety and performance, which encompass heuristics and constraints in the form of Linear Matrix Inequalities. Latter ones allow to define a convex optimisation problem to analyse their fulfilment, and require a polytopic description of the VIC, in this case, obtained from its formulation as a discrete-time Linear Parameter Varying system. With respect to the current state-of-art, this approach only limits the term definition obtained by the Learning from Demonstration technique to be continuous and function of exogenous signals, i.e. external variables to the robot. Therefore, using a solution-search method, the most suitable set of parameters according to assessment criteria can be obtained. Using a 7-DoF KinovaGen3 manipulator, validation and comparison against solutions with relaxed conditions are performed. The method is applied to generate Variable Impedance Controllers for a pulley belt looping task, inspired by the Assembly Challenge for Industrial Robotics in World Robot Summit 2018, to reduce the exerted force with respect to a standard (constant) Impedance Controller. These controllers fulfil a set of safety constraints, namely stability, bounds on task variables and maximum response overshooting; and their performance is determined by the User Preference heuristic, which allows to intuitively define the desired compliant behaviour along the task. In the context of the task, this is used to generate new controllers for one-off modifications of the nominal belt looping task setup without new demonstrations.

Data-Driven Tabulation for Chemistry Integration Using Recurrent Neural Networks.

Due to the wide range of time scales involved in the ordinary differential equations (ODEs) describing chemical reaction kinetics, multidimensional numerical simulation of chemical reactive flows using detailed combustion mechanisms is computationally expensive. To confront this issue, this article presents an economic data-driven tabulation algorithm for fast combustion chemistry integration. It uses the recurrent neural networks (RNNs) to construct the tabulation from a series of current and past states to the next state, which takes full advantage of RNN in handling long-term dependencies of time series data. The training data are first generated from direct numerical integrations to form an initial state space, which is divided into several subregions by the K-means algorithm. The centroid of each cluster is also determined at the same time. Next, an Elman RNN is constructed in each of these subregions to approximate the expensive direct integration, in which the integration routine obtained from the centroid is regarded as the basis for a storing and retrieving solution to ODEs. Finally, the alpha-shape metrics with principal component analysis (PCA) are used to generate a set of reduced-order geometric constraints that characterize the applicable range of these RNN approximations. For online implementation, geometric constraints are frequently verified to determine which RNN network to be used to approximate the integration routine. The advantage of the proposed algorithm is to use a set of RNNs to replace the expensive direct integration, which allows to reduce both the memory consumption and computational cost. Numerical simulations of a H₂/CO-air combustion process are performed to demonstrate the effectiveness of the proposed algorithm compared to the existing ODE solver.

Event-Triggered Distributed Stochastic Mirror Descent for Convex Optimization.

This article is concerned with the distributed convex constrained optimization over a time-varying multiagent network in the non-Euclidean sense, where the bandwidth limitation of the network is considered. To save the network resources so as to reduce the communication costs, we apply an event-triggered strategy (ETS) in the information interaction of all the agents over the network. Then, an event-triggered distributed stochastic mirror descent (ET-DSMD) algorithm, which utilizes the Bregman divergence as the distance-measuring function, is presented to investigate the multiagent optimization problem subject to a convex constraint set. Moreover, we also analyze the convergence of the developed ET-DSMD algorithm. An upper bound for the convergence result of each agent is established, which is dependent on the trigger threshold. It shows that a sublinear upper bound can be guaranteed if the trigger threshold converges to zero as time goes to infinity. Finally, a distributed logistic regression example is provided to prove the feasibility of the developed ET-DSMD algorithm.

Modified Gradient Projection Neural Network for Multiset Constrained Optimization

To solve nonlinear optimization problems under multiple set constraints, a modified gradient projection neural network (MGPNN) is proposed and investigated. Different from existing approaches specialized for linear constrained optimizations, such as the gradient-based recurrent neural network or dynamic-parameter zeroing neural network, the MGPNN is intrinsically designed from the perspective of the multiple set constrained optimization (MSCO), which is a more generalized form for the linear constrained optimization. The MGPNN is able to efficiently and conveniently provide a feasible solution to the MSCO problem. Ultimately, compared with existing solution methods, numerical simulations and applications to the control of an underactuated portal crane system are provided for verifications of the robust stability and preponderance of the proposed MGPNN model.

Current Fault Tolerance Control Strategy for 3-Phase Switched Reluctance Motor Combined With Position Signal Reconstruction

In this paper, a fault-tolerant control method for switched reluctance motors is proposed for the faults of power converters that occur during the operation of switched reluctance motors. The method combines position reconstruction strategies based on fault tolerant control. The fault detection scheme can accurately determine the faults component and faults type, and then and then maintain the motor drive through the current reconstruction. The position reconstruction scheme improves the traditional inductance detection method, which can improve the situation that the position signal fault leads to the current reconstruction fault. In addition, this paper sets constraints on the reconstruction of position signals to enhance the stability and robustness of position signals. Finally, the stability of the proposed method is verified by experiments.

Variations in the intermediate wind region of the blue supergiant 55 Cygni

Context. The quantitative near-infrared (NIR) spectroscopic synthesis is an important technique for determining wind properties of massive stars. The Brα line is an excellent mass-loss tracer and provides valuable information on the physical conditions of intermediate-wind regions. The knowledge of the wind properties gained by studying the NIR lines could provide extra ingredients to the theory of line-driven winds, mainly because the standard theory does not predict observed properties of blue supergiants, such as high values for the β parameter (β> 2), low terminal velocities, and mass-loss variability. Aims. We seek to enhance our understanding of the wind properties of B supergiants. To this end, we propose analysing their NIR spectra over different epochs to study wind variability and its connection with phenomena arising from regions close to the photosphere. Methods. We present the first sets of multi-epoch high-resolution K- and L-band spectra of 55 Cyg acquired with the Gemini Near-InfraRed Spectrograph (GNIRS). We measured line equivalent widths and modelled the Brα line to derive (unclumped) mass-loss rates. Synthetic line profiles were computed for a homogeneous spherical wind by solving the radiative transfer equations in the co-moving frame for a multi-level atom in non-local thermodynamic equilibrium (NLTE). Results. We observe variations in the spectral lines originating in the upper photosphere and the wind. The perturbations, on average, have periods of ~13 and ~23 days; the latter is similar to that found previously from optical data (22.5 days). The NIR lines observed in 2013 are described with the same wind structure used to model a quasi-simultaneous observation in Hα. By contrast, from observations taken in 2015, we derived a higher mean mass-loss rate. Variations in the mass-loss rate are also detected within a few weeks. Interestingly, we find that the profile shape of the Hu14 line sets constraints on the mass loss. Moreover, we find the Mg II doublet in emission, which suggests a tenuous circumstellar gas ring or shell. Conclusions. The variability detected in the NIR H emission lines of 55 Cyg is related to changes in the mass-loss rate, which doubled its value between 2013 and 2015. Furthermore, the short-term variability (within three weeks) in the spectral lines and mass loss supports the hypothesis of strange-mode oscillations. This pilot project demonstrates the importance of comprehensive monitoring of blue supergiants’ variability to deeply understand the physical properties of their stellar winds and the role of pulsations in recurrently enhancing mass loss.

Bi-level programming and multi-objective optimization for the distribution of resources in hierarchical organizations

The decision regarding the hierarchical distribution of resources is crucial for many organizations that want to allocate such resources efficiently. At the upper level of the hierarchy, a decision-maker may have an objective and a set of feasible solutions partially determined by the lower level. Nevertheless, the upper level can influence but not control the decision-maker at the lower level. Moreover, the objectives of the upper and lower levels are conflicting. Hence, the hierarchical distribution of resources can be formulated as a bi-level programming model. This paper reformulates the hierarchical allocation of resources. The Hooke-Jeeves (HJ) algorithm and the hybrid scatter search–Nelder-Mead (SSNM) method are proposed to solve the newly formulated problem. The problem is also analyzed from a multi-objective perspective to equally prioritize the upper and lower levels while subjecting each to an interdependent set of constraints; a multi-objective scatter search algorithm (MSSA) was developed for that purpose. Tailor-made instances were also designed for the implementation of the developed algorithms. The findings demonstrated that the HJ algorithm provides the best solution according to the upper-level objective function. However, it does not guarantee the best result for the lower level. In comparison, MSSA has a set of best possible solutions for both objectives. The hybrid SSNM method could not match the results provided by the former methods.

A fair consensus adjustment mechanism for large-scale group decision making in term of Gini coefficient

With social media and e-democracy development, decision-making environments and problems have become increasingly complex. Traditional group decision-making, including a small number of decision makers, cannot deal with many practical decision-making problems. Large-scale group decision-making has received extensive attention as an efficient measure to address this issue. This paper proposes a two-stage optimization consensus mechanism for large-scale group decision-making by the Gini coefficient, which considers both the consensus among decision makers’ opinions and the fairness of consensus adjustment. First, we introduce an improved density peak clustering method to cluster decision makers into subgroups that can avoid different clustering centers being too close. Then, we define the subgroups' weights by the silhouette coefficient, the subgroup judgment distance, and their sizes. Respecting the importance of minority opinions, we design a new method to identify and adjust the weights of the minority opinions, which considers more aspects than previous ones. Further, we use the Gini coefficient to measure the equity of consensus adjustment of the group and subgroup. Moreover, we present a two-stage Gini coefficient-based consensus mechanism to obtain the adjusted consensus individual decision matrices, which can ensure the minimization and fairness of consensus adjustment and allocation under the set constraints. Based on these results, we give a new large-scale group decision-making method. Notably, it is the first method that fully considers the opinions of minor subgroups and the fairness of allocation results. These characteristics are essential in decision-making problems such as resource allocation and urban public construction. Finally, we show the utility and validity of the new method through a case study and make the comparative analysis from numerical and principle aspects.

Soft Semi-Supervised Deep Learning-Based Clustering

Semi-supervised clustering typically relies on both labeled and unlabeled data to guide the learning process towards the optimal data partition and to prevent falling into local minima. However, researchers’ efforts made to improve existing semi-supervised clustering approaches are relatively scarce compared to the contributions made to enhance the state-of-the-art fully unsupervised clustering approaches. In this paper, we propose a novel semi-supervised deep clustering approach, named Soft Constrained Deep Clustering (SC-DEC), that aims to address the limitations exhibited by existing semi-supervised clustering approaches. Specifically, the proposed approach leverages a deep neural network architecture and generates fuzzy membership degrees that better reflect the true partition of the data. In particular, the proposed approach uses side-information and formulates it as a set of soft pairwise constraints to supervise the machine learning process. This supervision information is expressed using rather relaxed constraints named “should-link” constraints. Such constraints determine whether the pairs of data instances should be assigned to the same or different cluster(s). In fact, the clustering task was formulated as an optimization problem via the minimization of a novel objective function. Moreover, the proposed approach’s performance was assessed via extensive experiments using benchmark datasets. Furthermore, the proposed approach was compared to relevant state-of-the-art clustering algorithms, and the obtained results demonstrate the impact of using minimal previous knowledge about the data in improving the overall clustering performance.

Generation of Achievable Three-Dimensional Trajectories for Autonomous Wheeled Vehicles via Tracking Differentiators

Planning an achievable trajectory for a mobile robot usually consists of two steps: (i) finding a path in the form of a sequence of discrete waypoints and (ii) transforming this sequence into a continuous and smooth curve. To solve the second problem, this paper proposes algorithms for automatic dynamic smoothing of the primary path using a tracking differentiator with sigmoid corrective actions. Algorithms for setting the gains of the differentiator are developed, considering a set of design constraints on velocity, acceleration, and jerk for various mobile robots. When tracking a non-smooth primary path, the output variables of the differentiator generate smooth trajectories implemented by a mechanical plant. It is shown that the tracking differentiator with a different number of blocks also generates derivatives of the smoothed trajectory of any required order, taking into account the given constraints. Unlike standard analytical methods of polynomial smoothing, the proposed algorithm has a low computational load. It is easily implemented in real time on the on-board computer. In addition, simple methods for modeling a safety corridor are proposed, taking into account the dimensions of the vehicle when planning a polygon with stationary obstacles. Confirming results of numerical simulation of the developed algorithms are presented.

Constrained Density Peak Clustering

Clustering is a commonly used tool for discovering knowledge in data mining. Density peak clustering (DPC) has recently gained attention for its ability to detect clusters with various shapes and noise, using just one parameter. DPC has shown advantages over other methods, such as DBSCAN and K-means, but it struggles with datasets that have both high and low-density clusters. To overcome this limitation, the paper introduces a new semi-supervised DPC method that improves clustering results with a small set of constraints expressed as must-link and cannot-link. The proposed method combines constraints and a k-nearest neighbor graph to filter out peaks and find the center for each cluster. Constraints are also used to support label assignment during the clustering procedure. The efficacy of this method is demonstrated through experiments on well-known data sets from UCI and benchmarked against contemporary semi-supervised clustering techniques.