Inequality Constraints Research Articles

Model Predictive Control With Stability Guarantee for Second-Order DC/DC Converters

This paper proposes a continuous-control-set model-predictive control (CCS-MPC) designed explicitly for second-order dc/dc converters such as the boost, buck, buck-boost, and non-inverting buck-boost converters. These converters share a bilinear dynamic structure that allows designing a generalized control. In contradistinction to the conventional finite-control-set approach, the proposed control uses a pulse-width modulation that allows a continuous control set with constant switching frequency. The modulation index's saturation is considered an inequality constraint in the optimization model. The proposed control guarantees stability in the sense of Lyapunov. This property is proven by using the passive structure of these converters. Experimental results on the four converters mentioned above show the superior performance of the proposed control in terms of robustness, offset, computation time in the central processing unit and, of course, stability.

Relevant interval perturbation method for uncertainty analysis of structural–acoustic coupling model

The parameters in structural–acoustic coupling models are often coupled with each other, and the independence of the uncertain parameters is a necessary condition for using the traditional interval perturbation method. This article proposes a relevant interval perturbation method for the uncertainty analysis of a structural–acoustic coupling model with uncertain-but-bounded parameters. Aiming at the linear inequality constraint relationship in the uncertain parameters, this method introduces a sensitivity ranking mechanism for the uncertainty parameters, which overcomes the limitation of uncertain parameter independence in the traditional interval model, and effectively restrains the interval expansion phenomenon of the system response in the uncertainty analysis. Through numerical examples, it is demonstrated that the relevant interval perturbation analysis method can effectively solve a constrained problem when there are uncertain parameters, and the obtained response interval radius is smaller than that of the traditional method under constraint conditions.

Supervised Feature Selection via Collaborative Neurodynamic Optimization.

As a crucial part of machine learning and pattern recognition, feature selection aims at selecting a subset of the most informative features from the set of all available features. In this article, supervised feature selection is at first formulated as a mixed-integer optimization problem with an objective function of weighted feature redundancy and relevancy subject to a cardinality constraint on the number of selected features. It is equivalently reformulated as a bound-constrained mixed-integer optimization problem by augmenting the objective function with a penalty function for realizing the cardinality constraint. With additional bilinear and linear equality constraints for realizing the integrality constraints, it is further reformulated as a bound-constrained biconvex optimization problem with two more penalty terms. Two collaborative neurodynamic optimization (CNO) approaches are proposed for solving the formulated and reformulated feature selection problems. One of the proposed CNO approaches uses a population of discrete-time recurrent neural networks (RNNs), and the other use a pair of continuous-time projection networks operating concurrently on two timescales. Experimental results on 13 benchmark datasets are elaborated to substantiate the superiority of the CNO approaches to several mainstream methods in terms of average classification accuracy with three commonly used classifiers.

A time-fuel hybrid optimal guidance for micro/nano satellite’s glideslope approach

Aimed at the trajectory planning problem of micro/nano satellites’ approaching missions under multiple constraints, this paper proposes a V-bar glideslope maneuver optimization method considering both time and fuel consumption. A dual-objective trajectory optimization model is established based on the CW motion equation, considering the constraints of trajectory and impulse. The optimal solution set of the glideslope point position and maneuvering time is obtained by employing the NSGA2 algorithm. In order to reduce computational complexity, a method for simplifying trajectory optimization is proposed. By solving a constrained dual-impulse optimization problem, the feasible range of distance and transfer time between adjacent glideslope points are obtained, which transforms the constraint equations of the glideslope maneuver optimization into the search space of optimization variables. The problem is simplified to an unconstrained optimization problem. Finally, a simulation is conducted with the approaching mission of a typical micro/nano satellite. The results demonstrate that the proposed method can provide an optimal solution set considering both fuel and time consumption, offering a flexible maneuver planning strategy for on-orbit application.

Orbital constraints and efficiency modeling for calibration satellites in long-range radar applications

Orbit design is a critical aspect of the calibration satellite system, with its effective configuration being integral for successful satellite calibration. This paper addresses the challenge of designing satellite orbits while considering the constraints imposed by satellite-ground equipment performance and launch capabilities. Key factors constraining calibration satellite orbits are analyzed, and orbital constraint equations are developed, focusing on radar performance, calibration time, and regional coverage. The study also introduces a model to assess the efficiency of satellite orbits for calibration purposes, which is rigorously validated through simulation. This study uniquely addresses the integration of radar performance with satellite orbit design, an area not extensively explored in current literature.

Design of information management model based on multiobjective optimization algorithm in intelligent electric financial system

The electric power infrastructure is the cornerstone of contemporary society’s sustenance and advancement. Within the intelligent electric power financial system, substantial inefficiency and waste in information management persist, leading to an escalating depletion of resources. Addressing diverse objectives encompassing economic, environmental, and societal concerns within the power system helps the study to undertake a comprehensive, integrated optimal design and operational scheduling based on a multiobjective optimization algorithm. This article centers on optimizing the power financial system by considering fuel cost, active network loss, and voltage quality as primary objectives. A mathematical model encapsulates these objectives, integrating equations and inequality constraints and subsequently introducing enhancements to the differential evolutionary algorithm. Adaptive variation and dynamic crossover factors within crossover, variation, and selection operations are integrated to optimize algorithm parameters, specifically catering to the multiobjective optimization of the electric power system. An adaptive grid method and cyclic crowding degree ensure population diversity and control the Pareto front distribution. They experimentally validated the approach and the comparisons conducted against AG-MOPSO, INSGA-II, and NSDE algorithms across standard test functions: ZDT1, ZDT2, ZDT3, and DTLZ4. The convergence evaluation indices for this study’s scheme on ZDT1 and ZDT2 are 0.000938 and 0.0034, respectively. Additionally, distribution evaluation indices on ZDT1, ZDT2, ZDT3, and ZDT4 stand at 0.0018, 0.0026, 0.0027, and 0.0009, respectively. These indices indicate a robust convergence and distribution, facilitating the optimization of electric power financial information management and the intelligent handling of the electric power financial system’s information, thereby enhancing the allocation of material and financial resources.

Approaches for fast Brownian dynamics simulation with constraints

Constraints can be used to eliminate quickly fluctuating degrees of freedom in dynamical systems enabling solutions that resolve the behavior of the system with fewer time steps over long time scales. In Brownian dynamics models, these constraints lead to linear equations which represent saddle point problems describing the velocity tangent to the constraint manifold and the forces resulting from imposition of the constraint on the dynamics. Simulations of these constrained systems must solve these saddle point problems, which are often ill-conditioned and require clever preconditioning in order to facilitate large scale simulation using iterative methods like GMRES. In this work, we show that the projected conjugate gradient (PrCG) method, which applies conjugate gradients projected onto the tangent space of the constraint manifold, has certain advantages compared to preconditioned iterative methods like GMRES, without the need to devise and compute a preconditioner. Specifically, for simulations of beads interacting hydrodynamically with commonly encountered holonomic constraints, PrCG exhibits linear scaling with the number of beads similar to GMRES. However, because PrCG does not require a preconditioner it often requires less precomputational effort both in terms of computational complexity and storage, depending on the nature of the constraints. Furthermore, PrCG produces iterates which are always feasible, allowing for independent tuning of the error tolerance on the bead velocities from the error tolerance on the constraint equations. We demonstrate PrCG's efficiency by solving several example low Reynolds number systems involving rigid bodies, freely jointed chains, and immobile particles, as well as a mixed constraint problem with both a rigid body and immobile particles, comparing to GMRES and analytical solutions, when available.

Lie Symmetries of the Wave Equation on the Sphere Using Geometry

A semilinear quadratic equation of the form Aij(x)uij=Bi(x,u)ui+F(x,u) defines a metric Aij; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities Bi(x,u),F(x,u). From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric Aij, while the second set serves as constraint equations, which select elements from the conformal algebra of Aij. Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric Aij. In the present article, the nonlinear Poisson equation Δgu−f(u)=0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ∂t. It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general 1+(n−1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation Δgu−f(u)=0 admits Lie point symmetries only when f(u)=ku, and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom.

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields

Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics. The problem is formulated as a least-squares data fitting minimization problem with two sets of constraints: partial-differential equation (PDE) constraints governing elastic wave propagation and physical model constraints implementing prior information. The alternating direction method of multipliers is used to solve the problem, resulting in an iterative algorithm with well-conditioned subproblems. Although wavefield reconstruction is the most challenging part of the iteration, sparse linear algebra techniques can be used for moderate-sized problems and frequency domain formulations. The Hessian matrix is blocky with diagonal blocks, making model updates fast. Gradient ascent is used to update Lagrange multipliers by summing PDE violations. Various numerical examples are used to investigate algorithmic components, including model parameterizations, physical model constraints, the role of the Hessian matrix in suppressing interparameter cross-talk, computational efficiency with the source sketching method, and the effect of noise and near-surface effects.

Dealing with inequality constraints in large-scale semidefinite relaxations for graph coloring and maximum clique problems

Dealing with inequality constraints in large-scale semidefinite relaxations for graph coloring and maximum clique problems

Potential Energy Optimization Approach in the Form-Finding of Tensegrities and Cable–Strut Systems

A form-finding process for tensegrities and cable–strut systems is determined as a minimum problem of potential energy that is constructed considering initial estimated pre-stress states. Both the Jacobian and Hessian matrices of a potential energy function are derived explicitly regarding free nodal coordinates. Based on these derivations, both the first-order geodesic dynamic relaxation optimization method and the second-order optimization method of Newton can be applied to solve form-finding problems involving tensegrities and cable–strut structures. It should be noted that both the pre-stress state and the configuration of the systems are adjusted during the optimization process to satisfy self-equilibrium conditions. The geometrical constraints are then included in the proposed work as equality constraint conditions to find the final forms satisfying not only the self-equilibrium states but also the designers’ demands. Two well-known and complex systems are investigated to assess the efficiency of the proposed method, i.e. free-standing truncated icosahedral tensegrity and the Kiewitt Dome.

Gluing Small Black Holes into Initial Data Sets

We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document} spatial dimensions. We glue an ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-rescaling of an asymptotically flat data set (γ^,k^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{\\gamma }},{\\hat{k}})$$\\end{document} into the neighborhood of a point p∈X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}\\in X$$\\end{document} inside of another initial data set (X,γ,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\\gamma ,k)$$\\end{document}, under a local genericity condition (non-existence of KIDs) near p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document}. As the scaling parameter ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} tends to 0, the rescalings xϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{x}{\\epsilon }$$\\end{document} of normal coordinates x on X around p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document} become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document} on the other hand, the glued initial data converge back to (γ,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma ,k)$$\\end{document}. The initial data we construct enjoy polyhomogeneous regularity jointly in ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets (X,γ,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\\gamma ,k)$$\\end{document} and appropriate boosted Kerr initial data sets (γ^,k^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\hat{\\gamma }},{\\hat{k}})$$\\end{document} produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} black hole. The proof combines a variant of the gluing method introduced by Corvino and Schoen with geometric singular analysis techniques originating in Melrose’s work. On a technical level, we present a fully geometric microlocal treatment of the solvability theory for the linearized constraints map.

Novel solution strategies for multiparametric nonlinear optimization problems with convex objective function and linear constraints

AbstractThis paper expands the multiparametric quadratic programming (mp-QP) framework presented in Narciso et al. (Comput Chem Eng 164:107882, 2022. https://doi.org/10.1016/j.compchemeng.2022.107882) to the more general multiparametric nonlinear programming (mp-NLP) case. First, the vector of parameters in mp-NLP problems is recast so that a unique transformed parameter is implicitly assigned to each of the inequality constraints. Maps of critical regions in this transformed space of parameters feature a set of 1-dimensional parametric edges (two per inequality constraint), which then greatly facilitate solution calculation. In the mp-NLP case, however, parametric edges define nonlinear semi-infinite lines; this requires an adaptation to the mp-QP algorithm (deals with linear parametric edges only), to enable a suitable calculation path to the more general nonlinear case. Three routes are proposed to mp-NLPs: the first route delivers solutions in compact form (same format as in mp-QP) using a single reference point per edge; the second route delivers explicit solutions using a hybrid approach for critical region construction, where all active sets not detected in the parameters space are excluded from the solution (equivalent to first route concerning accuracy); the third route builds on the initial explicit solution and further partitions the parameters space until all solution fragments satisfy an error check. Five algorithms were coded for these routes, and tested in a large range of mp-NLP problems. These strategies enable significant improvements in terms of solution accuracy, algorithm efficiency, and interpretability when compared to the state-of-the-art mp-NLP algorithms.

Integral Sliding Mode Disturbance Observer-Based Preview Repetitive Control

Abstract This paper deals with the problem of integral sliding mode disturbance observer (ISDOB)-based preview repetitive control (PRC) for a class of nonlinear systems subject to external disturbances and nonlinearity. First, an ISDOB-based PRC law is presented to estimate and compensate the exogenous disturbances and nonlinearity. Next, by using a new equality constraint, a continuous-discrete two-dimensional (2D) system is established. Then, the stability of the combined controller–observer system is derived on the basis of a Lyapunov analysis, 2D system theory, and the singular-value-decomposition technique. By solving an linear matrix inequality (LMI), the gains of the controller and state observer can be obtained. Finally, two numerical examples are provided to illustrate the effectiveness and superiority of the proposed method.

Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares.

Combining multiple devices for localization has important applications in the military field. This paper exploits the land-based short-wave platforms and satellites for fusion localization. The ionospheric reflection height error and satellite position errors have a great impact on the short-wave localization and satellite localization accuracy, respectively. In this paper, an iterative constrained weighted least squares (ICWLS) algorithm is proposed for these two kinds of errors. The algorithm converts the nonconvex equation constraints to linear constraints using the results of the previous iteration, thus ensuring convergence to the globally optimal solution. Simulation results show that the localization accuracy of the algorithm can reach the corresponding Constrained Cramér-Rao Lower Bound (CCRLB). Finally, the localization results of the two methods are fused using Kalman filtering. Simulations show that the fused localization accuracy is improved compared to the single-means localization.

Distributed optimization for discrete time‐varying linear multi‐agent systems with event‐triggered communication

AbstractThis paper studies the distributed optimization problem (DOP) of discrete time‐varying linear multi‐agent systems (MASs), in which the global objective function is formed by a sum of local convex objective functions. Firstly, a DOP with discrete time‐varying MASs is considered, in which the time‐varying linear matrix satisfies a certain equality constraint. To solve this problem, a novel discrete‐time distributed optimization algorithm (DOA) with event‐triggered communication mechanism (ETCM) is proposed. Secondly, by constructing the error dynamical system and using a series of inequality techniques, some sufficient conditions for achieving consensus and obtaining the optimal solution are established. It is found that the considered MAS has generality and the proposed DOA has the advantage of reducing communication burden. Finally, a numerical simulation is presented to verify the validity of theoretical results.

A strategy based on integer programming for optimal dosing and timing of preventive hypoglycemic treatments in type 1 diabetes management

Background and objectivesOne of the major problems related to type 1 diabetes (T1D) management is hypoglycemia, a condition characterized by low blood glucose levels and responsible for reduced quality of life and increased mortality. Fast-acting carbohydrates, also known as hypoglycemic treatments (HT), can counteract this event. In the literature, dosage and timing of HT are usually based on heuristic rules. In the present work, we propose an algorithm for mitigating hypoglycemia by suggesting preventive HT consumption, with dosages and timing determined by solving an optimization problem. MethodsBy leveraging integer programming and linear inequality constraints, the algorithm can bind the amount of suggested carbohydrates to standardized quantities (i.e., those available in “off-the-shelf” HT) and the minimal distance between consecutive suggestions (to reduce the nuisance for patients). ResultsThe proposed method was tested in silico and compared with competitor algorithms using the UVa/Padova T1D simulator. At the cost of a slight increase of HT consumed per day, the proposed algorithm produces the lowest median and interquartile range of the time spent in hypoglycemia, with a statistically significant improvement over most competitor algorithms. Also, the average number of hypoglycemic events per day is reduced to 0 in median. ConclusionsThanks to its positive performances and reduced computational burden, the proposed algorithm could be a candidate tool for integration in a DSS aimed at improving T1D management.

Formulating and heuristic solving of contact problems in hybrid data-driven computational mechanics

In this work we consider the hybrid Data-Driven Computational Mechanics (DDCM) approach, in which a smooth constitutive manifold is reconstructed to obtain a well-behaved nonlinear optimization problem (NLP) rather than the much harder discrete-continuous NLP (DCNLP) of the direct DDCM approach. The key focus is on the addition of geometric inequality constraints to the hybrid DDCM formulation. Therein, the required constraint force leads to a contact problem in the form of a mathematical program with complementarity constraints (MPCC), a problem class that is still less complex than the DCNLP. For this MPCC we propose a heuristic quick-shot solution approach, which can produce verifiable solutions by solving up to four NLPs. We perform various numerical experiments on three different contact problems of increasing difficulty to demonstrate the potential and limitations of this approach.

On Classifying Continuous Constraint Satisfaction Problems

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical conditions. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical conditions) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

A collective neurodynamic approach to distributed resource allocation with event-triggered communication

To solve a distributed optimal resource allocation problem, a collective neurodynamic approach based on recurrent neural networks (RNNs) is proposed in this paper. Multiple RNNs cooperatively solve a global constrained optimization problem in which the objective function is a total of local non-smooth convex functions and is subject to local convex sets and a global equality constraint. Different from the projection dynamics to deal with local convex sets in the existing work, an internal dynamics with projection output is designed in the algorithm to relax the Slater’s condition satisfied by the optimal solution. To overcome continuous-time communication in a group of RNNs, an aperiodic communication scheme, called the event-triggered scheme, is presented to alleviate communication burden. It is analyzed that the convergence of the designed collective neurodynamic approach based on the event-triggered communication does not rely on global information. Furthermore, it is proved the freeness of the Zeno behavior in the event-triggered scheme. Two examples are presented to illustrate the obtained results