Karush-Kuhn-Tucker Research Articles

Reflux Power Optimization of a Dual-Active Hybrid Full-Bridge Converter Based on Active Disturbance Rejection Control

The dual-active hybrid full-bridge (H-FDAB) DC–DC converter has great potential in medium-voltage high-power photovoltaic power station applications by introducing a three-level bridge arm to increase the output voltage range. However, its mathematical model and optimum modulation schemes have not been fully explored. Under the traditional PI control, the H-FDAB DC–DC converter will produce significant reflux power, which will lead to a decrease in converter efficiency and output voltage fluctuation. On this basis, this paper proposes a reflux power optimization strategy for an H-FDAB DC-DC converter based on active disturbance rejection control (ADRC). Firstly, the structure and power characteristics of the H-FDAB DC–DC converter are analyzed, and the relationship among the reflux power, the transmission power, and the phase shift angle is derived. Secondly, to reduce the complexity of the control calculation, upon the foundation of dual phase-shifting modulation, the Karush–Kuhn–Tucker (KKT) condition is used to solve for the phase shift angle that corresponds to the minimum reflux power. Simultaneously, we develop an ADRC loop utilizing an extended state observer (ESO) for the real-time estimation of system states. We also consider the sudden changes in input voltage, load switching, and transmission power fluctuations caused by reflux power optimization strategies as system disturbances and compensate for them accordingly. Finally, the experiments conclusively validate the designed control strategy’s correctness and feasibility.

Fuzzy clustering with capacity constraints: Algorithm, convergence analysis and numerical experiments

In this paper we study the fuzzy clustering problem with capacity constraints. Despite of the fact that the fuzzy clustering approach is widely encountered in the literature, the inclusion of capacity constraints is recent and has several practical applications. We propose a general formulation of the clustering problem, where each point has an associated weight and the sum of the weights of the points that compose each group is established a priori. We discuss existence of solutions of the involved problems, providing a mathematical foundation for the established formulas. Besides, we propose a practical algorithm for solving this problem and present its convergence analysis. This algorithm follows an alternate minimization scheme, wherein a given iteration addresses the problem first in terms of the probabilities of each point xj belonging to each cluster i, denoted as uij, finding subsequently the position of the centroids, ci,i=1,…,g,j=1,…,n. This procedure is K-means-like, with the distinction that, as a point does not exclusively belong to a group, the computation of uij requires optimization techniques. In our case, this involves solving a linear system derived from the Karush–Kuhn–Tucker (KKT) conditions. With the aim of validating our algorithm, we present numerical tests with synthetic and real-world data to demonstrate its performance for the given problems. Since the proposal successfully solved these numerical tests within a reasonable computational time, it can be considered a valuable resource for addressing real-world applications.

A robust bi‐level programming approach for optimal planning an off‐grid zero‐energy complex

AbstractElectric power provision for all the customers is not always possible for distribution companies. Some customers are interested in serving their load through renewable resources regarding the climate situation. Electrically off‐grid zero‐energy building is an applicable concept, in which the electrical energy provision of the buildings is isolated from the power supply infrastructures. This paper introduces a robust bi‐level programming model to create a cost‐effective off‐grid zero energy complex in Kish Island under risk management. The upper level of the planning is composed of two components, the passive design of buildings within the complex and the design of a stand‐alone energy system. The passive design as an energy‐saving tool includes the selection of insulation material for building external walls and finding the optimum thickness. Also, the stand‐alone energy system design denotes the sizing of diesel generator, photovoltaic, and battery energy storage as the distributed energy resources. The lower‐level problem optimally handles the annual scheduling of these resources to meet the complex demand under the impact of passive cooling. The Karush–Kuhn–Tucker condition method is used to solve this bi‐level planning problem. Furthermore, the battery degradation is concerned via the throughput model to consider the replacement cost of the problem.

Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

A varying-parameter complementary neural network for multi-robot tracking and formation via model predictive control

In this paper, a varying-parameter complementary neural network (VPCNN) is designed and combined with model predictive control (MPC) to solve the multi-robot tracking and formation problems via a leader–follower strategy. First, multi-robot tracking and formation problems are transformed into quadratic programming (QP) problems employing an MPC approach. Second, a nonlinear complementary function approach, i.e., the Fischer–Burmeister function, is used to map the Karush–Kuhn–Tucker conditions of the QP problem with double-ended inequality constraints to a system of nonlinear equations. Finally, the VPCNN is designed to solve the multi-robot tracking and formation problem. The effectiveness of the proposed method is demonstrated by numerical simulations, and the advantage of VPCNN in terms of solution speed is indicated by comparisons with a primal–dual neural network.

Voltage Hierarchical Control Strategy for Distribution Networks Based on Regional Autonomy and Photovoltaic-Storage Coordination

High-penetration photovoltaic (PV) integration into a distribution network can cause serious voltage overruns. This study proposes a voltage hierarchical control method based on active and reactive power coordination to enhance the regional voltage autonomy of an active distribution network and improve the sustainability of new energy consumption. First, considering the reactive power margin and spatiotemporal characteristics of distributed photovoltaics, a reactive voltage modularity function is proposed to divide a distribution grid into voltage regions. Voltage region types and their weak points are then defined, and the voltage characteristics and governance needs of different regions are obtained through photovoltaic voltage regulation. Subsequently, a dual-layer optimal configuration model of energy storage that accounts for regional voltage regulation is established. The upper-layer model focuses on planned configurations to minimize the annual comprehensive operating cost of the energy storage system (ESS), while the lower-layer model focuses on optimal dispatch to achieve the best regional voltage quality. KKT conditions and the Big-M method are employed to convert the dual-layer model into a single-layer linear model for optimization and solution. Finally, an IEEE 33-node system with high-penetration photovoltaics is modeled using MATLAB (2022a). A comparative analysis of four scenarios shows that the comprehensive cost of an ESS decreased by 8.49%, total revenue increased by 19.36%, and the overall voltage deviation in the distribution network was reduced to 0.217%.

Large‐scale elasto‐plastic topology optimization

AbstractThis work presents large‐scale elasto‐plastic topology optimization for design of structures with maximized energy absorption and tailored mechanical response. The implementation uses parallel computations to address multi million element three‐dimensional problems. Design updates are generated using the gradient‐based method of moving asymptotes and the material is modeled using small strain, nonlinear isotropic hardening wherein the coaxiality between the plastic strain rate and stress is exploited. This formulation renders an efficient state solve and we demonstrate that the adjoint sensitivity scheme resembles that of the state update. Furthermore, the KKT condition is enforced directly into the path dependent adjoint sensitivity analysis which eliminates the need of monitoring the elasto‐plastic switches when calculating the gradients and provides a straight forward framework for elasto‐plastic topology optimization. Numerical examples show that structures discretized using several millions degrees of freedom and loaded in multiple load steps can be designed within a reasonable time frame.

Multiplier rules for Dini-derivatives in a topological vector space

We provide new results of first-order necessary conditions of optimality problem in the form of John's theorem and in the form of Karush–Kuhn–Tucker's theorem. We establish our result in a topological vector space for problems with inequality constraints and in a Banach space for problems with equality and inequality constraints. Our contributions consist in the extension of the results known for the Fréchet and Gateaux-differentiable functions as well as for the Clarke's subdifferential of Lipschitz functions to the more general Dini-differentiable functions. As consequences, we extend the result of B.H. Pourciau in [Modern multiplier rules. Amer Math Monthly. 1980;87(6):433–457, Theorem 6, p. 445] from the convexity to the Dini-pseudoconvexity.

Learning Hidden Markov Models with Structured Transition Dynamics

The hidden Markov model (HMM) provides a natural framework for modeling the dynamic evolution of latent diseases. The unknown probability matrices of HMMs can be learned through the well-known Baum–Welch algorithm, a special case of the expectation-maximization algorithm. In many disease models, the probability matrices possess nontrivial properties that may be represented through a set of linear constraints. In these cases, the traditional Baum–Welch algorithm is no longer applicable because the maximization step cannot be solved by an explicit formula. In this paper, we propose a novel approach to efficiently solve the maximization step problem under linear constraints by providing a Lagrangian dual reformulation that we solve by an accelerated gradient method. The performance of this approach critically depends on devising a fast method to compute the gradient in each iteration. For this purpose, we employ dual decomposition and derive Karush–Kuhn–Tucker conditions to reduce our problem into a set of single variable equations, solved using a simple bisection method. We apply this method to a case study on sports-related concussion and provide an extensive numerical study using simulation. We show that our approach is in orders of magnitude computationally faster and more accurate than other alternative approaches. Moreover, compared with other methods, our approach is far less sensitive with respect to increases in problem size. Overall, our contribution lies in the advancement of accurately and efficiently handling HMM parameter estimation under linear constraints, which comprises a wide range of applications in disease modeling and beyond. History: Accepted by Paul Brooks, Area Editor for Applications in Biology, Medicine, & Healthcare. Funding: This research was funded by [Grant R01DE028283] from the National Institute of Dental and Craniofacial Research, National Institutes of Health. G.-G. Garcia was also funded by the Georgia Clinical & Translational Science Alliance National Institutes of Health award [Grant UL1-TR002378]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0342 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0342 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Stochastic linear–quadratic control problems with affine constraints

This paper investigates the stochastic linear–quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, both the dual problem and the state feedback form of the solution are obtained for the primal problem. Under the Slater condition, the strong duality is proved between the dual problem and the primal problem, and the KKT condition is also provided for solving the primal problem. Moreover, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem. Finally, two numerical examples are provided to illustrate our main results.

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.

Low‐carbon economic operation of multi‐energy microgrid based on multi‐level robust optimisation

AbstractThe economic and low‐carbon operation strategy of multi‐energy microgrids (MEM) has become an important research topic in smart grids. The operation of MEM is affected by uncertain factors from renewable energy and internal load. To handle uncertainties from both source and load sides, the authors propose a novel worst‐case‐and‐probability uncertainty sets and a novel worst‐expectation min‐max‐max‐min two‐stage four‐level robust optimisation (RO) model considering stepped carbon trading for MEM. This four‐level RO model is a more generalised model compared with the traditional two‐stage RO and distributionally robust optimisation based models. First, the unsolvable original four‐level model is decoupled into a master problem and a sub‐problem (SP). Second, Karush–Kuhn–Tucker condition is applied to convert SP into a solvable problem. Third, column and constraint generation (C&CG) algorithm is employed to solve the reformulated four‐level RO. Finally, the effectiveness of the proposed model and solution method is verified by case studies. The results indicate that the convergence behaviour of this solution method is excellent. The MEM operator can select appropriate robustness factors and comprehensive norms to control its conservatism level. Besides, MEM could reduce carbon emissions by participating in carbon trading and installing carbon capture devices.

Joint optimization of offloading strategy and resource allocation for multi-user in dynamic vehicular edge computing systems

Internet of Vehicles (IoV) relies heavily on its computing capability to facilitate various vehicular applications. Since the cloud computing or mobile edge computing (MEC) only cannot satisfy the latency requirement due to the limitation of the resource coverage, the cloud–edge-end cooperative computing has become an emerging paradigm. A comprehensive IoV architecture is considered and a joint optimization problem is formulated to minimize the system function value. To optimize the resource allocation and the task offloading strategy, the simulated spring system algorithm (SSSA) is designed where the initial problem is decoupled into two sub-problems with priority. The first one is to allocate computing resources based on KKT conditions, thus the individual optimal solution is achieved. The second one is solved based on the idea of simulated spring system such that the task offloading strategy is obtained. Two sub-problems iterate mutually to update each other until finishing the binary tree traversal. Thus, the proposed solution adapts to various conditions and the computational complexity is also reduced compared with traditional methods. Simulation verifies that the proposed algorithm reduces the maximum system function value by about 31% compared with the benchmark methods and performs efficiently in various road conditions.

Lossless convexification and duality

The main goal of this paper is to investigate the strong duality of non-convex semidefinite programming problems (SDPs). In the optimization community, it is well-known that a convex optimization problem satisfies strong duality if Slater’s condition holds. However, this result cannot be directly generalized to non-convex problems. In this paper, we prove that a class of non-convex SDPs with special structures satisfies strong duality under Slater’s condition. Such a class of SDPs arises in SDP-based control analysis and design approaches. Throughout the paper, several examples are given to support the proposed results. We expect that the proposed analysis can potentially deepen our understanding of non-convex SDPs arising in the control community and promote their analysis based on KKT conditions.

A new bi-level model for the false data injection attack on real-time electricity market considering uncertainties

State Estimation (SE) plays a critical role in various applications of power systems including the electricity market. False Data Injection Attacks (FDIAs) are capable of manipulating the power system SE process aiming to gain financial profits by the players. In this paper, the potential economic influences of FDIA on SE and consequently the Real-Time (RT) electricity market are examined. Therefore, a new bi-level optimization framework is proposed to realistically model the FDIAs from the attacker perspective. At the upper-level, the attacker intends to intelligently choose the most critical congested transmission lines and manipulate their associated meter readings to maximize the expected financial profitability. At the lower-level, the RT market-clearing model is formulated under the influence of FDIA. Considering the incomplete attacker's prior knowledge of the network topology i.e., connection/disconnection of transmission lines, the angles of the phase-shifting transformers, and the real-time load uncertainty, the proposed formulation is extended to a new bi-level scenario-driven stochastic model. The developed attack model, which must remain undetectable by the system operator in all scenarios, is solved using the Karush–Kuhn–Tucker (KKT) approach. The correctness and effectiveness of the proposed optimization scheme have been validated in GAMS software using CONOPT and OQNLP solvers based on the IEEE 14-bus test system and also comparative results are presented to demonstrate the merits of the proposed formulation.

Model predictive path tracking control of intelligent vehicles based on dual-stage disturbance observer under multi-channel disturbances

Abstract Parameter fluctuations, unmodeled dynamics, speed variation, steering actuator faults, and other multi-channel uncertain disturbances are the key challenges faced by the path tracking control of intelligent vehicles, which will affect the accuracy and stability of the path tracking. Therefore, a model predictive control (MPC) method based on a dual-stage disturbance observer (DDOB) is proposed in this paper. First, a tracking error dynamics model considering multi-channel uncertain disturbances is constructed, based on which a model predictive controller is designed to obtain the nominal front wheel steering angle by the Karush–Kuhn–Tucker (KKT) condition. Furthermore, the DDOB is designed to enable real-time estimation of the system disturbances, and then the estimated disturbances are used as the compensation for the nominal front wheel steering angle, which establishes the MPC control law with parallel compensation of the DDOB. Finally, the error boundedness of the DDOB and the global stability of the model predictive controller are analyzed. The effectiveness and superiority of the proposed algorithm are verified through Carsim–Simulink simulation and hardware-in-the-loop experiments.

Physics-informed neural networks with hard linear equality constraints

Surrogate modeling is used to replace computationally expensive simulations. Neural networks have been widely applied as surrogate models that enable efficient evaluations over complex physical systems. Despite this, neural networks are data-driven models and devoid of any physics. The incorporation of physics into neural networks can improve generalization and data efficiency. The physics-informed neural network (PINN) is an approach to leverage known physical constraints present in the data, but it cannot strictly satisfy them in the predictions. This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints through projection layers derived from KKT conditions. Numerical experiments on Aspen models of a continuous stirred-tank reactor (CSTR) unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.

Optimality and KKT conditions for interval valued optimization problems on Hadamard manifolds

Recently, a new type of optimization problems, the so-called interval optimization problems on Hadamard manifolds, is introduced by the authors in Nguyen et al. [Interval optimization problems on Hadamard manifolds. J Nonlinear Convex Anal. 2023;24(11):2489–2511]. In this follow-up, we further offer the algorithmic bricks for these problems. More specifically, we characterize the optimality and KKT conditions for the interval valued optimization problems on Hadamard manifolds. For unconstrained problems, the existence of efficient points and the steepest descent algorithm are investigated. To the contrast, the KKT conditions and exact penalty approach are explored in the ones involving inequality constraints. These results pave the foundations for the solvability of interval valued optimization problems on Hadamard manifolds.

Learnable Graph Matching: A Practical Paradigm for Data Association.

Data association is at the core of many computer vision tasks, e.g., multiple object tracking, image matching, and point cloud registration. however, current data association solutions have some defects: they mostly ignore the intra-view context information; besides, they either train deep association models in an end-to-end way and hardly utilize the advantage of optimization-based assignment methods, or only use an off-the-shelf neural network to extract features. In this paper, we propose a general learnable graph matching method to address these issues. Especially, we model the intra-view relationships as an undirected graph. Then data association turns into a general graph matching problem between graphs. Furthermore, to make optimization end-to-end differentiable, we relax the original graph matching problem into continuous quadratic programming and then incorporate training into a deep graph neural network with KKT conditions and implicit function theorem. In MOT task, our method achieves state-of-the-art performance on several MOT datasets. For image matching, our method outperforms state-of-the-art methods on a popular indoor dataset, ScanNet. For point cloud registration, we also achieve competitive results.

Secure energy efficiency maximization for joint ITS and IRS assisted satellite downlink communications

AbstractSatellite communications (SatCom) have been viewed as a promising technique to achieve ubiquitous global coverage in next‐generation wireless systems. This article investigates the secure energy efficient downlink transmission for SatCom where an intelligent transmissive surface‐aided transmitter is deployed at the satellite to perform energy efficient beamforming and an intelligent reflecting surface as well as a cooperative jammer are deployed on the ground to enhance the secure performance. The aim is to maximize the secure energy efficiency by jointly designing ITS beamforming, IRS phase shift, jammer precoding vector, and transmit power allocation. To develop a low‐complexity solution,first, an approximated concave lower bound of the non‐concave objective function is derived by utilizing Dinkelbach's method and a novel successive convex approximation technique. Then the reformulated problem is decoupled using alternating optimization algorithm and the subproblems are solved by adopting Riemannian manifold optimization, element‐wise block coordinate descent, Lagrange dual method and Karush–Kuhn–Tucker conditions. At last, simulation results demonstrate the effectiveness of the proposed scheme.