Karush-Kuhn-Tucker Research Articles

Minimizing energy consumption for NOMA multi-drone communications in automotive-industry 5.0

The forthcoming era of the automotive industry, known as Automotive-Industry 5.0, will leverage the latest advancements in 6G communications technology to enable reliable, computationally advanced, and energy-efficient exchange of data between diverse onboard sensors, drones and other vehicles. We propose a non-orthogonal multiple access (NOMA) multi-drone communications network in order to address the requirements of enormous connections, various quality of services (QoS), ultra-reliability, and low latency in upcoming sixth-generation (6G) drone communications. Through the use of a power optimization framework, one of our goals is to evaluate the energy efficiency of the system. In particular, we define a non-convex power optimization problem while considering the possibility of imperfect successive interference cancellation (SIC) detection. Therefore, the goal is to reduce the total energy consumption of NOMA drone communications while guaranteeing the lowest possible rate for wireless devices. We use a novel method based on iterative sequential quadratic programming (SQP) to get the best possible solution to the non-convex optimization problem so that we may move on to the next step and solve it. The standard OMA framework, the Karush–Kuhn–Tucker (KKT)-based NOMA framework, and the average power NOMA framework are compared with the newly proposed optimization framework. The results of the Monte Carlo simulation demonstrate the accuracy of our derivations. The results that have been presented also demonstrate that the optimization framework that has been proposed is superior to previous benchmark frameworks in terms of system-achievable energy efficiency.

Multi-Objective LQG Design with Primal-Dual Method

The objective of this paper is to investigate a multi-objective linear quadratic Gaussian (LQG) control problem. Specifically, we examine an optimal control problem that minimizes a quadratic cost over a finite time horizon for linear stochastic systems subject to control energy constraints. To tackle this problem, we propose an efficient bisection line search algorithm that outperforms other approaches such as semidefinite programming in terms of computational efficiency. The primary idea behind our algorithm is to use the Lagrangian function and Karush–Kuhn–Tucker (KKT) optimality conditions to address the constrained optimization problem. The bisection line search is employed to search for the Lagrange multiplier. Furthermore, we provide numerical examples to illustrate the efficacy of our proposed methods.

Quadratic programming with ramp functions and fast online QP-MPC solutions

A novel method is proposed for solving quadratic programming problems arising in model predictive control. The method is based on an implicit representation of the Karush–Kuhn–Tucker conditions using ramp functions. The method is shown to be highly efficient on both small and fairly large Quadratic Program problems, can be implemented using simple computer code, and has modest memory requirements.

The impact of R&D expenditures and screening in an economic production rate (EPR) inventory model for a flawed production system with imperfect screening under an interval-valued environment

In this paper, a single product of a single manufacturing unit for an imperfect production inventory system is modelled to minimize the total cost of a production cycle. A large fraction of defective items is passed to customers by manufacturers due to imperfect screening method so manufacturer replace them with new ones while incurring penalties. This fraction depends on the investment of research and development (R&D). This paper optimizes the R&D cost to minimize the defective items produced in a production line. The demand function depends on the selling price, advertisement cost, and product quality. The quality of the produce items is measured by screening time effort and investment in R&D. Further, we considered all the costs of the production cycle in interval-valued form and elaborately discuss the application of interval metric in the total cost interval objective function. A necessary and sufficient conditions are derived for the proposed model with the help of modified KKT conditions. Results show that applying screening and R&D costs significantly reduces the number of defective items and simultaneously reduces the total cost. A numerical example is presented along with sensitivity analyses on different parameters with the help of MATLAB software.

Automatic differentiation rules for Tsoukalas–Mitsos convex relaxations in global process optimization

With increasing digitalization and vertical integration of chemical process systems, nonconvex optimization problems often emerge in chemical engineering applications, yet require specialized optimization techniques. Typical global optimization methods proceed by progressively refining bounds on the unknown optimal value, by strategically employing convex relaxations. This article constructs a general closed-form expression for the convex subdifferentials of recent “multivariate McCormick” convex relaxations of nontrivial composite functions, by solving a previous duality formulation in all cases using nonsmooth Karush–Kuhn–Tucker conditions. Based on this subdifferential expression, new automatic differentiation rules are developed to compute gradients and subgradients for multivariate McCormick relaxations, to ultimately generate useful bounds in global optimization. Unlike established differentiation techniques for these relaxations, our new rules are expressed in closed form, do not require solving separate dual optimization problems, are efficiently carried out, and are compatible with the reverse/adjoint mode of algorithmic differentiation. Our formulations become more straightforward when the relevant functions are either smooth or piecewise smooth.

A Bilevel Stochastic Optimization Framework for Market-Oriented Transmission Expansion Planning Considering Market Power

Market power, defined as the ability to raise prices above competitive levels profitably, continues to be a prime concern in the restructured electricity markets. Market power must be mitigated to improve market performance and avoid inefficient generation investment, price volatility, and overpayment in power systems. For this reason, involving market power in the transmission expansion planning (TEP) problem is essential for ensuring the efficient operation of the electricity markets. In this regard, a methodological bilevel stochastic framework for the TEP problem that explicitly includes the market power indices in the upper level is proposed, aiming to restrict the potential market power execution. A mixed-integer linear/quadratic programming (MILP/MIQP) reformulation of the stochastic bilevel model is constructed utilizing Karush−Kuhn−Tucker (KKT) conditions. Wind power and electricity demand uncertainty are incorporated using scenario-based two-stage stochastic programming. The model enables the planner to make a trade-off between the market power indices and the investment cost. Using comparable results of the IEEE 118-bus system, we show that the proposed TEP outperforms the existing models in terms of market power indices and facilitates open access to the transmission network for all market participants.

Strategic Participation of Integrated Thermal and Electrical Energy Service Provider in Natural Gas and Wholesale Electricity Markets

The accelerating interest in energy system models has influenced numerous revolutionized alterations in the wholesale gas and electricity market. This article proposes a novel bilevel multifollower optimization framework for the strategic behavior observation of an integrated (thermal and electrical) energy service provider (IESP) as a pricemaker in the wholesale electricity market (WEM) and natural gas market (NGM). At the upper level, the IESP submits offers/bids in WEM and NGM to procure electricity/gas to the customers. To this end, the IESP endeavors to minimize operational costs and influences the market-clearing price (MCP) by deploying demand-side flexibilities, i.e., elastic electrical and thermal loads. At the lower level, the WEM operator and NGM operator receive offers/bids from all market participants and clear the market with the goal of maximizing social welfare. The IESP is modeled via IEEE-33 bus active distribution system and an 8-node district heating system, whereas the WEM and NGMs are embodied by a 6-bus transmission network and a 21-node natural gas network, respectively. Karush–Kuhn–Tucker conditions are introduced to transform the multifollower bilevel optimization problem into a single-level problem, whereas the inherent nonlinearities of the problem are linearized using the theory of strong duality. Moreover, the intrinsic intermittencies of the renewable energy sources is dealt with by the risk-averse information gap decision theory. The results confirm that flexible electrical and thermal demands can diminish the MCP by as much as 4.1%.

Analysis and Optimal Modulation for 2/3-Level DAB Converters to Minimize Current Stress With Five-Level Control

A two-to-three-level dual-active-bridge (DAB) dc–dc converter has high potential to become a promising solution for medium-voltage high-power applications. However, its switching characteristics, modeling, and optimal modulation schemes have not been comprehensively explored yet. In order to avoid voltage and current distortions and ensure reliable operation, this article analyzes the operating constraints for the neutral-point-clamped (NPC) DAB converters. Based on the obtained operating constraints, the transferred power and current stress models under a five-level control scheme, which is considered as one of the most advantageous control strategies for the NPC DAB converters, are obtained with an equivalent-wave modeling method. Generic control strategies based on analytical solutions have not been fully explored for the multilevel DAB converters due to the increased number of control variables. With the above, this article proposes a minimum-current-stress control strategy employing analytical solutions in the entire power range, which are obtained by combining the Karush–Kuhn–Tucker conditions and numerical-solution analysis. With the proposed modulation scheme, the control variables can be automatically modified online, instead of relying on the offline calculation of the numerical control variables. Therefore, the control complexity can be reduced significantly, especially when the operating parameters change in a wide range. Finally, experimental tests verify the effectiveness of the analysis.

Novel enhancement of energy distribution for marine hybrid propulsion systems by an advanced variable weight decision model predictive control

Marine hybrid technology is attracting increasing research interest thanks to its power versatility and potential economic advantages. However, its overall quality performance is determined by the energy management strategy. This research deals with a variable weighted decision model predictive control (VWDMPC) for marine hybrid energy management, where the main task is to dynamically weight-tuning and optimally allocate energy to obtain the optimal trade-off in terms of fuel consumption, power performance, and energy allocation. For the sake of enhancing the real-time and adaptability of VWDMPC, we utilize the KKT condition to compress the optimal energy management problem and the weight tuning process in one, which cleverly simplifies the optimization process. Performance experiments are performed on a test bench and a real-time hardware execution platform. Different weighting decision comparison scenarios were designed to evaluate the power performance improvement and equivalent fuel consumption of the proposed VWDMPC and give the best proportional weights (0.33,0.27,0.2,0.2) that take both aspects into account. Moreover, this paper also provides validation and feedback based on the evaluation results, which show that the optimal combination of weights that take into account economy and dynamics ranges from (0.3,0.3,0.2,0.2)-(0.33,0.27,0.2,0.2) based on the radar plot drawn from the performance index parameters.

Joint Resource Allocation and Learning Optimization for UAV-Assisted Federated Learning

Aiming at the unmanned aerial vehicle (UAV)-assisted federated learning wireless-network scenario, and considering the influence of the UAV altitude on the coverage area, we propose a joint optimization algorithm of UAV placement, computation and communication resources. Considering the energy efficiency and federated learning performance, we defined the cost function of the system. Under the constraint of the total delay of federated learning completion, we formulated an optimization problem of minimizing the cost function to achieve the balance between the total energy consumption of users and the federated learning performance. Since the formulated optimization problem is a non-convex problem, in order to solve this problem, we decomposed it into three optimization subproblems: UAV horizontal placement, local accuracy and computation and communication resources. We used successive convex approximation (SCA), the Dinkelbach Method, the Bisection method and KKT condition, respectively, to solve the three subproblems, and finally obtain the optimal solutions through iteration of the three subproblems. Simulation results show that compared with the federated learning scenario under fixed UAV altitude, our proposed algorithm not only guarantees the learning performance, but also reduces more users’ total energy consumption.

FLIRRAS: Fast Learning With Integrated Reward and Reduced Action Space for Online Multitask Offloading

With the rapid development of edge data intelligence, task offloading (TO) and resource allocation (RA) optimization in multiaccess edge computing networks can significantly improve the Quality of Service (QoS). However, for the online scenario, traditional methods (e.g., game theory and numerical methods) cannot adapt to dynamic environments. Deep reinforcement learning (DRL) is applied to adjust the policy to get long-term rewards. Nevertheless, since the joint problem of TO and RA is nonconvex and NP-hard, existing DRL methods cannot guarantee high efficiency because of the large action space. To solve the above problem, we propose a fast learning with integrated reward and reduced action space-based DRL framework (FLIRRAS), which adopts a low-complexity approach to jointly optimize TO and RA strategies. The FLIRRAS framework combines DRL with numerical methods to iteratively pursues the discrete TO and continuous RA. Specifically, a deep neural network (DNN) is used to learn environmental information, which can get prior knowledge of the offloading decision. Furthermore, a novel reward integrating the utility of TO and RA is designed to motivate the agent to find the optimal policy. To solve the dilemma that the action space is too large, low-complexity convex optimization methods, i.e., subgradient projection and KKT condition, are used to supplement and adjust the decision, which reduces the network parameters and the decision space. In addition, given the dynamic online environment, we introduce the experience replay mechanism, where policy is updated regularly to reflect the best mapping between states. The experiment results show that the performance of FLIRRAS is better than greedy and other DRL approaches, and it outperforms the latest DRL method by over 18.0% in terms of execution time.

Pricing Strategy for a Virtual Power Plant Operator with Electric Vehicle Users Based on the Stackelberg Game

With the popularity and promotion of electric vehicles (EVs), virtual power plants (VPPs) provide a new means for the orderly charging management of decentralized EVs. How to set the price of electricity sales for VPP operators to achieve a win–win situation with EV users is a hot topic of current research. Based on this, this paper first proposes a Stackelberg game model in which the VPP participates in the orderly charging management of EVs as a power sales operator, where the operator guides the EV users to charge in an orderly manner by setting a reasonable power sales price and coordinates various distributed resources to jointly participate in the power market. Furthermore, taking into account the impact of wind power output uncertainty on VPP operation, a robust optimization method is used to extend the deterministic Stackelberg game pricing model into a robust optimization model, and a robust adjustment factor is introduced to flexibly adjust the conservativeness of the VPP operator’s bidding scheme in the energy market. The model is then transformed into a robust mixed-integer linear programming (RMILP) problem solved by Karush–Kuhn–Tucker (KKT) conditions and strong dyadic theory. Finally, the effectiveness of the solution method is verified in the calculation example, which gives the optimal pricing strategy for the VPP operator, the optimal charging scheme for EV users, and the remaining internal resources’ contribution plan, providing an important idea for the VPP to centrally manage the charging behavior of EVs and improve its own operating revenue.

A Stabilized Sequential Quadratic Programming Method for Optimization Problems in Function Spaces

In this paper, we propose a stabilized sequential quadratic programming (SQP) method for optimization problems in function spaces. A form of the problem considered in this paper can widely formulate many types of applications, such as obstacle problems, optimal control problems, and so on. Moreover, the proposed method is based on the existing stabilized SQP method and can find a point satisfying the Karush-Kuhn-Tucker (KKT) or asymptotic KKT conditions. One of the remarkable points is that we prove its global convergence to such a point under some assumptions without any constraint qualifications. In addition, we guarantee that an arbitrary accumulation point generated by the proposed method satisfies the KKT conditions under several additional assumptions. Finally, we report some numerical experiments to examine the effectiveness of the proposed method.

On the Fulfillment of the Complementary Approximate Karush–Kuhn–Tucker Conditions and Algorithmic Applications

On the Fulfillment of the Complementary Approximate Karush–Kuhn–Tucker Conditions and Algorithmic Applications

OPTIMIZATION OF A FUZZY INVENTORY MODEL WITH PENTAGONAL FUZZY NUMBERS

This paper explores an optimal replenishment strategy for a two-echelon inventory model which has been considered and analyzed in a fuzzy environment. In fuzzy environment, carrying cost, ordering cost and the replenishment processing cost are assumed to be pentagonal fuzzy numbers. The purpose of this model is to minimize the total inventory cost in fuzzy scenario. There are two inventory models proposed in this paper. Crisp models are developed with fuzzy total inventory cost but crisp optimal order quantity. Fuzzy model is also formulated with fuzzy total inventory cost and fuzzy optimal order quantity. Graded mean integration formula is employed to defuzzify the total inventory cost and the Kuhn–tucker condition is used to determine the optimal order quantity. Finally, we develop an algorithm to obtain the optimal order quantity. A comparison of fuzzy model with classical inventory model is being made. Numerical results highlighting the sensitivity of various parameters are also elucidated. The result illustrates that this fuzzy model can be quite useful in determining the optimal order quantity and minimum total inventory cost.

Multicycle disassembly-based decomposition algorithm to train multiclass support vector machines

Employing the classic optimization solver to train a multiclass support vector machine (SVM) requires prohibitive training time as the sample size and number of categories increase. It has been proposed to develop the corresponding decomposition algorithm (DA) as it is efficient for training SVMs. However, the dual problem of multiclass SVM comprises complex constraints that complicate DA design, so no corresponding DA has yet been developed. We propose a multicycle disassembly-based DA (MCD-DA) to efficiently solve the training problem of multiclass SVM. First, a graph model is constructed to re-express the constraints in multiclass SVM. Then, the original complex feasible region is partitioned into several simple sub-feasible regions, and multiple cycle-based disassembly strategies are designed to update the working variables analytically within each specific sub-feasible region. We mathematically verify that MCD-DA can stop within a finite number of cycle disassemblies and reach the τ-optimal solution satisfying relaxed Karush–Kuhn–Tucker conditions. Remarkably, MCD-DA as a universal decomposition algorithm can be used to solve many other SVM variants, including C-SVM, v-SVM, and one-class SVM. Experimental results using six UCI datasets demonstrate that MCD-DA outperforms typical optimization algorithms for more sample cases.

A proposed hedge-based energy market model to manage renewable intermittency

Renewable energy power producers are exposed to significant volatility in revenue due to intermittency associated with wind and solar energy. Moreover, higher penetration of variable renewable energy in the electric grids is significantly increasing the cost associated with procuring reserves, resulting in higher electricity costs to consumers. Hence, a market that can appropriately distribute the cost of procuring reserves whilst ensuring revenue stability for renewable energy producers is crucial. In this work, the authors propose a novel hedge-based energy market model that allows renewable generators to secure hedge contracts from flexible generating technologies as insurance against weather-driven energy deficits. The proposed model supplements a representative day-ahead market model and maximizes the revenue of market participants whilst diminishing the costs of procuring reserves and generating investment signals for green projects. A mathematical model is formulated to determine market equilibrium based on the Karush Kuhn Tucker (KKT) optimality conditions. Simulation studies are carried out to demonstrate the efficacy of the proposed model on a test network using MATLAB. The theoretical results are verified by simulation results and provide a feasible region in which mutually acceptable hedge contracts result in higher overall revenues. The results show that a hedge-based energy market model can be deployed to manage renewable intermittency in a day-ahead energy market model to address the risk management needs of renewable power producers.

Multi-view clustering guided by unconstrained non-negative matrix factorization

Multi-view clustering based on non-negative matrix factorization (NMFMvC) is a well-known method for handling high-dimensional multi-view data. To satisfy the non-negativity constraint of the matrix, NMFMvC is usually solved using the Karush–Kuhn–Tucker (KKT) conditions. However, this optimization method is poorly scalable. To this end, we propose an unconstrained non-negative matrix factorization multi-view clustering (uNMFMvC) model. First, the objective function was constructed by decoupling the elements of the matrix and combining the elements with a non-linear mapping function in a non-negative value domain. The objective function was then optimized using the stochastic gradient descent (SGD) algorithm. Subsequently, three uNMFMvC methods were constructed based on different mapping functions and detailed reasoning was provided. Finally, experiments were conducted on eight public datasets and compared with cutting-edge multi-view clustering methods. The experimental results demonstrate that the proposed model has significant advantages.

Multi-Microgrids Operation With Interruptible Loads in Local Energy and Reserve Markets

In the evolution of the power systems, a particular case is the presence of a number of microgrids (MGs) operated with mutual interconnection, but without connection to the main distribution system. The interconnected MGs form a structure in which the overall system operation and resource scheduling can be determined by considering centralized or decentralized approaches. This article introduces local energy and reserve markets (LERMs) in which the MG managers (MGMs) can meet their required energy and reserve with optimal scheduling of their resources, besides competing with the other MGs. To model such decision-making framework for MGMs, a bilevel optimization approach is developed in which the MGMs’ problem is modeled as the upper level problem and the LERMs clearing problem is modeled as the lower level problem. This model is transformed into a mathematical programming with equilibrium constraints (MPEC) using the primal-dual transformation. Then, the resulting MPEC for each MG is replaced with its Karush–Kuhn–Tucker conditions, obtaining an equilibrium problem with equilibrium constraints (EPEC) model. The nonlinear terms of the model are linearized through different approaches. Finally, the EPEC model is transformed into a mixed-integer linear problem considering the objective function of all MGMs. The model is applied to a test system with three interconnected MGs. Moreover, the sensitivity of the results to the probability of calling reserve is investigated.

Stochastic Strategic Participation of Active Distribution Networks With High-Penetration DERs in Wholesale Electricity Markets

With the increasing penetration of distributed energy resources (DERs), traditional distribution networks as load-serving entities in wholesale electricity markets, now evolve towards active distribution networks (ADNs) which can proactively participate in wholesale markets by optimally controlling the DERs in their networks. A stochastic bilevel optimization model is proposed in this paper for the strategic participation of ADNs and DERs to provide energy and grid services in wholesale electricity markets. The bilevel optimization model can capture the interactions between the ADN and the wholesale energy and ancillary service markets, considering the uncertainties of DERs in the ADN. In the upper-level model, the ADN makes optimal decisions on energy and reserve bidding considering the availability, uncertainties, and flexibility of DERs. The joint energy and reserve market-clearing of the independent system operator (ISO) is modeled as the lower-level problem. Using strong duality theory and Karush-Kuhn Tucker (KKT) conditions, the proposed bilevel optimization problem is reformulated as mathematical programming with equilibrium constraints (MPEC) problem and further converted into a computationally-solvable mixed-integer second-order-cone programming (MISOCP) model. The simulation results demonstrate the effectiveness of the model and the interactions between an ADN and wholesale electricity markets.