Tucker Conditions Research Articles

An efficient penalty function method for scenario-based uncertainty quantification problems

This paper proposes an efficient penalty function method to transform constrained optimization problems into unconstrained ones without introducing Lagrange multipliers or slack variables. The method designs new activation and loss penalty functions for inequality and equality functional models and establishes a non-Lagrangian-constrained optimization method. A novel optimality condition independent of any additional variables and equivalent to the Karush–Kuhn–Tucker (KKT) condition is introduced. Additionally, a scenario constraint handling method that does not rely on slack variables is proposed. Compared to soft constraint optimization methods, the proposed method can easily handle numerous scenario constraints. Two examples are used to compare the results with those in the literature, verifying the effectiveness and reliability of the proposed method. Finally, the method is applied to three scenario-based uncertainty quantification problems, including a trigonometric function affected by a noise term, the dynamic performance of a black-box system controller, and the frequency response of a damaged suspension arm. The results demonstrate that the proposed penalty function method can effectively solve scenario-based uncertainty quantification problems with many constraints and improve computational efficiency, providing a new method and means for addressing uncertainty quantification optimization problems in engineering.

Optimization Strategy for Shared Energy Storage Operators-Multiple Microgrids with Hybrid Game-Theoretic Energy Trading

To address the issue of low utilization rates, constrained operational modes, and the underutilization of flexible energy storage resources at the end-user level, this research paper introduces a collaborative operational approach for shared energy storage operators in a multiple microgrids (ESO-MGs) system. This approach takes into account the relation of electricity generated by MGs and the integration of diverse energy storage resources managed by ESO. A hybrid game-theoretic energy trading strategy is employed to address the challenges associated with energy trading and revenue distribution in this joint operational mode. Firstly, a multi-objective master–slave game optimization model is developed with the objective of maximizing the revenue earned by shared energy storage operators while simultaneously minimizing the operational costs of multiple microgrids. Secondly, acknowledging the peer-to-peer (P2P) energy sharing dynamics inherent in the multiple microgrid system, a non-co-operative game model is formulated. This model seeks to establish a multi-microgrid Nash equilibrium and equitable income allocation. Finally, leveraging the Karush–Kuhn–Tucker (KKT) conditions and drawing upon the principles of strong duality theory, precise dimensionality reduction is executed on the master–slave game model. The non-co-operative income is iteratively determined using the alternating direction multiplier algorithm. The empirical findings of this study indicate that the integration of electric vehicle clusters contributes to flexible storage resources for shared energy storage operators. Moreover, the proposed hybrid game optimization strategy enhances the overall benefits for shared energy storage operators and multiple microgrids, thereby affirming the economic viability and reliability of this innovative strategy.

Distributed continuous-time accelerated neurodynamic approaches for sparse recovery via smooth approximation to [formula omitted]-minimization

This paper develops two continuous-time distributed accelerated neurodynamic approaches for solving sparse recovery via smooth approximation to L1-norm minimization problem. First, the L1-norm minimization problem is converted into a distributed smooth optimization problem by utilizing multiagent consensus theory and smooth approximation. Then, a distributed primal–dual accelerated neurodynamic approach is designed by using Karush–Kuhn–Tucker (KKT) condition and Nesterov’s accelerated method. Furthermore, in order to reduce the structure complexity of the presented neurodynamic approach, based on the projection matrix, we eliminate a dual variable in the KKT condition and propose a distributed accelerated neurodynamic approach with a simpler structure. It is proved that the two proposed distributed neurodynamic approaches both achieve O(1t2) convergence rate. Finally, the simulation results of sparse recovery are given to demonstrate the effectiveness of the proposed approaches.

Social welfare and equality equilibrium based carbon tax subsidy incentive approach for biomass-coal co-firing towards carbon emissions

The lack of effective and scientifically supported policy measures has led to poor application of biomass-coal co-firing. A bi-level multi-objective model is developed to formulate a reasonable carbon tax subsidy policy and optimal biomass-coal co-firing plan, which considers the interactive relationship between authority and coal-fired power plants, and achieves a trade-off between social benefits, subsidy fairness, and environmental protection. A case study and scenarios demonstrate the feasibility and effectiveness of the proposed model. The ϵ-constraint method and Karush–Kuhn–Tucker conditions are employed to optimize the global equilibrium. The results indicate that model can achieve Stackelberg equilibrium between the hierarchical decision-makers with conflicting objectives, and the subsidy allocation is highly equitable. Sensitivity analyses show that the carbon tax subsidy policy can enhance biomass fuels utilization and encourage active participation of coal-fired power plants in achieving cleaner production. These findings provide a reference role for the authority to develop scientific subsidy mechanisms and coal-fired power plants to rationalize the planning of biomass-coal co-firing for power generation.

Outage performance and energy efficiency optimization of wireless-powered millimeter-wave sensor networks

AbstractWith the widespread use of wireless sensor networks, one of the most pressing concerns is extending the lifetime of the sensors. By deploying directional antenna arrays, millimeter wave (mmWave) is a possible candidate for wireless energy transfer (WPT). This paper investigates a beneficial combination of WPT and data transmission in a typical mmWave sensor network with Rayleigh channels, where a transmission interval can be divided into two sub-intervals. During the first sub-interval, one hybrid access point (HAP) employs beamforming techniques to transfer energy for serving multiple sensors within the service sector. The sensors then transmit their individual signal in turn to the HAP based on time division multiple address (TDMA) strategy by using the whole harvested energy. According to stochastic geometry, the exact and approximate expressions of beam outage probability for the considered system are determined, respectively. The optimal time allocation of energy harvesting and data transmission for sensors is examined in order to maximize the energy efficiency of the system. The optimization problem can be translated into corresponding parametric form, and the resulting optimization problem can be solved using the Lagrange dual method with Karush–Kuhn–Tucker (KKT) conditions. The numerical results show the variation trend of the beam outage probability under various parameters and verify the accuracy of the theoretical analyses. Furthermore, the simulation results illustrate that the proposed optimal time allocation strategy can significantly enhance the overall energy efficiency of the system compared with a similar scheme.

Distributed strategies for mixed equilibrium problems: Continuous-time theoretical approaches

In this paper, the mixed equilibrium problem is studied by employing a continuous-time multi-agent system, where the objective of agents is to cooperatively find a point from the feasible set such that the sum of bifunctions with a free-variable is nonnegative. Different from existing works on mixed equilibrium problems with only convex set constraints, we consider a more general case where the feasible set is constrained by a set of convex inequalities. Moreover, each agent only has access to the information of its own bifunction and inequality constraint function, as well as the information of a local convex set. To handle this problem, the Karush–Kuhn–Tucker condition that is suitable for continuous-time theoretical approaches is provided, and a continuous-time distributed primal–dual algorithm is proposed. Under mild assumptions on the graph and bifunctions, we prove that the states of all agents asymptotically converge to the common solution of the mixed equilibrium problem. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.

The charge‐discharge compensation pricing strategy of electric vehicle aggregator considering users response willingness from the perspective of Stackelberg game

AbstractWith the rapid increase of electric vehicle (EV) ownership, the impact of EV charging load on the power grid is becoming more and more prominent. To reasonably guide EV charging/discharging to participate in Demand Response (DR) and help the power grid achieve peak cutting and valley filling, the charge‐discharge compensation pricing strategy of EV Aggregator (EVA) considering user response willingness from the perspective of Stackelberg game is proposed. Firstly, EVA, as the leader, provides charge‐discharge compensation price, to maximise its income within a day, taking into account user satisfaction constraints. Secondly, a user response willingness model is established. User engagement is used to describe the change in the number of EV responses with the change of the charge‐discharge compensation price by EVA and select the random EV set that accepts EVA charge‐discharge guidance. Finally, EV, as a follower, conducts charging/discharging behaviour to minimise the charging cost. By using the Karush–Kuhn–Tucker (KKT) condition, strong duality theory and iterative method, the strategy equilibrium solution is solved. The results show that considering the user response willingness can effectively reduce the decision risk when EVA participates in bidding. Although EVA income slightly decreases considering the response willingness, the average user satisfaction increases by 0.1.

Solving Fuzzy Convex Programming Problems via a Projection Neural Network Framework

In this paper, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is given using a projection recurrent neural network (RNN) scheme. Since there is a few research for resolving of FNLOP by RNNs, we describe a new framework to solve the problem. By reformulating the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are obtained. Moreover, we utilize the KKT conditions into a RNN as a capable tool to solve the problem. Besides, the global convergence and the Lyapunov stability of the neuro-dynamic model are established. In the final step, some simulation examples are stated to validate the obtained results. Reported results are compared with some other previous neural networks.

On the gradient-enhanced damage model for a hyperelastic material

An approach for modeling damage in the material in a geometrically nonlinear setting is studied. We consider a gradient-enhanced damage approach where the damage variable is taken as an independent variable, and the fluxes conjugated to the damage gradient are computed. The concept of maximum dissipation is used to derive the global Kuhn–Tucker conditions. The total free energy function consists of two nonlocal terms corresponding to referential gradient of the nonlocal damage variable and a penalty term to bridge the local damage variable and the nonlocal damage variable. Based on the principle of minimum total potential energy, a coupled system of Euler–Lagrange equations is obtained and solved in weak form. These nonlinear system of equations is solved by a standard Newton–Raphson-type solution scheme. The regularization capabilities, as well as the accuracy and efficiency of the proposed damage models are demonstrated by some standard benchmark examples.

Modeling robust bi-level BCC production planning problem with uncertain carbon emission mechanism

This paper studies the biomass–coal co-firing (BCC) production planning problem under the carbon emission quota allocation (CEQA) mechanism. However, carbon emission parameters are uncertain due to many factors, such as the type of power units and coal quality. To address this challenge issue, this paper proposes a new globalized robust bi-level optimization model, where the uncertain parameters are characterized by a pair of uncertainty sets. In the proposed model, the upper government as the leader decides the CEQA mechanism, while the lower power plants as the followers develop the production planning according to the given CEQA mechanism. Moreover, based on the Lagrange duality theory and Karush–Kuhn–Tucker (KKT) conditions, the proposed model is equivalently converted into a computationally tractable single-level model. Finally, a practical case study in Shandong Province demonstrates that compared with the nominal bi-level and robust bi-level models, the proposed optimization model can not only effectively resist parameter uncertainty, but also be less conservative.

Another solution for some optimum allocation problem

We derive optimality conditions for the optimum sample allocation problem in stratified sampling, formulated as the determination of the fixed strata sample sizes that minimize the total cost of the survey, under the assumed level of variance of the stratified ? estimator of the population total (or mean) and one-sided upper bounds imposed on sample sizes in strata. In this context, we presume that the variance function is of some generic form that, in particular, covers the case of the simple random sampling without replacement design in strata. The optimality conditions mentioned above will be derived from the Karush-Kuhn- Tucker conditions. Based on the established optimality conditions, we provide a formal proof of the optimality of the existing procedure, termed here as LRNA, which solves the allocation problem considered. We formulate the LRNA in such a way that it also provides the solution to the classical optimum allocation problem (i.e. minimization of the estimator’s variance under a fixed total cost) under one-sided lower bounds imposed on sample sizes in strata. In this context, the LRNA can be considered as a counterparty to the popular recursive Neyman allocation procedure that is used to solve the classical problem of an optimum sample allocation with added one-sided upper bounds. Ready-to-use R-implementation of the LRNA is available through our stratallo package, which is published on the Comprehensive R Archive Network (CRAN) package repository.

Joint Optimization for Secure and Reliable Communications in Finite Blocklength Regime

To realize ultra-reliable low latency communications with high spectral efficiency and security, we investigate a joint optimization problem for downlink communications with multiple users and eavesdroppers in the finite blocklength (FBL) regime. We formulate a multi-objective optimization problem to maximize a sum secrecy rate by developing a secure precoder and to minimize a maximum error probability and information leakage rate. The main challenges arise from the complicated multi-objective problem, non-tractable back-off factors from the FBL assumption, non-convexity and non-smoothness of the secrecy rate, and the intertwined optimization variables. To address these challenges, we adopt an alternating optimization approach by decomposing the problem into two phases: secure precoding design, and maximum error probability and information leakage rate minimization. In the first phase, we obtain a lower bound of the secrecy rate and derive a first-order Karush–Kuhn–Tucker (KKT) condition to identify local optimal solutions with respect to the precoders. Interpreting the condition as a generalized eigenvalue problem, we solve the problem by using a power iteration-based method. In the second phase, we adopt a weighted-sum approach and derive KKT conditions in terms of the error probabilities and leakage rates for given precoders. Simulations validate the proposed algorithm.

Neural networks as an approximator for a family of optimization algorithm solutions for online applications

In this paper, we propose a sufficient condition at which a neural network can approximate a set of optimization algorithm solutions; we establish under which conditions a neural network can replace an optimization algorithm to solve a problem with the objective of safely deploying that network in a system where online solutions are necessary to simplify the hardware or allowing the processor to solve the optimization problem on time. To that end, first, we define the family of optimization problems to be addressed; then, we construct a vector with the parameters on which the solution depends, in order to propose a function based on the first-order Karush–Kuhn–Tucker conditions to find conditions under which the inverse of the proposed function maps the problem minimizer with respect to the constructed vector, we provide the sufficiency proof of, both, existence and feasibility of approximation by a neural network regarding the inverse function. Two case studies are proposed, one numerical case showing how a neural network can solve an optimization problem faster than popular solvers to illustrate how it can be implemented in applications where the computation time is tight, and the other case is a Model Predictive Control implementation with the optimization problem solver replaced by a neural network which allows a hardware downgrade; both cases are presented with time statistics comparisons.

Robust Optimization for the Two-Dimensional Strip-Packing Problem with Variable-Sized Bins

The two-dimensional strip-packing problem (2D-SPP) emerges as a notable variant of the cutting and packing (C&P) problem, aiming to optimize the arrangement of small rectangular items within unique strips with a fixed width and infinite height to minimize the usage of height. Despite extensive academic exploration, applying 2D-SPP solutions in industrial settings remains challenging. Two significant issues, often overlooked in academia yet frequently encountered in industrial contexts, are the uncertain demand for items, exacerbated by the bullwhip effect, and the need for diverse types of strips to cater to varying customer needs. Our paper addresses this academia–industry gap by proposing a robust optimization model for the uncertain 2D-SPP with variable-sized bins, aiming to manage the demand fluctuations within a box uncertainty set framework. Additionally, we employ the contiguous one-dimensional relaxation technique in conjunction with column generation to tighten the lower bound of the problem, thereby augmenting solution accuracy. Furthermore, we leverage the Karush–Kuhn–Tucker (KKT) condition to transform the model into a more tractable form, subsequently leading to an exact solution. Based on datasets from a real-life plastic-cutting company, comprehensive experiments validate the effectiveness and efficiency of our proposed relaxation method and algorithm, showcasing the potential for an improved industrial application of 2D-SPP solutions.

Optimization-based power management for battery/supercapacitor hybrid energy storage system with load estimation capability in a DC microgrid

This paper proposes a novel optimization-based power management strategy (PMS) for a battery/supercapacitor hybrid energy storage system (HESS) with a semi-active structure in a DC microgrid application. As the DC bus voltage regulation is the main purpose of the HESS, the supercapacitor control due to direct connection to the DC bus becomes more important. While, in the semi-active structure, there is no control over the supercapacitor. The proposed PMS solved this challenge by considering the supercapacitor current as a control target in determining the reference current of the battery. Extending the battery life span by drawing smooth current from the battery and responding the supercapacitor to load current changes, and charging the battery with a constant current as a new objective function, are the other optimization targets. This multi-objective problem is solved by the Karush–Kuhn–Tucker (KKT) condition which makes the proposed PMS suitable for real-time applications. In addition, in contrast to previous related studies, (i) the load current (grid excess/deficiency power) as one of the important inputs of PMS is estimated by the adaptive Kalman filter (AKF), and (ii) the converter control signal is determined by system average model instead of conventional PI controller. Finally, using simulation, the performance of the proposed PMS has been verified by comparing the results with fuzzy logic, droop control, and adaptive filter-based methods. As well, the presented control strategy has been implemented on an experimental setup, and its performance is investigated.

Generation side strategy and user side cost based on equilibrium analysis of the power market under the reliability option

The reliability option (RO) ensures generation capacity adequacy, drawing inspiration from financial product options. However, most recent RO studies overlook the gaming strategies employed by power producers. This paper proposes a bi-level model based on an equilibrium analysis of the power market under RO. The upper-level model aims to maximize the power producers’ profit, whereas the lower-level model focuses on market clearing. Profit calculation considers factors like electricity sales income, option fees income, fuel cost, expansion investment cost, refund cost, and penalties. The bi-level model is reformulated into a mathematical problem with equilibrium constraints (MPEC) for each producer using Karush–Kuhn–Tucker conditions of the lower-level model. We then apply a series of linearization methods. The market equilibrium is obtained through the dynamic game framework for all MPECs. We also provide a comprehensive purchase cost calculation and demonstrate its equivalence to user side surplus. We examine cases from various spot market construction maturities in a provincial area of China. Our results indicate that implementing the RO mechanism and aligning parameters with the proposed optimal RO parameter curve, one can reduce the comprehensive purchase cost.

Dynamic offloading strategy for computational energy efficiency of wireless power transfer based MEC networks in industry 5.0

Wireless power transfer (WPT) has emerged as a promising solution for delivering services to low-power Internet of Things (IoT) devices in a demand-driven manner. In this work, we consider the Wireless power-enabled Hybrid Mobile Edge Cloud (WPHMEC) network, which utilizes a dynamic offloading strategy (partial and binary) to maximize computational efficiency while minimizing device energy consumption. To address this challenge, we formulate a convex optimization problem to maximize the number of computational bits and minimize the energy consumption of the IIoT devices in the WPT-based MEC system for Industry 5.0 applications. To achieve this goal, we employ a reformulation approach based on block coordinate descent (BCD) to formulate an optimization problem that addresses the nonconvexity of the problem and propose a solution approach using the Karush–Kuhn–Tucker (KKT) conditions. To validate the effectiveness of the proposed scheme, extensive simulations are carried out using the Matlab software to evaluate the system’s performance and components. The results demonstrate that optimal resource allocation maximizes energy efficiency and enhances computational resource utilization, making WPHMEEC an ideal choice for addressing the evolving demands of Industry 5.0 applications.

Model and solution of sustainable bi-level emergency commodity allocation based on type-2 fuzzy theory

The allocation of emergency commodities is very important for efficient emergency response after a disaster. In this study, a sustainable multi-objective bi-level emergency commodity allocation model is established, providing a practical emergency commodity allocation strategy for humanitarian supply chains. First, type-2 fuzzy variables are used to describe uncertain transportation costs and handle them based on the conditional value-at-risk (CVaR) theory of type-2 fuzzy variables. Second, Karush-Kuhn–Tucker (KKT) conditions with complementary constraints are used, auxiliary variables are introduced, and global criteria are applied to achieve the equivalent transformation of the model. Finally, with the Wenchuan earthquake in China in 2008 as an example, we prove the necessity of introducing the CVaR for risk measurement and the practicability of using type-2 fuzzy variables to describe uncertain parameters by comparing this scenario with a risk-neutral situation and the proposed model with a model based on type-1 fuzzy theory. The novelties of this study include considering both the government and suppliers are decision-makers, exploiting CVaR to measure tail risk and using type-2 fuzzy theory to transform uncertain parameters. The contributions of this study include considering sustainability during the rescue process and proposing a CVaR formula based on type-2 theory.

Bilevel Programming for Manufacturers Operating in an Omnichannel Retailing Environment

In this article, we present a joint production, pricing, and inventory control problem that can help manufacturers manage revenue in an omnichannel environment. This article puts forward a multiperiod Stackelberg game between a manufacturer and a retailer, assuming the former to be the leader. The leader–follower game is formulated as a mixed-integer nonlinear bilevel optimization problem wherein both players seek to maximize their respective profits. The article envisions an omnichannel retailing environment where online, offline, direct, and drop-shipping channels coexist. It then investigates how enabling showrooming and webrooming on the drop-shipping channel account can affect the supply-chain profitabilities. The analyses suggest that when the customers are aided with showrooming and webrooming services and are allowed to place drop-shipping orders from retailers’ stores, additional profit can be generated in the supply chain. To solve the proposed bilevel optimization problem, we investigate single-level reduction using Karush–Kuhn–Tucker conditions and hierarchical optimization technique based on the simulated annealing and randomized decomposition solver. To assess the efficacy of the solution techniques, a comparative analysis is carried out. Thereafter, with the aid of numerical experiments and sensitivity analyses, this article draws the key managerial insights for manufacturers.

A novel distributed approach for event-triggered economic dispatch of energy hubs under ramp-rate limits integrated with sustainable energy networks

This paper investigates a new consensus-oriented distributed approach for the event-triggered (ET) economic dispatch problem (EDP) over a smart grid under ramp-rate limits (RRLs) integrated with green power sources (GPSs) such as solar and wind energy for demand response strategies over hybrid energy power systems. To address the RRL condition, the authors have transformed the RRLs as minimum and maximum bounds on the derivative of generation for a generator. Then, a Karush–Kuhn–Tucker (KKT) condition and a more practical approximate KKT condition are developed for determining the optimality conditions. A practical ET protocol is proposed over a topology between generators by application of the proposed approximate KKT condition. In contrast to existing distributed optimization methods, this paper provides both optimally condition and distributed optimization scheme for dealing with RRLs integrated with sustainable hybrid energy systems. In addition, a computationally efficient ET mechanism, eliminating Zeno behaviour, has been considered for dealing with the efficient utilization of communication resources. This study incorporates the real-time input data from thermal production plants and GPSs for experimental analysis. RETScreen software having data-set of over 6,700 local meteorological stations is applied to obtain input data for GPSs. Furthermore, Weibull and Beta distribution functions have been applied for dealing with uncertainty in wind and solar energy sources. Two case studies are examined (with and without GPSs), and simulation results demonstrated the suitable performance of the proposed distributed ET EDP approach.