Tucker Conditions Research Articles

An Active-Set Fischer–Burmeister Trust-Region Algorithm to Solve a Nonlinear Bilevel Optimization Problem

In this paper, the Fischer–Burmeister active-set trust-region (FBACTR) algorithm is introduced to solve the nonlinear bilevel programming problems. In FBACTR algorithm, a Karush–Kuhn–Tucker (KKT) condition is used with the Fischer–Burmeister function to transform a nonlinear bilevel programming (NBLP) problem into an equivalent smooth single objective nonlinear programming problem. To ensure global convergence for the FBACTR algorithm, an active-set strategy is used with a trust-region globalization strategy. The theory of global convergence for the FBACTR algorithm is presented. To clarify the effectiveness of the proposed FBACTR algorithm, applications of mathematical programs with equilibrium constraints are tested.

Modeling the Truck Appointment System as a Multi-Player Game

Background: Random truck arrivals at maritime container terminals are one of the primary reasons for gate congestion. Gate congestion negatively affects the terminal’s and drayage firms’ productivity and the surrounding communities in terms of air pollution and noise. To alleviate gate congestion, more and more terminals in the USA are utilizing a truck appointment system (TAS). Methods: This paper proposes a novel approach to modeling the truck appointment system problem. Unlike previous studies which largely treated this problem as a single-player game, this study explicitly models the interplay between the terminal and drayage firms with regard to appointments. A multi-player bi-level programming model is proposed, where the terminal functions as the leader at the upper-level and the drayage firms function as followers at the lower-level. The objective of the leader (the terminal) is to minimize the gate waiting cost of trucks by spreading out the truck arrivals, and the objective of the followers (drayage firms) is to minimize their own drayage cost. To make the model tractable, the bi-level model is transformed to a single-level problem by replacing the lower-level problem with its equivalent Karush–Kuhn–Tucker (KKT) conditions and the model is solved by finding the Stackelberg equilibrium in one-shot simultaneous-moves among players. For comparison purposes, a single-player version of the TAS model is also developed. Results: Experimental results indicate that the proposed multi-player model yields a lower gate-waiting cost compared to the single-player model, and that it yields higher cost savings for the drayage firms as the number of appointments per truck increases. Moreover, the solution of the multi-player model is not dependent on the objective function coefficients, unlike the single player model. Conclusions: This study demonstrates that a TAS is more effective if it considers how the assigned appointment slot affects a truck’s drayage cost. It is recommended that terminal operators and port authorities initiate conversations with their TAS providers about incorporating this element into their TAS.

Stochastic differential reinsurance and investment games with delay under VaR constraints⋆

This article investigates stochastic differential reinsurance and investment games with delay under Value-at-Risk (VaR) constraints. In our model, three players and two games are considered, i.e., a non-zero-sum stochastic differential game between two insurers and a stochastic Stackelberg differential game between the reinsurer and the two insurers are investigated. The reinsurer can set the reinsurance premium price and invest her surplus in a financial market consisting of a risk-free asset and a risky asset. The two insurers can purchase proportional reinsurance and invest in the same financial market. Since the delay is considered in this article, the wealth processes of three players are described by the stochastic delay differential equations. In this article, we establish two optimization problems, one is to maximize the expected utility of the reinsurer’s terminal wealth with delay, and the other is to maximize the expected utility of the combination of each insurer’s terminal wealth and relative performance with delay. Furthermore, in order to control risk, we incorporate the VaR constraints into the optimizations and derive the corresponding value functions and Nash equilibrium strategies by using optimal control theory, dynamic programming principle, backward induction, Lagrange function and Karush–Kuhn–Tucker condition. Finally, several numerical examples are given.

A Latency-Aware Offloading Strategy over Fiber-Wireless (FiWi) Infrastructures for Tactile Internet Services

With the emergence of the tactile internet, low-latency, even, real-time data transmission is indispensable for human-agent–robot teamwork. Offloading is considered a feasible approach. Determining the offloading solution according to the dynamic network circumstance is attractive. In this paper, we investigate the resource management issue in a three-tier, heterogeneous, fiber-wireless (FiWi) network with offloading. Based on the model of the wireless link, the fiber link, the data rate, and the offloading, a mixed-integer, non-linear problem is formulated to obtain the minimum total latency for tactile internet services. Through constraint relaxation, MINLP is converted to a linear problem (LP). A Lagrange multiplier method with Karush–Kuhn–Tucker (KKT) conditions is used to solve LP. Using the numerical simulation, the superiority of our work is evaluated and compared with the previous work.

Optimal Decisions and Implementation Efficiency of the Extended Producer Responsibility System for E-Waste under Take-Back Legislations

The collection and reuse of the e-wastes have attracted increasing attention all over the world. In order to incentivize the producer to collect and recover the e-waste, more and more countries have formulated the take-back legislation with the economic or administrative instruments based on the spirit of extended producer responsibility (EPR). Facing two different types of policy instruments (economic or administrative instrument), how the take-back legislation affects the collection and reuse of the e-waste and how to ensure the efficiencies of the two types of take-back legislation have attracted the public’s attention. Therefore, this paper establishes a stylized model consisting of a monopoly manufacturer who is responsible for collection and remanufacturing under two different types of the take-back legislation. The manufacturer’s optimal production, collection, remanufacturing decisions, and optimal profit are derived using Karush–Kuhn–Tucker (KKT) conditions. Sensitivity analysis shows that stricter mandated collection rate hurts the manufacturer’s profit, but the effects of the government’s subsidy and tax on the manufacturer’s optimal decisions rely on the cost of the new product. The firm needs to adjust the cost of the new products corresponding to the specific subsidy and tax. Finally, a neutral fiscal policy is proposed to determine the government’s optimal subsidy and tax in order to ensure the consistency of the efficiencies of the two types of take-back legislation. The government should set that the ratio of the optimal tax and subsidy under the neutral fiscal policy is exactly consistent with the mandatory collection target. This policy aims to guide the manufacturer’s voluntary collection rate to meet the government’s mandated collection target and thus change the mandatory collection mode to the voluntary collection mode.

GPU-Accelerated PD-IPM for Real-Time Model Predictive Control in Integrated Missile Guidance and Control Systems.

This paper addresses the problem of real-time model predictive control (MPC) in the integrated guidance and control (IGC) of missile systems. When the primal-dual interior point method (PD-IPM), which is a convex optimization method, is used as an optimization solution for the MPC, the real-time performance of PD-IPM degenerates due to the elevated computation time in checking the Karush–Kuhn–Tucker (KKT) conditions in PD-IPM. This paper proposes a graphics processing unit (GPU)-based method to parallelize and accelerate PD-IPM for real-time MPC. The real-time performance of the proposed method was tested and analyzed on a widely-used embedded system. The comparison results with the conventional PD-IPM and other methods showed that the proposed method improved the real-time performance by reducing the computation time significantly.

A gradient micromorphic modeling for plasticity softening

Localization of internal quantities, which induce material softening, potentially leads to an ill-posed global partial differential equation. Along with further developments of singularity issues in the simulations, an unwanted divergent numerical solution may occur. The work at hand attempts to formulate a ductile softening model by considering a size reduction of the plastic yield surface. An internal variable to reduce the size of the plastic yield surface is constituted by a gradient micromorphic approach. In detail, a degradation function induced by an internal softening quantity is employed to multiply the thermodynamic force of plastic hardening in the yield function. In this regard, a gradually shrinking plastic yield surface is constituted to model the overall softening response. The evolution of the internal softening variable is governed by a Kuhn–Tucker condition together with its non-local extension. The present constitutive model of the coupled softening problem is derived based on a thermodynamically consistent algorithm from a well-defined Helmholtz free energy potential, which is implemented into the context of a conventional Finite Element Method. A representative and meaningful numerical example is studied to demonstrate the capability of the present model.

Tractable model discrimination for safety–critical systems with disjunctive and coupled constraints

This paper considers the model discrimination problem among a finite number of models in safety–critical systems that are subjected to constraints that can be disjunctive and where state and input constraints can be coupled with each other. In particular, we consider both the optimal input design problem for active model discrimination that is solved offline as well as the online passive model discrimination problem via a model invalidation framework. To overcome the issues associated with non-convex and generalized semi-infinite constraints due to the disjunctive and coupled constraints, we propose some techniques for reformulating these constraints in a computationally tractable manner by leveraging the Karush–Kuhn–Tucker (KKT) conditions and introducing binary variables, thus recasting the active and passive model discrimination problems into tractable mixed-integer linear/quadratic programming (MILP/MIQP) problems. When compared with existing approaches, our method is able to obtain the optimal solution and is observed in simulations to also result in less computation time. Finally, we demonstrate the effectiveness of the proposed active model discrimination approach for estimating driver intention with disjunctive safety constraints and state–input coupled curvature constraints, as well as for fault identification.

Competitive facility location problem with foresight considering discrete-nature attractiveness for facilities: Model and solution

This paper addresses a bi-level mixed-integer nonlinear programming (MINLP) model for the competitive facility location problem in a closed-loop supply chain (CLSC), in which a firm (i.e., leader) aims at entering a market by locating new distribution and collection facilities, where a competitor (i.e., follower) already exists. The goal is to find the location and attractiveness of each facility going to be established by the leader who seeks to maximize its profit while also taking the follower’s response into account. The attractiveness of each facility is a function of integer variables related to the facility’s characteristics. Customer behavior is considered to be probabilistic based on the Huff gravity-based rule. To globally optimize the model, a procedure that handles the discrete decisions of the follower’s problem is proposed. Afterward, by replacing the inner level convex program with its corresponding Karush–Kuhn–Tucker (KKT) conditions, the bi-level MINLP is converted into a single-level MINLP model, optimized by an improved branch-and-refine algorithm. Numerical experiments on randomly generated instances are conducted to illustrate the model’s applicability. Moreover, through a computational analysis of the proposed model, the amount of gain the leader makes and the follower loses due to foresight in the competition are calculated.

Parametric Transient Stability Constrained Optimal Power Flow Solved by Polynomial Approximation Based on the Stochastic Collocation Method

To better respond to the impact of power system-uncertain parameters on transient stability, a novel model named the parametric transient stability constrained optimal power flow (parametric TSCOPF) is proposed. It seeks the optimal control scheme of transient stability constrained optimal power flow (TSCOPF) expressed by the function of uncertain parameters in power systems. The key difficulty to solve this model lies in that the relationship between the parametric TSCOPF solution and uncertain parameters is implicit, which is hard to derive generally. To this end, this paper approximates the optimal solution of parametric TSCOPF by polynomial expressions of uncertain parameters based on the stochastic collocation method. First, the parametric TSCOPF model includes both uncertain parameters and transient stability constraints, in which the transient stability constraint is constructed as a set of polynomial expressions using the SCM. Then, to derive the relationship between the parametric TSCOPF solution and uncertain parameters, the SCM is applied to the parametric Karush–Kuhn–Tucker (KKT) conditions of the parametric TSCOPF model, so that the optimal solution of the parametric TSCOPF is approximated by using polynomial expressions with respect to uncertain parameters. The proposed parametric TSCOPF model has been tested on a 3-machine, 9-bus system, and the IEEE 145-bus system, which verifies the effectiveness of the proposed method.

Leveraging the flexibility of electric vehicle parking lots in distribution networks with high renewable penetration

The ongoing rapid increase in the integration of variable and uncertain renewable energy sources calls for enhancing the ways of providing flexibility to power grids. To this end, we propose an optimal approach for utilizing electric vehicle parking lots to provide flexibility at the distribution level. Accordingly, we present a day-ahead scheduling model for distribution system operators, where they can offer discounts on the network tariff to electric vehicle parking lot operators. This way, they will be encouraged to exploit the potential flexibility of electric vehicle batteries to assist in alleviating the steep ramps of system net-load. To determine the optimal discounts, the distribution system operator minimizes the network operating costs considering the network operational constraints, while the electric vehicle parking lot operators try to maximize their profits. Due to the contradictory objectives and decision hierarchy, the problem is an instance of Stackelberg games and can be formulated as a bi-level program, which is linearized and converted to a single-level mixed-integer linear program using strong-duality theorem and Karush–Kuhn–Tucker conditions. To validate the proposed model, comprehensive simulation studies are performed on a test distribution network. The simulation results show that implementing the model can reduce the peak-off-peak difference and peak-to-average ratio of the network net-load by up to 15% and 24%, respectively.

A Fair and Cooperative MAC Protocol for Heterogeneous Cognitive Radio Enabled Vehicular Ad-Hoc Networks

Advancements in intelligent transportation necessitate a dependable medium access control (MAC) protocol to enable high-priority safety broadcasts. Implementing cognitive radio-based vehicular ad-hoc networks (CR-VANETs) alleviates the significant growth in spectrum demand. The architecture must deploy multiple networks to make CR-VANET viable; however, the MAC design is a substantial challenge. The hardware-limited vehicles have varied characteristics and do not attain the same performance when provided with equal resources, posing a fairness concern. The resource allocation should balance two conflicting objectives: performance should be increased, prioritizing a few users under better conditions, and fairness among users should be emphasized, which reduces the overall performance. This paper presents a Fair and Cooperative MAC protocol for heterogeneous CR-VANETs (FCCR-MAC) to find this balance. In the event of a primary user appearance, cooperation allows pro-active channel switching. In FCCR-MAC, mode selection and retransmission of unsuccessful broadcasts are enabled by including additional control signals. By assisting users in determining the identities of others, cooperation also improves the system’s fairness. Markov Chain and the Karush–Kuhn–Tucker (KKT) conditions are utilized to offer a unified mathematical explanation of the throughput versus fairness trade-off optimization problem. According to the findings, the FCCR-MAC protocol outperforms other protocols with better fairness.

Energy Efficiency Optimization Based on Power Allocation in Massive MIMO Downlink Systems

To solve the energy efficiency (EE) optimization in a multi-cell (MU) massive multiple-input multiple-output (MIMO) downlink system, of which channels are of symmetry in the Time-Division Duplex (TDD) protocol, we utilize a spatially correlated channel model and adopt the minimum mean-squared error (MMSE) estimator to tailor linear precoding vectors. Then, we derive the expression of downlink spectral efficiency (SE), taking interference into account. Subsequently, we establish the EE optimization function, which is defined as the average capacity divided by power consumption. In an interference-limited scenario, the EE optimization is of high complexity to solve globally as it is not jointly concave. To this end, we propose the Dinkelbach-like power allocation algorithm to obtain a suboptimal solution. We transform the EE problem in a fractional form into a subtractive optimization form called an auxiliary subproblem. Then, we relax the sub-problem to a concave problem by initializing the interference and omitting the dynamic power term about throughput. Lastly, we solve iteratively the Karush–Kuhn–Tucker (KKT) conditions by bisection search. Consequently, we obtain a sub-solution with modest complexity. The simulation results justify the rationality of the Dinkelbach-like algorithm and demonstrate that the proposal outperforms the reference schemes and effectively improves the performance metrics EE and SE.

Resource Allocation for IRS-Assisted Wireless-Powered FDMA IoT Networks

This article investigates intelligent reflecting surface (IRS)-assisted wireless-powered Internet of Things (IoT) networks. Specifically, multiple IoT devices first collect energy radiated from a power station (PS), then each device uses its harvested energy to support data transmission to an access point (AP) via frequency-division multiple access (FDMA). In addition, an IRS aims to improve wireless energy transfer (WET) and wireless information transfer (WIT) capabilities using passive reflection beamformers. The system sum throughput, as a performance metric, is maximized evaluate the overall performance of the considered model, which is subject to the constraints of IRS phase shifts, transmission time scheduling, and bandwidth allocation. This problem is not convex with respect to multiple coupled variables, and cannot be directly solved. To circumvent this nonconvexity, the transmission time scheduling and the bandwidth allocation are optimally designed in the closed form by the Lagrange dual method and the Karush–Kuhn–Tucker (KKT) conditions. Moreover, an alternating optimization (AO) algorithm is used to optimally design the IRS’s phase shifts during the WET and WIT phases in an alternating fashion. Specifically, we propose elementwise block coordinate decent (EBCD) and complex circle manifold (CCM) algorithms to iteratively derive the optimal phase shifts in the closed form. We also characterize the convergence behavior of the proposed algorithms. Finally, numerical results are presented to validate the performance of the proposed scheme, where the benefits of the IRS are highlighted in terms of sum throughput, transmission time scheduling, and energy harvesting, compared with the benchmark schemes.

On improving the efficiency of modifier adaptation via directional information

In real-time optimization, the solution quality depends on the model ability to predict the plant Karush–Kuhn–Tucker (KKT) conditions. In the case of non-parametric plant-model mismatch, one can add input-affine modifiers to the model cost and constraints as is done in modifier adaptation (MA). These modifiers require estimating the plant cost and constraint gradients. This paper discusses two ways of reducing the number of input directions, thereby improving the efficiency of MA in practice. The first approach capitalizes on the knowledge of the active set to reduce the number of KKT conditions. The second approach determines the dominant gradients using sensitivity analysis. This way, MA reaches near plant optimality efficiently by adapting the first-order modifiers only along the dominant input directions. These approaches allow generating several alternative MA schemes, which are analyzed in terms of the number of degrees of freedom and compared in a simulated study of the Williams–Otto plant.

Optimizing performance of a rover on partially known terrain

One of our earlier articles (Noble and Issac, 2019) proposed a method for finding optimal wheel torques for a six-wheeled rover for two types of objective functions, namely, (a) friction requirement and (b) torque requirement. In this technical note, an approach for seeking the optimal torque of a rover when the coefficient of friction under one wheel is proposed (Noble, 2021). The approach is demonstrated for a rover climbing a large step. The solutions are verified to be a local minima by checking Karush–Kuhn–Tucker conditions. The solutions indicate that when the coefficient of friction available in a patch is lower than the minimum required, the other two wheels require a higher coefficient of friction than the minimum required. In cases where the available friction is higher than the minimum required, there could be situations when two wheels need to apply only very low torques, or no torques at all, for the rover to climb.

Optimal marketing channel and strategy in social commerce

This paper aims to explore the optimal marketing channel of e-commerce enterprises with various consumer base size and product added-value in social commerce. By characterizing the consumer utility, we construct decision-making models of e-commerce enterprises under the traditional e-commerce platform (TECP) and two kinds of social commerce marketing channels involving “e-commerce platform + social” (ECPS) and “social media + business’ (SMB). Then, we obtain the optimal strategy of e-commerce enterprises for each channel by using the Karush–Kuhn–Tucker (KKT) conditions, and verify the effectiveness of the corollaries by numerical simulation. Results find that social commerce marketing channels (ECPS and SMB) are not necessarily better than TECP. For e-commerce enterprises with low consumer base size and low product added-value, the TECP channel offers the largest profit. Besides, SMB is the preferred marketing channel for most e-commerce enterprises.

Real-time pricing for smart grid considering energy complementarity of a microgrid interacting with the large grid

Currently, generating electricity relies primarily on fossil energy, which is not only limited in resources but also harmful to the environment as it emits carbon dioxide and other gases. For contributing to carbon peaking and carbon neutrality, building a resource-saving and environment-friendly society, clean energy should be used for generating electricity as much as possible. Additionally, the overproduction of electricity by suppliers and the instability of power grids, which are caused by the information asymmetry between customers and power suppliers, can be solved by the smart grid based on smart meters. Motivated by the two aspects, a smart power system combining a microgrid system based on clean energy generation with a supplier based on fossil energy generation is considered. For pursuing maximal social welfare under joint power supply, a real-time pricing model is established. This model, which is convex, does not explicitly contain the electricity price variable such that the optimal Lagrange multiplier of its Karush–Kuhn–Tucker condition, i.e., shadow price, is taken as the optimal electricity price. For solving the Karush–Kuhn–Tucker system, the smooth Levenberg–Marquardt algorithm is used to solve its approximate smooth system of equations. The simulation results demonstrate that the proposed pricing model, joint supply mechanism and the smooth Levenberg–Marquardt algorithm are reasonable, feasible and effective.

Asteroid Flyby Cycler Trajectory Design Using Deep Neural Networks

Asteroid exploration has been attracting more attention in recent years. Nevertheless, we have just visited tens of asteroids, whereas we have discovered more than 1 million bodies. As our current observation and knowledge should be biased, it is essential to explore multiple asteroids directly to better understand the remains of planetary building materials. One of the mission design solutions is utilizing asteroid flyby cycler trajectories with multiple Earth gravity assists. An asteroid flyby cycler trajectory design problem is a subclass of global trajectory optimization problems with multiple flybys, involving a trajectory optimization problem for a given flyby sequence and a combinatorial optimization problem to decide the sequence of the flybys. As the number of flyby bodies grows, the computation time of this optimization problem expands maliciously. This paper presents a new method to design asteroid flyby cycler trajectories utilizing a surrogate model constructed by deep neural networks approximating trajectory optimization results. Because one of the bottlenecks of machine learning approaches is the heavy computation time to generate massive trajectory databases, we propose an efficient database generation strategy by introducing pseudo-asteroids satisfying the Karush–Kuhn–Tucker conditions. The numerical result applied to Japan Aerospace Exploration Agency’s mission shows that the proposed method is practically applicable to space mission design and can significantly reduce the computational time for searching asteroid flyby sequences.

Decisions on sailing frequency and ship type in liner shipping with the consideration of carbon dioxide emissions

To find the optimal sailing frequency and optimal ship type of container liner operation under low carbon environment, a mixed integer nonlinear programming model is proposed, in which the ship load capacity constraint is considered. The objective of the proposed model is to minimize voyage total cost consisting of carbon dioxide emissions cost, voyage shipping cost(fuel costs, port charges, operating costs) as well as shipper’s inventory cost in a certain operation period. It is shown from theoretical analysis that the proposed model can be solved by Kuhn–Tucker condition and Enumeration method. A numerical example is given to demonstrate the effectiveness of the proposed model, and analyze the impacts of carbon tax levels, fuel prices, ship size and the sailing frequency on total cost and environmental cost. The results show that as the size of the ship increases, the ship total cost decreases first and then increases, the percentage of carbon emission cost in total ship cost increases. As the sailing frequency increases, the ship total cost decreases first and then increases, the percentage of carbon emission cost in total ship cost decreases. The results indicate that the shipping companies can improve economic performance and environment performance simultaneously by employing reasonable sailing frequency and ship type decisions.