Conditions For Optimal Solution Research Articles

On optimality for fuzzy optimization problems with granular differentiable fuzzy objective functions

In recent years, there has been a growing interest in optimization problems with uncertainty. Fuzzy optimization is one of the approaches for investigation of such real-world extremum problems that contain uncertain data and, therefore, that are not well defined. Nonetheless, there is not enough discussion on the optimality conditions for optimal solutions (with regard to distinct fuzzy numbers) in (nonconvex) granular differentiable fuzzy optimization problems and methods for solving such uncertain extremum problems. Although the convexity notion is a very important property of optimization models, there are many real-world extremum problems and processes with uncertainty that cannot be modeled as convex fuzzy optimization problems. In this work, therefore, a new notion of granular differentiable generalized convexity is defined. Namely, the definition of granular differentiable F-convexity is introduced for fuzzy optimization problems and some properties of this concept are analyzed. Further, both the Fritz John necessary optimality conditions and, under the Guignard constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions are established for an optimal solution (with respect to distinct fuzzy numbers) in a fuzzy extremum problem with granular differentiable fuzzy-valued objective function and both inequality and equality constraints. As main applications of the introduced concept of granular differentiable F-convexity, the sufficiency of the derived Karush-Kuhn-Tucker necessary optimality conditions are proved and a new approach based on the so-called modified F-objective function method is proposed for solving the investigated fuzzy extremum problem.

Minimax optimal control problems for an extensible beam equation with uncertain initial velocity

This paper is devoted to the problem of minimax optimal control problems of an extensible beam equation with distributed controls and initial velocity disturbances (or noises). The existence of optimal solutions for distributed control with fixed disturbance, namely the (P_{v}) problem, and the existence of minimax optimal solutions to (P) problem without restricting the initial disturbance are proved. We derive the necessary optimality conditions for optimal solutions of the (P_{v}) and (P) problems in the form of Pontryagin’s maximum principle.

On the use of adjoint gradients for time-optimal control problems regarding a discrete control parameterization

AbstractIn this paper, we discuss time-optimal control problems for dynamic systems. Such problems usually arise in robotics when a manipulation should be carried out in minimal operation time. In particular, for time-optimal control problems with a high number of control parameters, the adjoint method is probably the most efficient way to calculate the gradients of an optimization problem concerning computational efficiency. In this paper, we present an adjoint gradient approach for solving time-optimal control problems with a special focus on a discrete control parameterization. On the one hand, we provide an efficient approach for computing the direction of the steepest descent of a cost functional in which the costs and the error in the final constraints reduce within one combined iteration. On the other hand, we investigate this approach to provide an exact gradient for other optimization strategies and to evaluate necessary optimality conditions regarding the Hamiltonian function. Two examples of the time-optimal trajectory planning of a robot demonstrate an easy access to the adjoint gradients and their interpretation in the context of the optimality conditions of optimal control solutions, e.g., as computed by a direct optimization method.

Towards energy-efficient and time-sensitive task assignment in cross-silo federated learning

Federated learning (FL) is a novel distributed learning framework in which clients with models can collaboratively train a high-quality global model while maintaining the privacy of their local data. The heterogeneity of datasets makes the global aggregation and local training requirements of each client in the cross-silo scenario inconsistent. Although many efforts are dedicated to improving the efficiency of cross-silo FL, the task assignment design of cross-silo FL remains an open question. To optimize the efficiency of energy consumption and training time of cross-silo FL, we develop a novel Energy-efficient and Time-sensitive Task Assignment (ETTA) mechanism. Specifically, we model clients’ requirements for energy efficiency and training time in cross-silo FL as an optimization problem, and design a utility transfer function to maintain the honesty of clients and the stability of ETTA. In terms of energy efficiency maximization, we analyze the conditions of optimal solutions based on the equilibrium of marginal cost and income, and design an approximate algorithm for cross-silo FL scenarios that require lightweight mechanisms. In terms of training time minimization, we model this objective as a min–max problem and give its analytical solution with Pareto optimality. To ensure that the task assignment achieves incentive compatibility, we further design a utility transfer selection function to incentivize honest collaboration among selfish clients. Finally, we conduct extensive experiments on the performance of ETTA to demonstrate that it outperforms the state-of-the-art and can maintain stability in a dishonest environment.

Price Competition, Experiential Quality and Online Marketing Decisions of a Tourism Supply Chain

ABSTRACT This study examines the price competition, experiential quality and online marketing decisions of a tourism supply chain, consisting of a tourism operator and two online retailers. The equilibrium solutions and the conditions for unique optimal solutions are derived in three supply chain strategies. This study also investigates performance improvement measures, including the operator’s involvement in online marketing and online reviews for quality improvement. An experimental analysis is conducted to compare supply chain performance and provide tourism managerial insights. The investigation provides a reference for tourism policies that could assist in improving experiential quality and operation performance.

Research on the Mechanism of Multi-Domain Coupling Centrifugal Electrostatic Blowing Flying Deposition.

The centrifugal electrostatic blowing process proposed in this paper solves the difficult continuous and stable deposition problem in the traditional centrifugal electrostatic spinning process. By establishing a flight deposition model of the centrifugal electrostatic spraying process, CFD is used to simulate and analyze the electrohydrodynamic effect of centrifugal jets, and the driving mechanism is explored. Subsequently, MATLAB is used to obtain the optimal solution conditions, and finally, the establishment of a two-dimensional flight trajectory model is completed and experimentally verified. In addition, the deposition model of the jet is established to clarify the flight trajectory under the multi-field coupling, the stable draft area of the jet is found according to this, and the optimal drafting station is clarified. This research provides new ideas and references for the exploration of the deposition mechanism of the centrifugal electrostatic blowing and electrostatic spinning process.

Modeling the transmission dynamics and vaccination strategies for human papillomavirus infection: An optimal control approach

Human papillomavirus (HPV) vaccines have been introduced in several countries and have shown positive results in reducing HPV infection and related diseases. Nevertheless, immunization programs remain sub-optimal and more effort is needed to design efficient vaccination deployment. We formulate a two-sex deterministic mathematical model that incorporates the most important epidemiological features of HPV infection and associated cancers. To assess the population-level impact of HPV immunization programs, the model incorporates school-based vaccine delivery for juveniles and catch-up vaccination for adults. The dynamics of the model are rigorously analyzed using the next-generation operator, the center manifold theorem, and normal forms theory. We formulate an optimal control problem to determine the best deployment strategy for HPV vaccination for several plausible scenarios. We establish the existence of solutions of the optimal control problem, and use Pontryagin’s Maximum Principle to characterize the necessary conditions for optimal control solutions. The findings suggest that if girls-only programs are complemented with catch-up vaccination for adult females, such program has the potential to achieve HPV-associated cancers eradication even if boys and males do not receive the vaccine. We also find that the optimal vaccine deployment, in terms of minimizing HPV associated diseases and the cost of vaccination, is to allocate as much vaccines as possible at the initial phase of the epidemic and once a high vaccination coverage is reached then gradually decrease vaccination rates.

Optimality conditions for fuzzy optimization problems under granular convexity concept

In this paper, under the condition of granular differentiation, we consider the fuzzy optimization problems with the general fuzzy function as the objective function. Firstly, we introduce the concept of granular convexity, and propose the properties of the granular convex fuzzy functions. Secondly, we present the Karush-Kuhn-Tucker type optimality conditions of the fuzzy relative optimal solution of more general fuzzy programming problems and some test examples. Finally, the relationships between a class of variational inequalities and the fuzzy optimization problems are established.

Optimization of Controlled Free-Time Sweeping Processes With Applications to Marine Surface Vehicle Modeling

This letter is devoted to a free-time optimal control problem for sweeping processes. We develop a constructive finite-difference approximation procedure that allows us to establish necessary optimality conditions for discrete optimal solutions and then show how these optimality conditions are applied to solving a controlled marine surface vehicle model.

Strong and weak conditions of regularity and optimality

Nondegenerate optimality conditions for Pareto and weak Pareto optimal solutions to multiobjective optimization problems with inequality and multi-equality constraints determined by Fréchet differentiable functions are established. First, weak and strong

ROLE OF MEDIA COVERAGE IN MITIGATING AN EPIDEMIC OUTBREAK: AN OPTIMAL CONTROL MODEL

Flu is an acute respiratory infection caused by the influenza virus. The outbreak usually occurs every year in temperate region during the fall and winter seasons, but it is present year-round in tropics. Perceived risk of infection, affordability and lack of awareness among the population results in a low level of vaccination coverage. To control disease transmission and promote vaccination, public health officials use media coverage to spread awareness on vaccine safety, vaccine coverage, disease prevalence in the population through public health websites, advertisements, and other social media web pages. Media coverage acts as an incentive as it helps to decrease overall transmission potential and also at the same time increases the vaccination coverage in the population. Since the public health department has a limited budget, it needs to make optimum allocation of its effort to reduce the total cost of infection. Our paper investigates the effect of media coverage using SIR model of disease transmission. We look at three possible functional relationships — linear, exponential, and hyperbolic — the way media coverage may affect the disease transmission and vaccination rate. We derive necessary conditions of optimal solution using Optimal Control Theory and Pontryagin Maximum Principle (PMP) to minimize the total cost for infection. Analysis of our paper demonstrates that the cost of optimal management is four times less than the cost of constant control effort, and putting more effort into reducing transmission is optimal rather than an effort to increase vaccination at the beginning of the outbreak. Analysis of the role of media coverage under three different scenarios may help in formulating policies for public health programs in mitigating the influenza outbreak.

Process Optimization for Manufacturing PAN-Based Conductive Yarn with Carbon Nanomaterials through Wet Spinning.

This study aimed to manufacture PAN-based conductive yarn using a wet-spinning process. Two types of carbon nanomaterials, multiwall carbon nanotubes (MWCNT) and carbon nanofiber (CNF), were used alone or in a mixture. First, to derive the optimal composite solution condition for the wet spinning process, a composite solution was prepared with carbon nanomaterials of the same total mass weight (%) and three types of mechanical stirring were performed: mechanical stirring, ultra-sonication, and ball milling. A ball milling process was finally selected by analyzing the viscosity. Based on the above results, 8, 16, 24, and 32 wt% carbon nanomaterial/PAN composite solutions were prepared to produce wet spinning-based composite films before preparing a conductive yarn, and their physical and electrical properties were examined. By measuring the viscosity of the composite solution and the surface resistance of the composite film according to the type and content of carbon nanomaterials, a suitable range of viscosity was found from 103 cP to 105 cP, and the electrical percolation threshold was from 16 wt% carbon nanomaterial/PAN, which showed a surface resistance of 106 Ω/sq or less. Wet spinning was possible with a PAN-based composite solution with a high content of carbon nanomaterials. The crystallinity, crystal orientation, tenacity, and thermal properties were improved when CNF was added up to 24 wt%. On the other hand, the properties deteriorated when CNTs were added alone due to aggregation. Mixing CNT and CNF resulted in poorer properties than with CNF alone, but superior properties to CNT alone. In particular, the electrical properties after incorporating 8 wt% CNT/16 wt% CNF into the PAN, 106 Ω/cm was similar to the PAN-based conductive yarn containing 32 wt% CNF. Therefore, this yarn is expected to be applicable to various smart textiles and wearable devices because of its improved physical properties such as strength and conductivity.

Optimal Denial-of-Service Attack Strategy on State Estimation Over Infinite-Time Horizon

Security issues of cyber-physical systems have become a timely and hot topic in recent years. In this brief, we investigate the power-constrained optimal denial-of-service (DoS) attack strategy with multiple power levels on state estimation over infinite-time horizon. We start from the two-level case and explicitly derive out the basic form of optimal attack strategy. In order to handle the intractability brought about by the larger number of levels, we develop an equivalent property for general multi-level cases that the multi-level scheduling can be viewed as a feasible combination of several sub-schedules with a designed stochastic-triggered manner, which provides a necessary condition of optimal solution. Moreover, based on the practical communication model, we give the condition under which the optimal schedule can be uniquely and explicitly determined for the two-pick strategy. Numerical simulations are accomplished to verify the effectiveness of our proposed methods.

Material profile reconstruction using plane electromagnetic waves in PML-truncated heterogeneous domains

This paper discusses a full-waveform inversion method for reconstructing the spatial distribution of permittivity in heterogeneous infinite domains, using measured electric field intensities at sparse sensor locations. In solving the electromagnetic wave problems numerically, perfectly matched layer (PML) absorbing boundaries are used to truncate the originally infinite extent to a finite computational domain of interest without introducing significant reflections. The full-waveform inversion is based on an optimization scheme where Maxwell’s equations endowed with the PML for plane electromagnetic waves are imposed as constraints. The approach seeks the optimal solution of permittivity profile to minimize the objective functional comprising the L2-norm of a misfit between calculated and measured electric fields. For casting the problem to an unconstrained optimization problem, a Lagrangian is constructed augmenting the objective functional with the PML-endowed Maxwell’s equations via Lagrange multipliers. Enforcing the stationarity of the Lagrangian yields time-dependent state, adjoint, and time-invariant control problems, which constitute Karush-Kuhn-Tucker (KKT) conditions for optimal solutions. The permittivity profile of the PML-truncated domain is iteratively updated by solving the KKT conditions in the reduced space of the control variable. A conjugate gradient method with inexact line search is used to update the permittivity profile in each inversion iteration. Tikhonov and total variation regularization schemes are explored to relieve the ill-posedness of the inverse problem. Through a set of numerical results, it is shown that both smooth and sharply-varying permittivity profiles can be recovered successfully using the proposed inversion method.

Effects of solution and aging treatments on the microstructure and mechanical properties of cold rolled 2024 Al alloy sheet

The effects of solution and aging on the microstructure evolution, mechanical properties and conductivity of cold rolled 2024 Al sheet were clarified. The results showed that the 2024 Al sheet mainly had the second phases of Al7Cu2Fe, S (Al2CuMg) and θ (Al2Cu). During the solution stage, S and θ phases gradually dissolved into Al matrix, resulting in a great improvement of mechanical properties and conductivity. The static recrystallization and grain growth also occurred during solution, while the grain growth was inhibited by T (Al20CuMn3) particles. GPB zones and S phases precipitated during artificial aging, which acted as the main strengthening phases. Before peak-aging, the amount of GPB zones and S phases increased, and the strengthening effect was enhanced. The precipitates reduced the number of electron scattering sources in the matrix and thus enhanced the conductivity. The optimal solution condition was determined as 500 °C × 70 min and the optimal aging condition was determined as 180 °C × 16 h, by which the excellent ultimate tensile strength of 527 MPa, yield strength of 411 MPa, elongation of 12.5%, hardness of 148.9 HV and conductivity of 37.4%IACS were achieved.

주문취소가 발생하는 구매종속적 수요에 대한 EOQ 접근에 관한 고찰

We investigate the inventory models for purchase-dependent demands in the case in which customer orders are canceled if all items in the order are not delivered all at once. The exact objective function of an inventory model is quite difficult to derive when purchase dependence exists. Determining the optimal order quantity and reorder point is also a challenging task. Thus, we first analyze the properties of optimal solutions using an EOQ model to derive the condition of optimal solution. Then, we extend the condition to continuous review models. Through computer simulations, we demonstrate the usefulness of this study’s approach. In general, the inventory model, which considers the purchase dependence, reduces the loss of sales by holding more inventories through more frequent orders, thus eventually minimizing the total cost.

Solving Optimization Problems over the Weakly Efficient Set

In this paper, we study the optimization problem (PWE) of minimizing a convex function over the set of weakly efficient solutions of a convex multiobjective problem. This is done by using the fact that each lower semicontinuous convex function is an upper envelope of its affine minorants together with a generalized cutting plane method. We give necessary conditions for optimal solutions of the problem Moreover, a novel algorithm for solving the problem (PWE) together with numerical results are presented. We also prove that the proposed algorithm terminates after a finite numbers of iterations, and the algorithm is coded in MATLAB language and evaluated by numerical examples.

Fractional optimal control dynamics of coronavirus model with Mittag–Leffler law

Most countries around the world are battling to limit the spread of severe acute respiratory syndrome-coronavirus 2 (SARS-CoV-2). As the world strives to get an effective medication to control the disease, appropriate control measures for now remains one of the effective measures to reduce the spread of the disease. In this study, a fractional optimal control model is formulated in Atangana-Baleanu-Caputo derivative sense. The reproduction number and steady state of disease free of the Coronavirus model are examined and found to be globally stable. The existence and uniqueness of solution of the fractional Coronavirus model is established by using the Banach fixed point theorem approach. Three controls are considered in the model and Pontryagins Maximum Principle is used to establish the necessary conditions for optimal control solution. The numerical solution suggests that the best strategy is found to be the utilization of all three controls at the same time.

Optimality results for a specific fractional problem

<p style='text-indent:20px;'>In this paper, one minimizes a fractional function over a compact set. Using an exact separation theorem, one gives necessary optimality conditions for strict optimal solutions in terms of Fréchet subdifferentials. All data are assumed locally Lipschitz.</p>

Optimization and discrete approximation of sweeping processes with controlled moving sets and perturbations

This paper addresses a new class of optimal control problems for perturbed sweeping processes with measurable controls in additive perturbations of the dynamics and smooth controls in polyhedral moving sets. We develop a constructive discrete approximation procedure that allows us to strongly approximate any feasible trajectory of the controlled sweeping process by feasible discrete trajectories and also establish a W1,2-strong convergence of optimal trajectories for discretized control problems to a given local minimizer of the original continuous-time sweeping control problem of the Bolza type. Employing advanced tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete optimal solutions under fairly general assumptions formulated entirely in terms of the given data. The obtained results give us efficient suboptimality (“almost optimality”) conditions for the original sweeping control problem that are illustrated by a nontrivial numerical example.