Unique Optimal Solution Research Articles

Age-replacement policies under age-dependent replacement costs

We consider a stochastically deteriorating system with self-announcing failures that require immediate reactive replacement. For such a system, we consider an age-replacement policy (without minimal repair) under which the system is replaced at failure (reactive replacement) or at a prescribed replacement time (preventive replacement), whichever occurs first. Motivated by factors such as decreasing salvage value or increasing costs associated with obtaining spare parts, we assume that replacement costs are non-decreasing in system age. We formulate a long-run expected cost-rate minimization model with instantaneous replacements that captures this dependency, and provide conditions under which there exists a unique optimal solution. We provide analytical and numerical results that compare the cost-rate minimizing optimal replacement policy, and its performance, to those for the case in which replacement costs are assumed to be constant. Finally, we also consider non-instantaneous replacements, and compare cost-rate minimizing and availability maximizing policies.

Smoothness Regularized Multiview Subspace Clustering With Kernel Learning

Multiview subspace clustering has attracted an increasing amount of attention in recent years. However, most of the existing multiview subspace clustering methods assume linear relations between multiview data points when learning the affinity representation by means of the self-expression or fail to preserve the locality property of the original feature space in the learned affinity representation. To address the above issues, in this article, we propose a new multiview subspace clustering method termed smoothness regularized multiview subspace clustering with kernel learning (SMSCK). To capture the nonlinear relations between multiview data points, the proposed model maps the concatenated multiview observations into a high-dimensional kernel space, in which the linear relations reflect the nonlinear relations between multiview data points in the original space. In addition, to explicitly preserve the locality property of the original feature space in the learned affinity representation, the smoothness regularization is deployed in the subspace learning in the kernel space. Theoretical analysis has been provided to ensure that the optimal solution of the proposed model meets the grouping effect. The unique optimal solution of the proposed model can be obtained by an optimization strategy and the theoretical convergence analysis is also conducted. Extensive experiments are conducted on both image and document data sets, and the comparison results with state-of-the-art methods demonstrate the effectiveness of our method.

Fuzzy best-worst method based on triangular fuzzy numbers for multi-criteria decision-making

In this paper, we propose a new fuzzy best-worst method (BWM) based on triangular fuzzy numbers for multi-criteria decision-making (MCDM). Aimed at the Best-to-Others vector and the Others-to-Worst vector in the form of triangular fuzzy numbers, this paper regards consistency equations as fuzzy equations. The derivation of optimal fuzzy weights of criteria is formulated as a fuzzy decision-making problem, where a mathematical programming model is constructed to derive optimal fuzzy weights of criteria to build a normalized triangular fuzzy weight vector. Then, we propose four linear programming models based on the obtained mathematical programming model for the optimistic decision maker, the pessimistic decision maker and the neutral decision maker, respectively. Through a proper selection of the values of tolerance parameters, each of the linear programming models certainly has a unique global optimal solution. Moreover, this paper proposes the concept of fuzzy consistency index and the concept of fuzzy consistency ratio. Several application examples are used to validate the proposed fuzzy BWM. The proposed fuzzy BWM provides us with a very useful way for MCDM in fuzzy environments.

Optimal risk-sharing across a network of insurance companies

Risk transfer is a key risk and capital management tool for insurance companies. Transferring risk between insurers is used to mitigate risk and manage capital requirements. We investigate risk transfer in the context of a network environment of insurers and consider capital costs and capital constraints at the level of individual insurance companies. We demonstrate that the optimisation of profitability across the network can be achieved through risk transfer. Considering only individual insurance companies, there is no unique optimal solution and, a priori, it is not clear which solutions are fair. However, from a network perspective, we derive a unique fair solution in the sense of cooperative game theory. Implications for systemic risk are briefly discussed.

A method for tackling primal multiplicity of solutions of dynamic flux balance models

A method is presented to tackle primal multiplicity of Dynamic flux balance analysis (DFBA) which is a Linear Programming (LP) based modeling approach that assumes that the cell distributes fluxes such as to maximize a specific biological objective. When the LP problem has multiple optima, the LP solvers usually only report the first optimum that it is reached which may not fit well the experimental data. To tackle this primal multiplicity problem, the weighted primal-dual method with auxiliary parameters is used to calculate a unique time trajectory for a given set of initial conditions. Through tuning of these auxiliary parameters, a unique optimal solution can be obtained and calibrated to fit available experimental data. Beyond its capability to tackle multiplicity, the algorithm is shown to significantly improve the prediction of some metabolites in a case study of the fed-batch fermentation of Bordetella pertussis.

Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

SummaryThe main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.

Optimal extended warranty length with limited number of repairs in the warranty period

A policy plan is proposed for the warranty length of a repairable system which consists of a number of minimal repairs and duration of service. For this purpose, two total expected costs of the manufacturer are considered as the objective of the optimization model and the conditions for finding unique optimal solutions are investigated. To illustrate the performance of the theoretical results, numerical computations and graphical analysis are given when the lifetime of the original system follows an exponential distribution or a Weibull distribution. Finally, a real data set corresponds to the time between failures of a compressor is used to demonstrate the results.

Optimal Design of Antenna Arrays Focused on Multiple Targets

This article presents an optimal design method for antenna arrays focused on multiple targets. The optimal design process is characterized by a quadratically constrained quadratic programming (QCQP). The complicated QCQP problem is solved by introducing a weighted diagonal matrix. The radiated power from the transmitting antenna array can be distributed among the multiple targets according to predetermined pattern by adjusting the weighted diagonal matrix. Once the positions of the targets and the predetermined field focusing pattern are all set, the test receiving antennas may be introduced to replace the targets, and the power transmission efficiency between the transmitting antenna array and the test receiving antennas is then maximized, which guarantees the maximum possible power delivery to each target in a focused manner. A unique optimal solution for the distribution of excitations can be found explicitly by analytically solving the QCQP. Several systems are designed for focusing the power to multiple targets in closed and open regions to demonstrate the focusing performance of the transmitting array excited with the optimized distributions of excitations obtained from QCQP.

Optimal Control of the Double Integrator with Minimum Total Variation

We study the well-known minimum-energy control of the double integrator, along with the simultaneous minimization of the total variation in the control variable. We derive the optimality conditions and obtain the unique optimal solution to the combined problem, where the initial and terminal boundary points are specified. We study the problem from a multi-objective optimal control viewpoint, constructing the Pareto front. We show that the unique asymptotic optimal control function, for the minimization of the total variation alone, is piecewise constant with one switching at the midpoint of the time horizon. For any instance of the boundary conditions of the problem, we prove that the asymptotic optimal total variation is exactly 2/3 of the total variation of the minimum-energy control. We illustrate the results for a particular instance of the problem and include a link to a video which animates the solutions while moving along the Pareto front.

Distributed Optimization for Two Types of Heterogeneous Multiagent Systems.

This article studies distributed optimization algorithms for heterogeneous multiagent systems under an undirected and connected communication graph. Two types of heterogeneities are discussed. First, we consider a class of multiagent systems composed of both continuous-time dynamic agents and discrete-time dynamic agents. The agents coordinate with each other to minimize a global objective function that is the sum of their local convex objective functions. A distributed subgradient method is proposed for each agent in the network. It is proved that driven by the proposed updating law, the agents' position states converge to an optimal solution of the optimization problem, provided that the subgradients of the objective functions are bounded, the step size is not summable but square summable, and the sampling period is bounded by some constant. Second, we consider a class of multiagent systems composed of both first-order dynamic agents and second-order dynamic agents. It is proved that the agents' position states converge to the unique optimal solution if the objective functions are strongly convex, continuously differentiable, and the gradients are globally Lipschitz. Numerical examples are given to verify the conclusions.

Adaptive visual servoing with an uncalibrated camera using extreme learning machine and Q-leaning

In this paper, a novel image-based visual servoing (IBVS) method using Extreme Learning Machine (ELM) and Q-learning is proposed to solve the problems of complex modeling and selection of the servo gain. First, the pseudoinverse of the interaction matrix is approached by ELM which avoids the singularity of the interaction matrix effectively and is robust to interferences such as feature noises and camera calibration errors. Second, a reinforcement learning method, Q-learning, is adopted to adaptively adjust the servo gain in order to improve the convergence speed and stability. Compared with other methods, ELM has better generalization performance, faster operation speed and a unique optimal solution. Also, Q-learning has self-learning ability without experience in advance. The effectiveness of the proposed method is validated by simulations and experiment on a 6-DOF robot with eye-in-hand configuration.

Problem Formulation and Solution Methodology for Energy Consumption Optimization in Bernoulli Serial Lines

As the main force of energy consumption, machines consume a huge amount of energy in some production systems. To operate the systems in an energy-efficient way, the efficiencies of the machines should be elaborately optimized so that the total energy consumption in the production lines is minimized while maintaining the required production rate. For this purpose, this article investigates the energy consumption optimization problem in Bernoulli serial lines having more than two machines. Specifically, this problem is first formulated as a nonlinear programming with a production rate constraint; due to difficulties in solving this constrained nonlinear programming, structural characteristics of the problem and properties of the objective function, which are inspired by the results and insights obtained in two-machine lines, are analyzed. Finally, based on the structural characteristics and properties explored, a solution methodology is developed to recursively solve the unique optimal solution of the energy consumption optimization problem. Extensive numerical experiments show that the solution provided by this method, which has the reversibility property and is better than the one provided by an existing method in the literature, is numerically optimal (note that the theoretically optimal solution is extremely hard to obtain if not totally impossible). Note to Practitioners-Machines consume a huge amount of energy in energy-intensive (i.e., high energy-consuming) production systems in, e.g., automobile, semiconductor, and steel companies. Due to economic, social, and environmental concerns, it is a high priority to consider ways to reduce the energy consumption in these production systems. In this article, the problem of minimizing the total energy consumption in Bernoulli serial lines while maintaining the required production rate is investigated. Based on the aggregation method developed for performance analysis of long lines and the optimization results obtained for two-machine lines, the energy consumption optimization problem for long Bernoulli serial lines is elaborately analyzed and a method is proposed to recursively solve its unique optimal solution (i.e., the optimal machine efficiencies). From the point of view of system operation, this research, which makes the system operate in the most energy-efficient manner, offers a new managerial paradigm for the production systems.

Incentive-Based Integrated Demand Response for Multiple Energy Carriers Considering Behavioral Coupling Effect of Consumers

Incentive-based demand response (DR) has been recognized as a powerful tool to mitigate supply-demand imbalance in electricity market. However, it cannot be directly applied into integrated demand response (IDR) for multi-energy systems where there are energy substitution effect among different types of energy carriers and behavioral coupling effect of consumers. In this paper, an incentive-based IDR model for multiple energy carriers considering behavioral coupling effect of consumers is proposed. The proposed model not only effectively deals with the above two effects but also enhances applicability of incentive-based DR into the scenario where curtailment DR and absorbing DR are planned simultaneously. Furtherly, the proposed IDR model is also extended to multi-period coupling IDR considering the effect of energy storage unit. The solution methodologies for the proposed model and extended model are also presented. The existence and uniqueness of optimal solution are identified. Finally, simulation results verify merits of the proposed model in cutting down total cost of multi-energy aggregator (MEA), reducing dissatisfaction of consumers, improving accuracy of optimal solution, and improving the utilization of energy storage unit.

A Stabilizing Control Algorithm for Asynchronous Parallel Quadratic Programing via Dual Decomposition

Abstract This paper proposes a control algorithm for stable implementation of asynchronous parallel quadratic programing (PQP) through a dual decomposition technique. In general, distributed and parallel optimization requires synchronization of data at each iteration step due to the interdependency of data. The synchronization latency may incur a large amount of waiting time caused by an idle process during computation. We aim to mitigate this synchronization penalty in PQP problems by implementing asynchronous updates of the dual variable. The price to pay for adopting asynchronous computing algorithms is the unpredictability of the solution, resulting in a tradeoff between speedup and accuracy. In the worst case, the state of interest may become unstable owing to the stochastic behavior of asynchrony. The stability condition of asynchronous PQP problems is investigated by employing the switched system framework. A formal algorithm is provided to ensure the asymptotic stability of dual variables. Furthermore, it is shown that the implementation of the proposed algorithm guarantees the uniqueness of optimal solutions, irrespective of asynchronous behavior. To verify the validity of the proposed methods, simulation results are presented.

A dynamic advertising model in a vaccination campaign

Vaccines save thousands of lives every year, but many people remain unvaccinated because serious adverse neurological disorders are wrongly attributed to vaccination. The “urban myth” of a relevant vaccine-associated risk is sustained by anti-vaccination groups and it is spread by word-of-mouth communication. We face the problem of increasing the vaccination coverage using an approach which draws some elements from the theory of dynamic advertising models. We propose a dynamic model for the evolution of the number of unvaccinated people and assume that a policy-maker can control this dynamics through advertising. From a mathematical point of view, we state and analyze an optimal control problem with a pure state constraint. We find the unique optimal solution, which minimizes a cost functional, but may fail to be satisfactory from the different viewpoint of moving towards eradication of the disease. Our analysis suggests that we modify the problem statement in order to consider explicitly the goal of reducing the number of unvaccinated people, to a level which guarantees the herd immunity. Hence we introduce an upper bound to the final number of unvaccinated people. From the solutions to the two problems we obtain some prescriptions for the policy-maker.

Optimal Control of Nonclassical Diffusion Equations with Memory

In this paper we consider an optimal control problem of nonclassical diffusion equations with memory. We investigate the existence and uniqueness of optimal solutions. The necessary and sufficient optimality conditions are also studied. The main novelty of our result is to establish the optimality conditions for the parabolic optimal control problem with memory.

A novel image classification model based on adversarial training for pulsar candidate identification

Pulsars are highly magnetized, rotating neutron stars with small volume and high density. The discovery of pulsars is of great significance in the fields of physics and astronomy. With the development of artificial intelligent, image recognition models based on deep learning are increasingly utilized for pulsar candidate identification. However, pulsar candidate datasets are characterized by unbalance and lack of positive samples, which has contributed the traditional methods to fall into poor performance and model bias. To this end, a general image recognition model based on adversarial training is proposed. A generator, a classifier, and two discriminators are included in the model. Theoretical analysis demonstrates that the model has a unique optimal solution, and the classifier happens to be the inference network of the generator. Therefore, the samples produced by the generator significantly augment the diversity of training data. When the model reaches equilibrium, it can not only predict labels for unseen data, but also generate controllable samples. In experiments, we split part of data from MNIST for training. The results reveal that the model not only behaves better classification performance than CNN, but also has better controllability than CGAN and ACGAN. Then, the model is applied to pulsar candidate dataset HTRU and FAST. The results exhibit that, compared with CNN model, the F-score has increased by 1.99% and 3.67%, and the Recall has also increased by 6.28% and 8.59% respectively.

Trade-off-based multi-objective optimisation of a simultaneous saccharification and fermentation process

The demand for sustainable replacements for fossil-based products is steadily increasing, especially now that the effects of climate change are becoming more prominent. Lignocellulose, which is a sustainable and abundant carbon source, is dubbed to be the perfect replacement. Lignocellulose consists of lignin, hemicellulose, and cellulose. During the Simultaneous Saccharification and Fermentation (SSF) of cellulose, the hydrolysis and fermentation of the produced C6-sugars occurs simultaneously in the same vessel. The SSF process has mainly been developed to circumvent inhibitory effect and increase the overall product yield. Although the concept of the SSF process is promising, the applications are still limited. This contribution presents the trade-off-based multi-objective optimisation of an SSF process. Multi-objective optimisation allows for optimising (bio-)process with respect to multiple, and often conflicting, objectives. These optimisation problems do not render a unique optimal solution but instead an infinite set of so-called Pareto-optimal solutions, the Pareto front. From the Pareto front, the decision maker should select one working point. To aid decision makers in this selection process, the application of a novel genetic optimisation algorithm is presented in this contribution, i.e., tDOM, that is capable of filtering solutions using t-domination. This results in a less dense Pareto front that only contains solutions that are of interest for the decision maker. Additionally, by extending the t-domination concept to two subsequent solution populations, a novel problem-relevant stopping criterion is developed, resulting in a significant gain in the required computational time. A comparison to the well known NSGA-II is provided.

Optimal Frequency Control for Inverter-Based Micro-Grids Using Distributed Finite-Time Consensus Algorithms

This paper addresses the distributed frequency control and economic dispatch problem of micro-grid systems. The optimal frequency control problem is formulated considering both frequency regulation and generation cost minimization, by establishing an equivalent nonlinear second-order dynamic model. A distributed finite-time consensus protocol is proposed to solve the optimal frequency regulation problem based on the equivalent model. Despite the nonlinear dynamics, the system frequency converges to the nominal value and the economic dispatch is achieved both in finite time. It is proved that the equilibrium point of the closed-loop system is the unique optimal solution to the associated economic dispatch problem. Simulation results verify the theoretical analysis and demonstrate the effectiveness of the proposed control method.

Analysis of two substitute products newsvendor problem with a budget constraint

This paper presents a substitute product newsvendor problem with a budget constraint: the newsvendor sells two substitute products and determines the optimal order quantity and selling price for each product. We present a nonlinear programming model with a budget constraint and solve the model mainly according to the Karush-Kuhn-Tucker (KKT) theorem. The resulting insights are as follows: (1) We prove the existence of unique optimal solutions in a stochastic demand scenario with a general demand probability density function. (2) When the substitute products’ information is symmetric, the demand variability, store-level factor and competitive factor have no effect on optimal order quantities, and the optimal prices are the same as long as the value of store-level factor minus competitive factor remains the same, when other factors are fixed. However, this result is not observed when the substitute products’ information is asymmetric. (3) If the budget constraint is tight, releasing the non-negative constraints of the decision variables will lead to negative order quantities, which mean the retailer has to cancel the corresponding order from the purchasing list. In addition, the retailer should decrease the optimal retail price in order to obtain more expected revenue if the budget increases.