Pareto Optimization Problem Research Articles

Analysis for the Weakly Pareto Optimum in Multiobjective-Based Hyperspectral Band Selection

Band selection refers to finding the most representative channels from hyperspectral images. Usually, certain objective functions are designed and combined via regularization terms. A possible drawback of these methods is that they can only generate one solution in a single run with a given band number. To overcome this problem, multiobjective (MO)-based methods, which were able to simultaneously obtain a series of subsets with different band numbers, were investigated for band selection. However, because the range of band selection problem is discrete, recently proposed weighted Tchebycheff (WT)-based MO methods may suffer weakly Pareto optimal problem. In this case, the solutions for each band number will be nonunique and no optimal solution exists. Decision makers have to manually select a unique solution for each band number. In this paper, we provide a theoretical analysis about the weakly Pareto optimal problem in band selection, and quantitatively give the boundary conditions. Moreover, we further summarize the suggestions which will help users avoid the weakly Pareto optimal problem. According to these criteria, we develop a new adaptive-penalty-based boundary intersection (APBI) framework to improve the MO algorithm in hyperspectral band selection. APBI mainly includes two advantages: 1) avoiding weakly Pareto optimum and 2) reducing the sensibility of the penalty factor. The theoretical analysis is further validated by contrast experiments. The results demonstrate that the weakly Pareto optimal solutions really exist in WT methods, while APBI can overcome this problem.

Multi-agent scheduling in a no-wait flow shop system to maximize the weighted number of just-in-time jobs

ABSTRACTThis article studies a multi-agent scheduling problem on a set of m machines in a no-wait flow shop system, where each agent's objective function is to maximize its own weighted number of just-in-time jobs. Two variants of the problem are investigated. One is the constrained optimization problem and the other is the Pareto optimization problem. When the number of agents is arbitrary, both problems are proved to be strongly -hard. When the number of agents is fixed, pseudo-polynomial time algorithms are first designed to solve them, respectively, then an -approximation algorithm is provided for the former problem, and an -approximate Pareto-optimal frontier is constructed for the latter problem.

Multi-Objective Pareto Optimization of Electromagnetic Devices Exploiting Kriging With Lipschitzian Optimized Expected Improvement

This paper focuses on resolving the storage issue of correlation matrices generated by kriging surrogate models in the context of electromagnetic optimization problems with many design variables and multiple objectives. The suggested-improved kriging approach incorporating a direct algorithm is able to maintain memory requirements at a nearly constant level while offering high efficiency of searching for a global optimum. The feasibility and efficiency of this proposed methodology are demonstrated using an example of a classic two-variable analytic function and a new proposed benchmark TEAM multi-objective Pareto optimization problem.

Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time

This paper investigates the scheduling of family jobs with release dates on an unbounded parallel-batch machine. The involved objective functions are makespan and maximum flow time. It was reported in the literature that the single-criterion problem for minimizing makespan is strongly NP-hard when the number of families is arbitrary, and is polynomially solvable when the number of families is fixed. We first show in this paper that the single-criterion problem for minimizing maximum flow time is also strongly NP-hard when the number of families is arbitrary. We further show that the Pareto optimization problem (also called bicriteria problem) for minimizing makespan and maximum flow time is polynomially solvable when the number of families is fixed, by enumerating all Pareto optimal points in polynomial time. This implies that the single-criterion problem for minimizing maximum flow time is also polynomially solvable when the number of families is fixed.

Mining parametric temporal logic properties in model-based design for cyber-physical systems

One of the advantages of adopting a Model Based Development (MBD) process is that it enables testing and verification at early stages of development. However, it is often desirable to not only verify/falsify certain formal system specifications, but also to automatically explore the properties that the system satisfies. In this work, we present a framework that enables property exploration for Cyber-Physical Systems. Namely, given a parametric specification with multiple parameters, our solution can automatically infer the ranges of parameters for which the property does not hold on the system. In this paper, we consider parametric specifications in Metric or Signal Temporal Logic (MTL or STL). Using robust semantics for MTL, the parameter mining problem can be converted into a Pareto optimization problem for which we can provide an approximate solution by utilizing stochastic optimization methods. We include algorithms for the exploration and visualization of multi-parametric specifications. The framework is demonstrated on an industrial size, high-fidelity engine model as well as examples from related literature.

Joint Downlink Power and Time-Slot Allocation for Distributed Satellite Cluster Network Based on Pareto Optimization

In this paper, we design a novel architecture of distributed satellite cluster network (DSCN). In order to achieve a good trade-off between the energy consumption and the total capacity, we investigate the joint downlink power and time-slot allocation problem, taking into account the limitation of resource, collaborative coverage of multi-satellite, and dynamism, which is proved to be a Pareto optimization and NP-hard problem. Different from the existing 1-D multi-objective optimization algorithm (1D-MOA) based on meta-heuristics, such as immune clonal algorithm (ICA), we propose an improved 2-D dynamic immune clonal algorithm (TDICA) to search the solution space for approaching the Pareto front. From simulation results, several important concluding remarks are obtained as follows: 1) the proposed TDICA can obtain more non-dominated solutions in each iteration with better accuracy than existing algorithms; 2) with inter-satellite resource optimization, the total capacity can be improved; 3) compared with 1D-MOAs, the 2D-MOA can save more energy and achieve higher total capacity; d) MOAs can be transferred into multiple single-objective optimization algorithms (SOAs) under certain conditions.

Linear quadratic Pareto optimal control problem of stochastic singular systems

This paper is concerned with the finite horizon linear quadratic (LQ) Pareto optimal control problem of stochastic singular systems. By means of the square completion technique, for the finite horizon LQ optimal control of stochastic singular systems, we establish a new kind of generalized differential Riccati equations (GDREs) and present the existence condition of the solution of the GDREs. Then, for the finite horizon LQ Pareto optimal control, it is shown that under the solvability of the corresponding GDREs, all Pareto candidates can be obtained by solving a weighting sum optimal control. Finally, an example is provided to show the effectiveness of our main results.

A Multi Objective Teacher-Learning-Artificial Bee Colony (MOTLABC) Optimization for Software Requirements Selection

Background/Objectives: To select optimal software requirements by introducing Multi-objective Teacher-Learning- Artificial Bee Colony Optimization. Methods/Statistical Analysis: Teaching learning based optimization for the multi-objective software requirements selection has two objectives of minimized cost and maximum client satisfaction. Similarly the constraints namely interaction constraints and cost threshold constraints are considered. However, the efficiency of the software product development can be improved further when more efficient optimization techniques is used for the selection of software requirements along with consideration of more objectives and more constraints in larger real datasets. Findings: In this article, a hybrid optimization technique named Multi-Objective Teacher-Learning Artificial Bee Colony Optimization (MOTLABC) is proposed with set of multiple objectives and constraints. The objectives are minimum cost, maximum client satisfaction, minimum time consumption and maximum reliability. The constraints such as time threshold constraint, interaction constraints and cost threshold constraints are considered. The hybrid approach of MOTLABC with the above objectives improves the collection of set of needs for the development of the software. The Pareto optimal problem occurs in multi objective optimization solutions is resolved by the use of Pareto tournament function. Improvements/Applications: The experimental consequences prove that they obtained results perform improved than algorithms proposed in the literature.

On Hadamard well-posedness of families of Pareto optimization problems

This paper deals with the well-posedness of families of finite dimensional vector optimization problems ordered by components (Pareto problem). For this problem, two Hadamard well-posedness concepts are introduced. These concepts involve the existence and uniqueness of efficient/weak efficient solutions, and also the continuous behavior of these solutions with respect to perturbations of the data. The perturbations in the last property are formulated through a variational convergence notion of the objective functions and by considering approximate solutions of the perturbed problems. Necessary and sufficient conditions for the well-posedness of Pareto optimization problems are obtained in general, and also under convexity and quasiconvexity assumptions. To do this, asymptotic mathematical tools are employed. Finally, it is proved that the convex Pareto optimization problems are “essentially” well-posed in the sense of category theory.

Mechanical design, multiple criteria decision making and Pareto optimality gap

Purpose We present an approach to multiple criteria mechanical design problems, for cases where problem complexity precludes derivation of the whole Pareto front. For such problems we propose to limit search, and hence also derivation, of the Pareto front exclusively to regions of the direct designer’s interest, thus saving on computing efforts and gaining on tractable problem sizes. Design/methodology/approach To achieve the purpose, we frame the decision making process (design) into a combination of three specific concepts, namely decision maker's preference capture, local Pareto front search and approximate multiobjective optimization with assessments of the Pareto optimality gap. We illustrate the approach with two small design problems, namely Pareto optimal round tube beam and Pareto optimal pneumatic high speed machine drive selection. We solve these problems in a setting which can be regarded as representative for problem solving in real environment. Findings On the decision making side, the proposed approach has turned out to be a versatile tool for selecting designs from the Pareto suboptimal ones, where each such a Pareto suboptimal design has an explicit assessment of the Pareto optimality gap. On the technical (optimization) side, it has been demonstrated that the approach seamlessly works with evolutionary computations, structured to the specific needs of the approach. Research limitations/implications It has been shown that the navigation over the Pareto front can be achieved with limited effort, both on the cognitive and the computing side. Moreover, navigation over the Pareto front can be focused from the very beginning of the design selection process on the regions of the Pareto front which are of the direct designer’s interest. This eliminates the need to derive (or only approximate) the whole Pareto front, a tangible asset as the derivation of that set is the main factor precluding scalability of design selection problems to higher dimensions (to higher problem sizes). Practical implications Because of the general formulation of the Pareto optimal design selection problem considered in the paper, the absence of any assumptions on its form and easiness of implementation of the underlying procedure of the proposed approach, the paper offers a clear option to approaches based on classical optimization computations. Originality/value The approach offers derivation of Pareto suboptimal designs with assessments of the Pareto optimality gap, whereas currently available multiobjective evolutionary optimization algorithms which derive Pareto suboptimal designs as well, offer no such assessments. Thus, the approach provides a firm ground to valuate designs resulting from approximate multiobjective optimization computations.

Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation

Our goal is to study the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. First we consider a mathematical model of the beam equation which was obtained by S. Woinowsky-Krieger in 1950. Next we consider the Pareto optimal control problem based on this equation. Further, we describe the approximation of this system. We use the Galerkin method to approximate the solution of this control problem with respect to a spatial variable. Based on the standard finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation point of the solutions of the discrete optimal control problems exist and each of these points is the solution of the original Pareto optimal control problem.

A note on unbounded parallel-batch scheduling

This paper revisits the scheduling problem on an unbounded parallel batch machine for minimizing a maximum cost fmax. It was reported in the literature that the decision version of the problem is solvable in O(n2+nlog⁡P) time, where n is the number of jobs and P is the total processing time of jobs. This implies that the optimization version for minimizing fmax can be solved in weakly polynomial time. But a strongly polynomial-time algorithm has not been provided for this problem. In this paper, we present an O(n4)-time algorithm for the Pareto optimization problem for minimizing Cmax and fmax, where Cmax is the maximum completion time of jobs. Consequently, the problem for minimizing fmax can also be solved in O(n4) time.

Multiobjective Pareto Optimal Design of a Clutch System

Optimum design of a clutch and dual mass flywheel system is completed. Although the clutch systems are exposed to both rotational and axial vibrations, they are generally designed by considering rotational vibrations of engines and they do not have any component to damp axial vibrations due to the following two reasons: first, the package area of the clutch system is very limited and there is no available space to put additional springs and dampers to reduce these vibrations. Second, axial vibrations are normally insignificant to be considered in the analyses and such vibrations are supposed to be damped by the diaphragm spring and cushion springs. Nonetheless, axial vibrations may lead to some unexpected problems in the power train such as the rattle noise that is examined in this study by using global optimization techniques. The components in the clutch system are modeled analytically. Then, by considering the pedal characteristics and vibrations of pressure plate as objective functions, a multi-objective Pareto optimization problem is solved. It is shown that analytical models agree well with the experimental measurements and vibrations in the clutch system can be reduced significantly by choosing the design parameters with optimization tools.

Emerging trends in vibration control of wind turbines: a focus on a dual control strategy.

The paper discusses some of the recent developments in vibration control strategies for wind turbines, and in this context proposes a new dual control strategy based on the combination and modification of two recently proposed control schemes. Emerging trends in the vibration control of both onshore and offshore wind turbines are presented. Passive, active and semi-active structural vibration control algorithms have been reviewed. Of the existing controllers, two control schemes, active pitch control and active tendon control, have been discussed in detail. The proposed new control scheme is a merger of active tendon control with passive pitch control, and is designed using a Pareto-optimal problem formulation. This combination of controllers is the cornerstone of a dual strategy with the feature of decoupling vibration control from optimal power control as one of its main advantages, in addition to reducing the burden on the pitch demand. This dual control strategy will bring in major benefits to the design of modern wind turbines and is expected to play a significant role in the advancement of offshore wind turbine technologies.

Springback Reduction in Tailor Welded Blank with High Strength Differential by Using Multi-Objective Evolutionary and Genetic Algorithms

The springback behavior on two sides of TWIP (twinning-induced plasticity) and mild steels tailor welded blank (TWB) is considerably different due to the large difference in strength. To reduce springback defects in a U-draw bending process of such TWB, different non-constant blank holding force (BHF) profiles on two sides of the blank are applied in this study. A systematic approach to obtain optimal BHF-stroke profiles, which helps to reduce the springback is proposed. Control of the variable BHF aims at increasing the uniformity in through-thickness of the stress component along the stretching direction, while decreasing the risk of failure due to over-stretching. Therefore, the optimal condition would require satisfying two conflicting objectives simultaneously: (i) minimize springback deformation and (ii) minimize the forming severity, leading to a Pareto-optimal problem. The optimization procedure consists of sampling design, finite element (FE) simulations, metamodeling, and finally the calculation of a Pareto-frontier. Generated outputs from FE simulations on statistically significant sampling points are used for the construction of metamodels of optimum accuracy and complexity, which, in turn, are used to evaluate the output for any set of inputs, replacing the computing intensive FE simulations. A novel genetic algorithms based multi-objective optimization technique is subsequently applied for optimization.

Pareto optimization scheduling of family jobs on a p-batch machine to minimize makespan and maximum lateness

This paper studies the Pareto optimization scheduling problem of family jobs on an unbounded parallel-batching machine to minimize makespan and maximum lateness. In the problem, the jobs are partitioned into families and scheduled in batches, where each batch is a set of jobs belonging to a common family and the processing time of a batch is defined to be the longest processing time of the jobs in the batch. The objective is to find all Pareto optimal points for minimizing makespan and maximum lateness and, for each Pareto optimal point, provide a corresponding Pareto optimal schedule. We present an algorithm to solve this Pareto optimization problem. Our algorithm is of polynomial-time when the number of families is a constant.

Mobile Node Localization via Pareto Optimization: Algorithm and Fundamental Performance Limitations

Accurate estimation of the position of network nodes is essential, e.g., in localization, geographic routing, and vehicular networks. Unfortunately, typical positioning techniques based on ranging or on velocity and angular measurements are inherently limited. To overcome the limitations of specific positioning techniques, the fusion of multiple and heterogeneous sensor information is an appealing strategy. In this paper, we investigate the fundamental performance of linear fusion of multiple measurements of the position of mobile nodes, and propose a new distributed recursive position estimator. The Cram\'er-Rao lower bounds for the parametric and a-posteriori cases are investigated. The proposed estimator combines information coming from ranging, speed, and angular measurements, which is jointly fused by a Pareto optimization problem where the mean and the variance of the localization error are simultaneously minimized. A distinguished feature of the method is that it assumes a very simple dynamical model of the mobility and therefore it is applicable to a large number of scenarios providing good performance. The main challenge is the characterization of the statistical information needed to model the Fisher information matrix and the Pareto optimization problem. The proposed analysis is validated by Monte Carlo simulations, and the performance is compared to several Kalman-based filters, commonly employed for localization and sensor fusion. Simulation results show that the proposed estimator outperforms the traditional approaches that are based on the extended Kalman filter when no assumption on the model of motion is used. In such a scenario, better performance is achieved by the proposed method, but at the price of an increased computational complexity.

Uncertainties in Road Vehicle Suspensions

Road vehicles are subject to random excitation by the unevenness of the road. For a dynamical analysis, vehicle models of the vertical vibrations as well as guideway models of the road unevenness are required. The fundamental dynamics of vehicle suspensions can be already modeled by a quarter car featuring the decoupling of the car body motion and the wheel motion. This suspension model is characterized by five design parameters where two of them, the shock absorber and the tire spring, are highly uncertain due to wear and poor maintenance. For the assessment of the vehicles performance three criteria have to be used: ride comfort, driving safety and suspension travel. These criteria depend on all the five design parameters resulting in a conflict or a pareto-optimal problem, respectively. In this paper, the uncertainties of the parameters are projected into a criteria space in order to support the decision to be made on the basis of a pareto-optimal problem. Simulations with uncertainties support the robust suspension design. It is shown that controlled suspension parameters remain uncertain due to the unpredictable decisions made by the driver.

Pareto optimal control for mixed Neumann infinite-order parabolic system with state-control constraints

A distributed Pareto optimal control problem for an infinite order parabolic system is considered. The performance index has a vector form with two components in integral form. Constraints on controls and on states are imposed. To obtain optimality conditions for the Neumann problem, the generalization of the Dubovitskii–Milyutin theorem was applied.

Strength Pareto Multigroup Search Optimizer for Multiobjective Optimal Reactive Power Dispatch

Multiobjective optimal reactive power dispatch (MORD) is a highly constrained nonlinear nondifferential multimodal Pareto optimization problem with a mixture of continuous and discrete variables. This paper presents a novel strength Pareto multigroup search optimizer (SPMGSO) for solving this challenging problem. In the proposed algorithm, a number of enhancement mechanisms, such as ring-migration synergistic cooperation, crowding entropy constraint treatment, and chaotic logistic dispersion, were developed to effectively strengthen the diversity and distribution of the resulting nondominated front (NDF). The final best compromise solution is then extracted from the NDF using a hierarchical clustering and an equilibria-based multicriteria decision-making scheme. In addition, the multicore processing was introduced to parallelize the multigroup search so as to greatly improve the execution speed of the algorithm. Computational studies on the benchmark IEEE 30-bus and 162-bus power systems have confirmed the superior performance and improvement of the proposed algorithm for solving this Pareto problem, and demonstrated its applicability and capability to cope with the high-dimensional MORD problems with multiple operational objectives.