Pareto Points Research Articles

A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces

The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from Gebken and Peitz (J Optim Theory Appl 188:696–723, 2021) is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the Clarke subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.

A dynamic multiple proximal bundle algorithm for locally Lipschitz multiobjective optimization problems

We propose a method for nonsmooth nonconvex multiobjective optimization problems that is based on a bundle-type approach. The proposed method employs an extension of the redistributed proximal bundle algorithm and uses an augmented improvement function to handle different objectives. In which, at each iteration, a common piecewise linear model is used to approximate the augmented improvement function. Contrary to many existing multiobjective optimization methods, this algorithm works directly with objective functions, without using any kind of a priori chosen parameters or employing any scalarization. Under appropriate assumptions, we discuss convergence to points which satisfy a necessary condition for Pareto optimality. We provide numerical results for a set of nonsmooth convex and nonconvex multiobjective optimization problems in the form of tables, figures and performance profiles. The numerical results confirm the superiority of the proposed algorithm in the considered test problems compared to other multiobjective solvers, in the computational effort necessary to compute weakly Pareto stationary points.

Pareto navigation for multicriteria building energy supply design

The design of a building’s energy supply is a difficult task, where uncertainty in the requirements has to be taken into account and different conflicting objectives have to be optimized. To date, there exist various studies that consider different objective functions, different concepts of dealing with uncertainty, and different approximation techniques to find efficient solutions. However, there is no approach that gives the decision maker a complete overview of all trade-off solutions with their immediate consequences for the design configuration. This paper addresses the question of how much money and carbon emissions can be saved by allowing a certain amount of undersupply in the heating and cooling of an office building. To explore the trade-offs between the three objectives of minimizing cost, carbon emissions and inconvenience, the Pareto front of the multicriteria linear program is approximated with a Sandwiching algorithm. Then, it is explored using Pareto navigation. This allows to see all (approximate) Pareto points and also to visually observe the behavior of the decision variables by navigating across the front by pulling sliders in a specific software. In a case study, the navigation is demonstrated and ways to overcome modeling limitations that arise from using a fully linear programming approach are shown.

Efficient demand management for the on-time arrival problem: A convexified multi-objective approach assuming macroscopic traffic dynamics

This work proposes a novel solution approach to address the On-Time Arrival (OTA) problem, considering macroscopic traffic dynamics. In this context, all drivers who intend to use the road infrastructure are required to communicate their origin–destination pair and desired arrival time to a central scheduler prior to starting their trip. In response, the scheduler assigns a departure time and a multi-regional route to each driver to make possible their on-time arrival at the destination. The OTA problem is formulated as a nonconvex, nonlinear, multi-objective optimization problem considering two objective criteria. The first criterion aims at minimizing the travel time of all drivers in the network to prevent congestion, while the second criterion seeks to minimize the discrepancy between the desired and actual arrival time. The proposed formulation is solved efficiently through an approximated convex solution that leverages the Normal Boundary Intersection (NBI) method to efficiently generate a representative sample of the Pareto Front. Additionally, two solution methodologies based on “knee” solution and the Nash Bargaining Game is proposed which can be used to select a unique solution across all the Pareto points, that offers a reasonable trade-off between the two objective criteria. Finally, simulation results demonstrate that the proposed solution can significantly reduce congestion while ensuring that most drivers will arrive at their destination on their desired time.

Multi-objective optimization and uncertainty quantification for inductors based on neural network

PurposeThe prolonged training time of the neural network (NN) has sparked considerable debate regarding their application in the field of optimization. The purpose of this paper is to explore the beneficial assistance of NN-based alternative models in inductance design, with a particular focus on multi-objective optimization and uncertainty analysis processes.Design/methodology/approachUnder Gaussian-distributed manufacturing errors, this study predicts error intervals for Pareto points and select robust solutions with minimal error margins. Furthermore, this study establishes correlations between manufacturing errors and inductance value discrepancies, offering a practical means of determining permissible manufacturing errors tailored to varying accuracy requirements.FindingsThe NN-assisted methods are demonstrated to offer a substantial time advantage in multi-objective optimization compared to conventional approaches, particularly in scenarios where the trained NN is repeatedly used. Also, NN models allow for extensive data-driven uncertainty quantification, which is challenging for traditional methods.Originality/valueThree objectives including saturation current are considered in the multi-optimization, and the time advantages of the NN are thoroughly discussed by comparing scenarios involving single optimization, multiple optimizations, bi-objective optimization and tri-objective optimization. This study proposes direct error interval prediction on the Pareto front, using extensive data to predict the response of the Pareto front to random errors following a Gaussian distribution. This approach circumvents the compromises inherent in constrained robust optimization for inductance design and allows for a direct assessment of robustness that can be applied to account for manufacturing errors with complex distributions.

Metaheuristic approach to multi-objective ergonomic workforce scheduling for Bi-hazard industrial tasks

ABSTRACT This paper considers the daily scheduling of workers who are simultaneously exposed to two ergonomic hazards when performing their assigned tasks. Safe daily rotating work schedules are generated to minimize the maximum exposure amounts of both hazards simultaneously. A non-dominated sorting genetic algorithm II (NSGA-II) method is employed to generate near-optimal workforce scheduling solutions. A numerical example shows that for a small multi-objective problem, the NSGA-II is able to yield the same set of non-dominated Pareto points as those obtained from a multi-objective optimization model with ε-constraint method. A computational experiment further demonstrates that the NSGA-II can yield acceptable non-dominated Pareto points in reasonable computation times irrespective of the problem size.

Abrasive waterjet machining of titanium alloy using an integrated approach of taguchi-based passing vehicle search algorithm

AbstractThe titanium alloy, Ti6Al4V, is a popularly used material in aerospace and medical applications due to its specific attributes, such as excellent strength-to-weight ratio and resistance to corrosion. Ti6Al4V is challenging to machine using conventional methods due to its poor thermal conductivity, which necessitates the use of unconventional machining methods like abrasive water jet machining (AWJM). In this work, AWJM was preferred for machining Ti6Al4V, considering three key process variables: nozzle traverse speed (Tv), abrasive flow rate (Af), and stand-off distance (Sd). The performance of the AWJM process was assessed using three main metrics: Material removal rate (MRR), Surface roughness (SR), and Kerf taper angle (θ), which were crucial for evaluating the effectiveness of the AWJM. Taguchi’s L9 array, a design of experiments method, was used to plan the experiments. The adequacy of the developed models was assessed by analysis of variance (ANOVA). ANOVA results have shown that Tv, Af, and Sd were found to have a significant effect on MRR, SR, θ with contributions of 73.15%, 49.72%, and 78.35% respectively. The Passing Vehicle Search algorithm was adopted to find the global optimal solution. Simultaneous optimization results using the PVS algorithm have shown the optimal MRR, SR, and θ values of 0.17 g/min, 3.28 μm, and 2.45, respectively, at Tv of 237 mm/min, Af of 450 g/min, and Sd of 2.0 mm. The optimization results with Pareto points will help to achieve desired outcomes by selecting appropriate input conditions.

Multi-objective optimization of transpiration cooling for high pressure turbine vane

Transpiration cooling is a promising cooling measure for high-temperature components in gas turbines. Although the flow and heat transfer characteristics of transpiration cooling have been investigated in several experimental and numerical researches, the design optimization of transpiration cooling is less studied until now. In this paper, a multi-objective optimization framework based on the ANN surrogate model and the NSGA-Ⅱ is built to optimize the transpiration cooling of C3X turbine vane. The transpiration cooled C3X vane consists of multiple porous zones whose porosities could be different and are treated as the design parameters. The optimization objectives are minimum area-averaged vane surface temperature and minimum aerodynamic loss. A total of 960 CFD simulations are conducted to prepare the sample dataset. After the validation of the surrogate-based optimization framework, the Pareto front is obtained as an output. Three representative Pareto-optimal points are chosen from the Pareto front, considering the best cooling performance, the lowest aerodynamic loss, and a balance between them. The porosity distributions and coolant allocations, the cooling performances, and the aerodynamic losses for the three Pareto points are discussed, and the involved flow and heat transfer characteristics are analyzed. The results reveal that to reach a balance between the cooling efficiency and the aerodynamic loss for an optimal transpiration cooled C3X vane, coolant should be appropriately distributed to the leading edge, as well as zones immediately downstream of the leading edge.

Combining artificial intelligence and computational fluid dynamics for optimal design of laterally perforated finned heat sinks

The efficient design of heat sinks is a severe challenge in thermo-fluid engineering. A creative and innovative way is applying lateral perforations to parallel finned heat sinks. The significance of achieving an optimal design for perforated finned heat sinks (PFHSs) has inspired the present authors to introduce a novel hybrid designing approach that combines computational fluid dynamics (CFD), machine learning (ML), multi-objective optimization (MOO), and multi-criteria decision-making (MCDM). The design variables considered include the size (0.25<φ<0.5) and shape (square, circular, and hexagonal) of the perforations, as well as the airflow Reynolds number (2000<Re<5000). The design objectives have been redefined as dimensionless parameters to assess heat dissipation, pressure drop, and heat sink weight. These modified objectives encompass thermal performance (TP), thermo-hydraulic performance (THP), and thermo-volumetric performance (TVP). The modeling process showed that both stepwise mixed selection (SMS) and GMDH-NN techniques exhibited comparable performance in most modeling scenarios. Nevertheless, the SMS approach demonstrated more reliability in modeling diverse objectives. Furthermore, the optimization results demonstrated that the optimal size of the perforations is strongly dependent on their shapes. In PFHSs with square perforations, approximately 54% of the Pareto points had a φ-value greater than 0.45. Meanwhile, in PFHSs based on circular perforations, more than 50% of the optimal points have φ less than 0.4. The MCDM-based analysis on various real-world scenarios indicated that using PFHSs with square-shaped perforations with Reynolds numbers around 2000 and considering a wide range of perforations' sizes could result in optimal designs.

An asymptotic approach to centrally planned portfolio selection

AbstractWe formulate a centrally planned portfolio selection problem with the investor and the manager having S-shaped utilities under a recently popular first-loss contract. We solve for the closed-form optimal portfolio, which shows that a first-loss contract can sometimes behave like an option contract. We propose an asymptotic approach to investigate the portfolio. This approach can be adopted to illustrate economic insights, including the fact that the portfolio under a convex contract becomes more conservative when the market state is better. Furthermore, we discover a means of Pareto improvement by simultaneously considering the investor’s utility and increasing the manager’s incentive rate. This is achieved by establishing the collection of Pareto points of a single contract, proving that it is a strictly decreasing and strictly concave frontier, and comparing the Pareto frontiers of different contracts. These results may be helpful for the illustration of risk choices and the design of Pareto-optimal contracts.

Stability and thermodynamic property of TiO2/R141b nanorefrigerants by multi‐objective optimization

AbstractUtilizing nanorefrigerants as working fluids can significantly enhance the energy efficiency of low‐temperature waste heat recovery systems (≤ 50°C). Refrigerants with low viscosity and density require a substantial amount of surfactant to maintain a stable suspension of nanoparticles. However, the excessive use of surfactants, which have a notably low thermal conductivity, could lead to foam generation and reduce heat transfer coefficient. High viscosity lubricating oils with small amount of surfactant can prolong the stable suspension time and produce repulsive force. Therefore, a new combination of them improves the stability of TiO2/R141b nanorefrigerants. Additionally, viscosity and thermal conductivity of the nanorefrigerants were optimized using an implementation of a modified non‐dominated sorting genetic algorithm (NSGA‐II). The results show that adding lubricating oil inhibits aggregation of the nanoparticles leading to a stable suspension for more than 6 h at volumetric mixing ratios (lubricating oil: refrigerant) greater than 1:30. The best dispersion stability was achieved at surfactant polyvinyl pyrrolidone (PVP) mass ratio of 0.5, and the average absorbance value was increased by 65.45%. Compared with pure refrigerants, the thermal conductivity of TiO2/R141b (0.15 vol.%) nanorefrigerant was enhanced by up to 12.59% under the optimum mixing ratio. Moreover, the studied nanorefrigerants exhibited shear thickening behavior throughout the studied shear rate range, with increased non‐Newtonianization with decreasing temperature. Finally, the Pareto points were divided into three representative groups based on thermal conductivity and viscosity. These findings suggest enhanced high heat transfer efficiency with pumping power of nanorefrigerant in the waste heat recovery systems.

Inexact Exponential Penalty Function with the Augmented Lagrangian for Multiobjective Optimization Algorithms

This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.

The Multi-Objective Polynomial Optimization

The multi-objective optimization is to optimize several objective functions over a common feasible set. Because the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the geometry for (weakly) Pareto values and represent Pareto front as the boundary of a convex set. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them and how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check whether a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points. Funding: J. Nie is partially supported by the National Science Foundation [Grant DMS-2110780].

Single and multi-objective dynamic optimization study of an industrial scale fed batch reactor

Abstract The present work focuses on obtaining optimal operational policies of an industrial scale penicillin fed-batch fermentation process using dynamic optimization (DO). The three process objectives considered in this work include, maximization of total penicillin formed (J 1), minimization of total biomass formed (J 2), and minimization of fed-batch operation time (J 3). The control variables for the optimization study include, the feed flowrates of sugar, soyabean oil, and phenylacetic. We perform a single-objective optimization study with J 1 as the objective function and showcased the importance of DO. Consequently, we solve three multi-objective optimization (MOO) problems corresponding to all the pairwise combinations of the aforementioned three objectives. Outcomes of MOO problems are presented in terms of several non-dominated Pareto-optimal solutions. Furthermore, descriptive analysis of three representative Pareto points for all the MOO problems is presented, which highlights the importance of MOO study of the penicillin bio-reactor.

Deterministic Optimization Approach for High-Performance Externally Fired Biomass-Fueled Micro-Gas Turbines

Abstract This work focuses on a dual-objective optimization of a 100 kWe externally fired micro-gas turbine utilizing the producer gas from a biomass gasifier. Although externally fired micro-gas turbines are convenient for resolving operability issues in biomass combined heat and power applications, these configurations are still lacking in efficiency compared to the commercial natural-gas fired microturbines. The main cause is the material temperature limitations in the recuperator and the current uneconomical use of high-temperature resistance materials. Toward the achievement of higher efficiency by keeping system economic viability, an optimization process is followed based on the Normal Constraint Method, which generates evenly distributed solutions of a Pareto front. The selected method can determine high-performance solutions, being unidentified by one-dimensional approaches, providing information about the distribution of critical cycle parameters, across the complete objective space by the evaluation of a relatively small set of Pareto points. These critical parameters are the pressure ratio, the recuperator temperature difference, and maximum temperature. The exergetic efficiency and the relative recuperator cost are the optimization objectives. The deterministic Nelder–Mead algorithm is used for the acquisition of Pareto solutions, along with a penalty-based method to perform the constrained optimization. The implemented optimization method can identify superior solutions compared to one-dimensional approaches, as the latter result in higher recuperator costs around 41–112% at the same exergetic efficiency, revealing that high-performance is not only restricted by the recuperator but also by the compressor operating range.

Technical Note—Characterizing and Computing the Set of Nash Equilibria via Vector Optimization

What is the relation between the notion of Nash equilibria and Pareto-optimal points? It is well known that Nash equilibria do not need to be Pareto optimal, and Pareto points do not need to be Nash equilibria. However, the paper “Characterizing and Computing the Set of Nash Equilibria via Vector Optimization” by Feinstein and Rudloff takes a deeper look at the relation. It is shown that it is possible to characterize the set of all Nash equilibria as the set of all Pareto-optimal solutions of a certain vector optimization problem. This is accomplished by carefully designing the objective function and the ordering cone of the vector optimization problem such that both notions coincide. This characterization holds for all noncooperative games (nonconvex, convex, linear). It opens up a new way of computing Nash equilibria, as one can now use techniques and algorithms from vector optimization to compute the set of all Nash equilibria, which is in contrast to the classical fixed-point iterations that find just a single Nash equilibrium.

A Barzilai-Borwein descent method for multiobjective optimization problems

The steepest descent method proposed by Fliege and Svaiter has motivated the research on descent methods for multiobjective optimization, which has received increasing attention in recent years. However, empirical results show that the Armijo line search often results in a very small stepsize along the steepest descent direction, which decelerates the convergence seriously. This paper points out the issue is mainly due to imbalances among objective functions. To address this issue, we propose a Barzilai-Borwein descent method for multiobjective optimization (BBDMO), which dynamically tunes gradient magnitudes using Barzilai-Borwein’s rule in direction-finding subproblem. We emphasize that the BBDMO produces a sequence of new descent directions compared to Barzilai-Borwein’s method proposed by Morovati et al. With monotone and nonmonotone line search techniques, we prove that accumulation points generated by BBDMO are Pareto critical points, respectively. Furthermore, theoretical results indicate that the Armijo line search can achieve a better stepsize in BBDMO. Finally, comparative results of numerical experiments are reported to illustrate the efficiency of BBDMO and verify the theoretical results.

Supply Chain Optimization under Risk and Uncertainty using Nondominated Sorting Genetic Algorithm II for Automobile Industry

In recent decades, the use of supply chain management is essential for developing new technologies and setting the ground in expanding major global markets to integrate suppliers, manufacturers, warehouses, and stores effectively. In addition, the growing competition in the modern business environment has created an increasing trend of new products and improved product quality for attracting more consumers. However, an increase in the related costs and uncertainty in these innovations requires the development of algorithms to solve optimization problems. Therefore, this study is aimed to implement a multi-product supply chain including raw material suppliers, factories, distributors, and customers to maximize the quality and minimize costs. To this aim, supplier quality, distribution centers, and manufacturing products were considered for the quality model, while warehousing costs, product production, transportation, defective raw materials, and the like were regarded for the cost model. Then, the NSGAII algorithm was used for solving the created optimization problem, and accordingly, the optimal Pareto points were calculated. Based on the results, the proposed model can give the manufacturer the ability to decide on a multi-product and multi-time supply chain by involving cost and quality variables. Thus, the owner can manage the supply chain of the factory effectively.

Performance of Six Metaheuristic Algorithms for Multi-Objective Optimization of Nonlinear Inelastic Steel Trusses

This paper presents a multi-objective optimization of steel trusses using direct analysis. The total weight and the inter-story drift or displacements of the structure were two conflict objectives, while the constraints relating to strength and serviceability load combinations were evaluated using nonlinear inelastic and nonlinear elastic analyses, respectively. Six common metaheuristic algorithms such as nondominated sorting genetic algorithm-II (NSGA-II), NSGA-III, generalized differential evolution (GDE3), PSO-based MOO using crowding, mutation, and ε-dominance (OMOPSO), improving the strength Pareto evolutionary algorithm (SPEA2), and multi-objective evolutionary algorithm based on decomposition (MOEA/D) were applied to solve the developed MOO problem. Four truss structures were studied including a planar 10-bar truss, a spatial 72-bar truss, a planar 47-bar powerline truss, and a planar 113-bar truss bridge. The numerical results showed a nonlinear relationship and inverse proportion between the two objectives. Furthermore, all six algorithms were efficient at finding feasible optimal solutions. No algorithm outperformed the others, but NSGA-II and MOEA/D seemed to be better at both searching Pareto and anchor points. MOEA/D was also more stable and yields a better solution spread. OMOPSO was also good at solution spread, but its stability was worse than MOEA/D. NSGA-III was less efficient at finding anchor points, although it can effectively search for Pareto points.

A comparative study of reverse flow chromatographic reactor and fixed bed reactor: A multi-objective optimization approach

In this study, we have carried out a comparative analysis of a fixed bed reactor (FBR) and reverse flow chromatographic reactor (RFCR) for a series reaction. This comparison has been carried out using mathematical model based single and multi-objective optimization. In addition, since the model merely approximates the actual process, there is an inherent presence of uncertainty in it. To account for the uncertainty, optimization is done considering the uncertainty in the key model parameters. Three objective functions are used in this work for the comparative study, namely yield, selectivity, and conversion. The decision variables used in the optimization problem are the inlet concentration, Damkohler number, and dimensionless switching time. The multi-objective optimization results have been analysed in terms of optimal Pareto fronts for three bi-objective and one tri-objective optimization problems. A detailed analysis of the Pareto fronts for both the reactors is presented for five distinct representative Pareto points. The results of the work indicated the superiority of RFCR over FBR. In particular, a representative Pareto point solution of 3-objective problem in RFCR corresponds to 51.36% higher yield, 27.06% higher selectivity and 19.15% higher conversion as compared to FBR.