Optimization Of Multiple Objective Functions Research Articles

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

A parallel variable population multi-objective optimizer for accelerator beam dynamics optimization

The simultaneous optimization of multiple objective functions is needed in many particle accelerator applications. In this paper, we present a parallel evolution based multi-objective optimizer that uses a variable population from generation to generation and an external storage to save good solutions. Two heuristic optimization methods, one uses the unified differential evolution and the other uses the real-coded genetic algorithm, are included in the optimizer to generate next generation candidate solutions, and are compared in the test examples. As an application, we applied this optimizer to the beam dynamics design optimization of a photoinjector and attained the optimal front solutions after 200 generations with the unified differential evolution offspring production scheme.

A Method for Integration of Preferences to a Multi-Objective Evolutionary Algorithm Using Ordinal Multi-Criteria Classification

Most real-world problems require the optimization of multiple objective functions simultaneously, which can conflict with each other. The environment of these problems usually involves imprecise information derived from inaccurate measurements or the variability in decision-makers’ (DMs’) judgments and beliefs, which can lead to unsatisfactory solutions. The imperfect knowledge can be present either in objective functions, restrictions, or decision-maker’s preferences. These optimization problems have been solved using various techniques such as multi-objective evolutionary algorithms (MOEAs). This paper proposes a new MOEA called NSGA-III-P (non-nominated sorting genetic algorithm III with preferences). The main characteristic of NSGA-III-P is an ordinal multi-criteria classification method for preference integration to guide the algorithm to the region of interest given by the decision-maker’s preferences. Besides, the use of interval analysis allows the expression of preferences with imprecision. The experiments contrasted several versions of the proposed method with the original NSGA-III to analyze different selective pressure induced by the DM’s preferences. In these experiments, the algorithms solved three-objectives instances of the DTLZ problem. The obtained results showed a better approximation to the region of interest for a DM when its preferences are considered.

Twenty years of continuous multiobjective optimization in the twenty-first century

The survey highlights some of the research topics which have attracted attention in the last two decades within the area of mathematical optimization of multiple objective functions. We give insights into topics where a huge progress can be seen within the last years. We give short introductions to the specific sub-fields as well as some selected references for further reading. Primarily, the survey covers the progress in the development of algorithms. In particular, we discuss publicly available solvers and approaches for new problem classes such as non-convex and mixed integer problems. Moreover, bilevel optimization problems and the handling of uncertainties by robust approaches and their relation to set optimization are presented. In addition, we discuss why numerical approaches which do not use scalarization techniques are of interest.

Multi-Objective Function for SWIPT System by SADDE

Simultaneous wireless information and power transfer (SWIPT) optimization with multiple objective function optimization is presented in the millimeter band in this paper. Three different objective functions that are used for harvest power (HP), capacity, and bit error rate (BER) were studied. There are three different nodes in real environment for wireless power transfer (WPT) and SWIPT. The channel estimation calculated by shooting and bouncing ray/image techniques includes multi-path, fading effect, and path-loss in the real environment. We applied beamforming techniques at the transmitter to focus the transmitter energy in order to reduce the multi-path effect and adjust the length of the feed line on each array element in order to find the extremum of the objective functions by the self-adaptive dynamic differential evolution (SADDE) method. Numerical results showed that SWIPT node cannot achieve good performance by single objective function, but wireless power transfer (WPT) can. Nevertheless, both WPT and SWIPT nodes can meet the criteria by the multiple objective function. The harvesting power ratio as well as the BER and capacity can be improved by the multiple objective function to an acceptable level by only reducing a little harvesting energy compared to the best harvesting energy for the single objective function. Finally, the multiple optimization function cannot merely provide good information quality for SWIPT node but achieve good total harvesting power for WPT and SWIPT node as well.

Multiobjective Optimisation of Bevel Gear Pair design using NSGA-II

Abstract Spiral bevel gears mesh with additional overlapping tooth action compared to straight bevels. Because of this, the transmission of motion will be smoother than that by straight bevel gears or zero bevel gears. As a result they have reduced noise and vibration, particularly occuring at high speeds. Weight reduction with simultaneous increase in transmission efficiency, by optimization method, is very much essential for gear applications which transmit very high mechanical power, with a significant reduction in speed. In many instances, in order to satisfy technical, economic and security requirements, it is essential to undertake a long development process. This research paper aims to optimize the design of a pair of spiral bevel gears, using NSGA-II, a non-dominated sorting genetic algorithm for optimization of multiple objective functions.

Pareto-front shape in multiobservable quantum control

Many scenarios in the sciences and engineering require simultaneous optimization of multiple objective functions, which are usually conflicting or competing. In such problems the Pareto front, where none of the individual objectives can be further improved without degrading some others, shows the tradeoff relations between the competing objectives. This paper analyzes the Pareto-front shape for the problem of quantum multiobservable control, i.e., optimizing the expectation values of multiple observables in the same quantum system. Analytic and numerical results demonstrate that with two commuting observables the Pareto front is a convex polygon consisting of flat segments only, while with noncommuting observables the Pareto front includes convexly curved segments. We also assess the capability of a weighted-sum method to continuously capture the points along the Pareto front. Illustrative examples with realistic physical conditions are presented, including NMR control experiments on a $^{1}\mathrm{H}\ensuremath{-}^{13}\mathrm{C}$ two-spin system with two commuting or noncommuting observables.

SU‐F‐J‐02: Flexible Training of MR‐Guided Laser Ablation Models Via Global Optimization

Purpose:MR‐guided laser ablation is currying interest in neurosurgery as a minimally invasive procedure. The procedure is planned via an assumed lesion overlaid on various anatomic and functional images. Computational models of heat transfer may further inform planning the intervention, reducing surgical morbidity and ensuring therapeutic effect. These models require physical parameter data, via less relevant literature values. Using commercialized laser ablation equipment combined with MR temperature imaging (MRTI) feedback, recent work has demonstrated the use of inverse problems as a means to gain the necessary information via retrospective analysis. Here, performance of a global inverse problem solution is explored that allows for simultaneous optimization of multiple objective functions.Methods:A steady state bioheat model is trained on MRTI patient datasets (n=20); the MRTI regions exceeding a 57°C threshold are considered lethally damaged. Optical and blood perfusion parameters are densely sampled, allowing for global optimization of the model. There are two objective functions to optimize with respect to damage prediction: Dice similarity coefficient (DSC) and Hausdorff distance (HD). DSC is an overlap metric; HD is the distance‐to‐agreement in physical distance. Predictive performance is simulated via leave‐one‐out cross‐validation (LOOCV) and is compared to the same model using naive literature values.Results:Post‐optimization and during LOOCV, mean and median DSC are 0.824 and 0.850 respectively; mean and median HD are 4.64mm and 4.03mm respectively. These compare favorably to the model using literature values. Mean and median DSC are 0.6685 and 0.6564 respectively; mean and median HD are 5.85mm and 5.58mm respectively. A two‐tailed paired Student's t‐test for the mean DSC reported p=0.0002. For mean HD, p=0.0064.Conclusion:Clinical MRTI feedback provides useful information for designing optimized laser ablation models. Global optimization facilitates simultaneous optimization of multiple objective functions. Trained models are considerably more accurate than naive literature based models as measured by DSC and HD.

Multi-objective optimization for reservoir modelling and seismic data matching: proof of concept and field application

We present a new method to generate reservoir models by combining geostatistical simulation and optimization of multiple objective functions; including seismic data matching (i.e. a reservoir model seismic matching loop). Our method is used to estimate static reservoir models by simultaneously integrating several datasets including well logs, geologic information and various seismic attributes. The key advantage of our proposed method is that we can define multiple objective functions for a variety of data types and constraints, and simultaneously minimize the data misfits. Using our optimization method, the resulting models converge towards Pareto fronts, which represent the sets of best compromise model solutions for the defined objectives. We test our new approach on a 3D object-oriented reservoir model, where variogram-based simulation techniques typically fail to produce realistic models. Our results indicate that improved reservoir facies and porosity models and flow-unit connectivity can be obtained with this new multi-objective optimization approach.

Simultaneous optimization of multiple objective functions for reservoir modeling

We have developed a new method, a synthetic validity test, and a field data demonstration for constructing reservoir models that simultaneously match available seismic, borehole, and geologic data in an optimal way. We combine geostatistical simulation with simultaneous nonlinear stochastic optimization of multiple objective functions, and we do not require the selection of weighting schemes or multiple optimization computational runs. Because each geologic and geophysical data set has its own strengths and weaknesses, realistic models are best obtained by simultaneously matching all of the multiple data constraints. We define a set of objective functions for each of the multiple data sets that are used to constrain the reservoir-modeling optimization. In the examples, we specify an [Formula: see text]-norm misfit to match inverted seismic attribute volumes, an [Formula: see text]-norm to match lithofacies and porosity logs, and the Hausdorff metric to match image patterns in geologic map-based information. We use a geostatistical technique to initialize a population (ensemble) of starting models and run a nonlinear evolutionary algorithm to drive the ensembles of model solutions toward the Pareto optimal front, along which represents the best-compromise set of model solutions that simultaneously satisfy all of the multiple objectives. We tested our method on a 3D synthetic object-oriented reservoir model for which variogram-based simulation techniques typically fail to reproduce realistic models. We applied our method on a producing reservoir located offshore Western Australia. Our results indicate that the new method is capable of producing optimal models of reservoir properties and flow-unit connectivity that are consistent with known reservoir information, including dynamic production data such as pressure interference tests, and 4D seismic monitoring data.

On the design of shared memory approaches to parallelize a multiobjective bee-inspired proposal for phylogenetic reconstruction

Current efforts in solving computationally demanding optimization problems in bioinformatics rely on the combination of bioinspired computing and parallelism. The multiobjective nature shown by a wide variety of these problems represents an additional challenge, as the optimization of multiple objective functions involves growing computational requirements. In this work, a multiobjective metaheuristic inspired by honey bees is applied to tackle the phylogenetic inference problem. For this purpose, two parallel implementations for shared memory architectures are proposed: a synchronous generational model and a novel non-generational design inspired by the asynchronous behaviour of bees in nature. Experiments on six real biological data sets and comparisons with other parallel biological methods point out the relevance of applying nature-inspired parallelization strategies, addressing the performance pitfalls shown by traditional parallelization schemes.

Parallel Multiobjective Metaheuristics for Inferring Phylogenies on Multicore Clusters

The development of efficient parallel algorithms based on mixed mode programming represents one of the most popular lines of research in current bioinformatics. By exploiting hardware resources at inter-node/intra-node level, we can address grand computational challenges which involve the optimization of multiple objective functions simultaneously. In this sense, the inference of evolutionary trees represents one of the most difficult NP-hard problems in the field. Tackling such a problem requires efficient parallel designs to take advantage of the characteristics of modern multicore clusters. In this paper, we aim to solve the phylogenetic inference problem by applying MPI/OpenMP schemes to two multiobjective metaheuristics: fast non-dominated sorting genetic algorithm and multiobjective firefly algorithm. In order to assess the performance achieved by these proposals under different system and problem sizes, we have conducted experiments on six real nucleotide data sets according to a statistical methodology. Our parallel and multiobjective metrics point out the relevance of combining hybrid programming and novel bioinspired designs with regard to other parallel and biological approaches from the literature.

Containing Small Hydropower and Wind Power System Optimal Spinning Reserve

When larger scale of small hydropower and wind power exist in power system, the traditional reserve capacity algorithm is no longer applicable at the same time of reducing the carbon emission. Aiming at the volatility and non-regulatory of small hydropower and wind power, considering the forecast error of small hydropower, wind power and load,the unit outage rate and other factors, combined with the actual operation parameters of one regional power system in Southwest of China, the wind-hydro – thermal power coordinated operation of multiple objective function optimization calculation model of reserve capacity is established. Using the weighted coefficient method of unified objective method, it transforms the multi-objective optimization problem into a single objective optimization one. Applying the exterior penalty function method, the constrained optimization problem is changed into a non-constrained optimization one. The improved particle swarm optimization algorithm is got by introducing the particle concentration cognition into traditional particle swarm optimization algorithm. The simulation results verify the validity of the model. To Compare them under different strategies, this method can get less fuel expenses of thermal power units in the case of lower loss of load probability (LOLP).With the small hydropower and wind power output size to optimize the spinning reserve capacity. The model and the algorithm are helpful for drawing up optimization strategies of the reserve capacity in those regions in which larger scale small hydropower and wind power exists.

Generating optimized configurable business process models in scenarios subject to uncertainty

ContextThe quality of business process models (i.e., software artifacts that capture the relations between the organizational units of a business) is essential for enhancing the management of business processes. However, such modeling is typically carried out manually. This is already challenging and time consuming when (1) input uncertainty exists, (2) activities are related, and (3) resource allocation has to be considered. When including optimization requirements regarding flexibility and robustness it becomes even more complicated potentially resulting into non-optimized models, errors, and lack of flexibility. ObjectiveTo facilitate the human work and to improve the resulting models in scenarios subject to uncertainty, we propose a software-supported approach for automatically creating configurable business process models from declarative specifications considering all the aforementioned requirements. MethodFirst, the scenario is modeled through a declarative language which allows the analysts to specify its variability and uncertainty. Thereafter, a set of optimized enactment plans (each one representing a potential execution alternative) are generated from such a model considering the input uncertainty. Finally, to deal with this uncertainty during run-time, a flexible configurable business process model is created from these plans. ResultsTo validate the proposed approach, we conduct a case study based on a real business which is subject to uncertainty. Results indicate that our approach improves the actual performance of the business and that the generated models support most of the uncertainty inherent to the business. ConclusionsThe proposed approach automatically selects the best part of the variability of a declarative specification. Unlike existing approaches, our approach considers input uncertainty, the optimization of multiple objective functions, as well as the resource and the control-flow perspectives. However, our approach also presents a few limitations: (1) it is focused on the control-flow and the data perspective is only partially addressed and (2) model attributes need to be estimated.

Multicomponent pre-stack seismic waveform inversion in transversely isotropic media using a non-dominated sorting genetic algorithm

Inversion of band- and offset-limited single component (P wave) seismic data does not provide robust estimates of subsurface elastic parameters and density. Multicomponent seismic data can, in principle, circumvent this limitation but adds to the complexity of the inversion algorithm because it requires simultaneous optimization of multiple objective functions, one for each data component. In seismology, these multiple objectives are typically handled by constructing a single objective given as a weighted sum of the objectives of individual data components and sometimes with additional regularization terms reflecting their interdependence; which is then followed by a single objective optimization. Multi-objective problems, inclusive of the multicomponent seismic inversion are however non-linear. They have nonunique solutions, known as the Pareto-optimal solutions. Therefore, casting such problems as a single objective optimization provides one out of the entire set of the Pareto-optimal solutions, which in turn, may be biased by the choice of the weights. To handle multiple objectives, it is thus appropriate to treat the objective as a vector and simultaneously optimize each of its components so that the entire Pareto-optimal set of solutions could be estimated. This paper proposes such a novel multi-objective methodology using a non-dominated sorting genetic algorithm for waveform inversion of multicomponent seismic data. The applicability of the method is demonstrated using synthetic data generated from multilayer models based on a real well log. We document that the proposed method can reliably extract subsurface elastic parameters and density from multicomponent seismic data both when the subsurface is considered isotropic and transversely isotropic with a vertical symmetry axis. We also compute approximate uncertainty values in the derived parameters. Although we restrict our inversion applications to horizontally stratified models, we outline a practical procedure of extending the method to approximately include local dips for each source-receiver offset pair. Finally, the applicability of the proposed method is not just limited to seismic inversion but it could be used to invert different data types not only requiring multiple objectives but also multiple physics to describe them.