Extended Objective Function Research Articles

Uncertainty Analysis of Time-Integrated Activity Coefficient in Single-Time-Point Dosimetry Using Bayesian Fitting Method.

Calculation of the uncertainty of the individual time-integrated activity coefficient (TIACs) is desirable in molecular radiotherapy. However, the calculation of TIAC's uncertainty in single-time-point (STP) method has never been reported in the literature. This study presents a method based on the Bayesian fitting (BF) to calculate the standard deviation (SD) of individual TIACs in the STP dosimetry. Biokinetic data of 177Lu-DOTATATE in kidneys were obtained from PMID33443063. BF methods with extended objective function, which optimize the fitting using prior knowledge of the function's parameters, were used. Reference TIACs (rTIACs) were calculated by fitting a mono-exponential function to the all-time-point data. The goodness of fit was checked based on the visual inspection and the coefficient of variations (CV) of the fitted parameters < 0.5. BF with relative (BFr) and absolute-based (BFa) variance methods were used to obtain the calculated TIACs (cTIACs) from the STP dosimetry. Performance of the STP method was obtained by calculating the relative deviation (RD) between cTIACs and rTIACs. Visual inspection showed a good fit for all patients with CV of fitted parameters less than 50%. The mean ± SD of cTIAC's %RD were 7.0 ± 25.2 for BFr and 2.6 ± 8.9 for BFa. The range of %CV of the individual cTIAC's SD for BFr and BFa methods was 36-78% and 22-33%, respectively, while the %CV of the rTIAC SD was 0.8-49%. We introduce the BF method to calculate the SD of individual TIACs in STP dosimetry. The presented method might be used as an alternative method for uncertainty analysis in STP dosimetry. The online version contains supplementary material available at 10.1007/s13139-024-00851-8.

Error bounds for extended source inversion applied to an acoustic transmission inverse problem

A simple inverse problem for the wave equation requires determination of both the wave velocity in a homogenous acoustic material and the transient waveform of an isotropic point radiator, given the time history of the wavefield at a remote point in space. The duration (support) of the source waveform and the source-to-receiver distance are assumed known. A least squares formulation of this problem exhibits the ‘cycle-skipping’ behavior observed in field scale problems of this type, with many local minima differing greatly from the global minimizer. An extended formulation, dropping the support constraint on the source waveform in favor of a weighted quadratic penalty, eliminates this misbehaviour. With proper choice of the weight operator, the velocity component at any local minimizer of this extended objective function differs from the global minimizer of the least-squares formulation by less than a linear combination of the source waveform support radius and data noise-to-signal ratio.

Analysis and Algorithms for Some Compressed Sensing Models Based on L1/L2 Minimization

Recently, in a series of papers [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649--A3672; C. Wang, M. Tao, J. Nagy, and Y. Lou, SIAM J. Imaging Sci., 14 (2021), pp. 749--777; C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660--2669; P. Yin, E. Esser, and J. Xin, Commun. Inf. Syst., 14 (2014), pp. 87--109], the ratio of $\ell_1$ and $\ell_2$ norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing $\ell_1$/$\ell_2$ subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649--A3672] satisfies the Kurdyka--Łojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660--2669] (see equation 11) under mild assumptions. We next extend the $\ell_1$/$\ell_2$ model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [S. A. Vavasis, Derivation of Compressive Sensing Theorems from the Spherical Section Property, University of Waterloo, 2009; Y. Zhang, J. Oper. Res. Soc. China, 1 (2013), pp. 79--105] and extend the algorithm in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660--2669] (see equation 11) by incorporating moving-balls-approximation techniques [A. Auslender, R. Shefi, and M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232--3259] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by $ \ell_2 $ norm or Lorentzian norm via solving the corresponding $\ell_1$/$\ell_2$ models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy.

Constrained optimization by improved particle swarm optimization with the equivalent penalty coefficient method

The penalty function method has been widely used to solve constrained optimization problems. In the method, an extended objective function, which is the sum of the objective value and the constraint violation weighted by the penalty coefficient, is optimized. However, it is difficult to control the coefficient properly because the proper control depends on each problem. Recently, the equivalent penalty coefficient (EPC) method, which is a new adaptive penalty method for population-based optimization algorithms (POAs), has been proposed. The EPC method can be applied to POAs where a new solution is compared with the old solution. The EPC value, which makes the two extended objective values of the solutions the same, is used to control the coefficient. In this study, we propose to apply the EPC method to particle swarm optimization (PSO) where a new solution is compared with the best solution found so far. To improve the performance of constrained optimization, a mutation operation is also proposed. The proposed method is examined using two topologies of PSO. The advantage of the proposed method is shown by solving well-known constrained optimization problems and comparing the results with those obtained by PSO with a standard constraint-handling technique.

Infinite-Horizon Optimal Switching Regions for a Pair-Trading Strategy with Quadratic Risk Aversion Considering Simultaneous Multiple Switchings: A Viscosity Solution Approach

Very few studies have explored the structure of optimal switching regimes. We extend the existing research on the infinite-horizon multiple-regime switching problem with an arbitrary number of switch options by replacing the linear running reward function with a quadratic function in the objective function. To make our analysis more rigorous, we establish the theoretical basis for the application of the simultaneous multiple-regime switches to the problem with the extended objective function, and provide the sufficient condition under which each certain separated region in the space includes, at most, one single connected optimal switching region, which determines the structure of the optimal switching regions, and we identify the structure of the optimal switching regions for the particular problem.

Artificial bee colony–based Kalman filter hybridization for three–dimensional position estimation of a quadrotor

Purpose The purpose of this paper is to introduce an artificial bee colony-based Kalman filter algorithm along with an extended objective function to ensure the optimality of the estimator of the quadrotor in the presence of unknown measurement noise statistics. Design/methodology/approach Six degree-of-freedom mathematical model of the quadrotor is derived. Position controller for the quadrotor is designed. Kalman filter-based estimation algorithm is implemented in the sensor feedback loop. Artificial bee colony-based hybrid algorithm is used as an optimization method to handle the unknown noise statistics. Existing objective function is extended with a penalty term. Mathematical proof of the extended objective function is derived. Results of the proposed algorithm is compared with de facto genetic algorithm-based Kalman filter. Findings Artificial bee colony algorithm-based Kalman filter and extended objective function duo are able to optimize the measurement noise covariance matrix with an absolute error as low as 0.001 [m2]. Proposed method and function is capable of reducing the noise from 2 to 0.09 [m] for x-axis, 3.4 to 0.14 [m] for y-axis and 3.7 to 0.2 [m] for z-axis, respectively. Originality/value The motivation behind this paper is to bring a novel optimization-based solution for the estimation problem of the quadrotor when the measurement noise statistics are unknown along with an extended objective function to prevent the infeasible solutions with mathematical convergence analysis.

Chromatic Correlation Clustering

We study a novel clustering problem in which the pairwise relations between objects are categorical . This problem can be viewed as clustering the vertices of a graph whose edges are of different types ( colors ). We introduce an objective function that ensures the edges within each cluster have, as much as possible, the same color. We show that the problem is NP -hard and propose a randomized algorithm with approximation guarantee proportional to the maximum degree of the input graph. The algorithm iteratively picks a random edge as a pivot, builds a cluster around it, and removes the cluster from the graph. Although being fast, easy to implement, and parameter-free, this algorithm tends to produce a relatively large number of clusters. To overcome this issue we introduce a variant algorithm, which modifies how the pivot is chosen and how the cluster is built around the pivot. Finally, to address the case where a fixed number of output clusters is required, we devise a third algorithm that directly optimizes the objective function based on the alternating-minimization paradigm. We also extend our objective function to handle cases where object’s relations are described by multiple labels. We modify our randomized approximation algorithm to optimize such an extended objective function and show that its approximation guarantee remains proportional to the maximum degree of the graph. We test our algorithms on synthetic and real data from the domains of social media, protein-interaction networks, and bibliometrics. Results reveal that our algorithms outperform a baseline algorithm both in the task of reconstructing a ground-truth clustering and in terms of objective-function value.

Estimation of Relation between Financial Variables and Gold Price Using Fuzzy Multiple Linear Regression

Price forecasting is an integral part of economic decision making. Forecasts may be used in numerous ways; specifically, individuals may use forecasts to try to earn income from speculative activities, to determine optimal government policies or to make business decisions. The importance of this topic is caused by instability in the world economy and stock markets; there is a growing interest in gold from investors and the current bull market of gold. It is believed that gold can be used to predict, to some degree, the stock market trend. However, the current forecasting algorithm offers poor precision in solutions for non-linear problems. In India, gold is one of the foundation assets for Indian households in the form of investment. It is viewed as secure and liquid investment. In this paper, a forecasting model for gold price using Multiple fuzzy linear regression using shape preserving operations were proposed with an extended objective function .Keywords: Fuzzy linear regression, linear programming, Gold price, Financial variables

Evaluating geo-environmental variables using a clustering based areal model

Global regression models do not accurately reflect the spatial heterogeneity which characterises most geo-environmental variables. In analysing the relationships between such variables, an approach is required which allows the model parameters to vary spatially. This paper proposes a new framework for exploring local relationships between geo-environmental variables. The method is based on extended objective function based fuzzy clustering with the environmental parameters estimated through on a locally weighted regression analysis. The case studies and prediction evaluations show that the fuzzy algorithm yields well-fitted models and accurate predictions. In addition to an increased accuracy of prediction relative to the widely-used geographically weighted regression (GWR), the proposed algorithm provides the search radius (bandwidth) and weights for local estimation directly from the data. The results suggest that the method could be employed effectively in tackling real world kernel-based modelling problems.

An optimal strategy for leukemia therapy: a multi-objective approach

In this work we introduce a multi-objective optimization problem using the example of a leukemia treatment model. We believe that treatment affects not only leukemia cells, but also the healthy cells. The treatment effect is modelled as a therapy function. The optimization problem con- sists of two objective functions that are in conflict: on the o ne hand, minimizing the leukemia cells and on the other hand, maximizing the number of healthy cells. We reduce this multi-objective prob- lem by using the e-constraint method. With the aid of Pontryagin's Maximum Pr inciple we give an analytical solution to this reduced problem. In order to sol ve the problem using the epsilon-constraint- method, the restriction of a threshold value for the number of healthy cells is separately considered as an optimization problem with a new extended objective function. For the most relevant parameters the maximum dose of chemotherapeutics should be administered as long as the predetermined restrictions are not violated. Furthermore, the case in which singular control may occur during the therapy process is analysed. In this case, the optimal control is also determ ined.

Decision making in livestock conservation

In this article, theory and practical implementations of decision making in livestock conservation are reviewed. It is argued that the objective of livestock conservation is a composite of maintaining between and within breed diversity and single breeds of recognised value. Assuming that this goal can be reflected in an objective function, decision making requires maximisation of this objective function with or without restrictions. Different strategies have been proposed in the literature to achieve this goal. While the diversity measure suggested by Weitzman (1992) [Weitzman, M.L., 1992. On diversity. Quart. J. Economics, CVII: 363-405.] is not fully appropriate to be applied to breed conservation within livestock species, the approach of maximising expected diversity Weitzman (1993) [Weitzman, M.L., 1993. What to preserve? An application of diversity theory to crane conservation. Quart. J. Economics, CVII: 157-183.] is seen as a fundamental concept in this area. Although highly elaborate methods of decision making are discussed in the scientific literature, the concepts presently used by national or international authorities or non-government organisations are rather simplistic, mainly risk-related and based on simple functions of population size. It is argued that decision making has to account for the global diversity of a species and therefore decisions and conservation activities should be coordinated on an international level. The use of more appropriate decision rules will strongly increase the cost efficiency of conservation investments. Livestock conservation should be based on an extended objective function reflecting the expected future value of the conserved set of breeds, which encompasses within and between breed diversity as well as specific traits and cultural or scientific value of the main breeds. A critical issue is the derivation of the required parameters like the breed value or the risk status of a breed. Since it will generally be difficult to obtain exact values, decision making under uncertainty will be the usual challenge and Bayesian decision theory might be an option. The optimum allocation pattern and implementation of conservation activities can be derived based on Weitzman's expected diversity concept. Approximate decision rules for breed prioritisation based on the extinction probability of a breed and its marginal objective function are given. Even if the respective parameters are not perfectly known, the use of the suggested approaches has the potential to double the cost efficiency, in terms of maintained diversity per conservation dollar spent, compared to the simplistic approaches that are used today.

Inverse modeling of unsaturated heterogeneous media by two-phase pressure derivative matching

Inverse modeling by the pressure derivative matching method for two-phase fluid flow was developed and applied to construct the heterogeneous permeability distribution of a vertical sandstone plate using air injection test data under unsaturated conditions. To reduce the computing effort, the adjoint state method was used to calculate gradients of the extended objective function which includes the squared sum of differences between calculated pressure change rate and observed one. The method can also treat capillary pressure. The quasi-Newton method was used as an optimization algorithm. Laboratory tests were carried out to obtain transient pressure data at multiple points in the unsaturated zone of the sandstone.We compared the pressure derivative matching method with a conventional matching method. The results show that the pressure derivative matching method is effective for transient pressure data obtained by the laboratory test.

拡張目的関数によるモデル化誤差の影響を考慮した逆解析

拡張目的関数によるモデル化誤差の影響を考慮した逆解析

Nonlinear image recovery with half-quadratic regularization

One popular method for the recovery of an ideal intensity image from corrupted or indirect measurements is regularization: minimize an objective function that enforces a roughness penalty in addition to coherence with the data. Linear estimates are relatively easy to compute but generally introduce systematic errors; for example, they are incapable of recovering discontinuities and other important image attributes. In contrast, nonlinear estimates are more accurate but are often far less accessible. This is particularly true when the objective function is nonconvex, and the distribution of each data component depends on many image components through a linear operator with broad support. Our approach is based on an auxiliary array and an extended objective function in which the original variables appear quadratically and the auxiliary variables are decoupled. Minimizing over the auxiliary array alone yields the original function so that the original image estimate can be obtained by joint minimization. This can be done efficiently by Monte Carlo methods, for example by FFT-based annealing using a Markov chain that alternates between (global) transitions from one array to the other. Experiments are reported in optical astronomy, with space telescope data, and computed tomography.

Optimal Control of Stochastic Non-linear Models

The use of large macroeconometric models in policy formulation leads inevitably towards specifying a set of objectives and trying to meet these objectives as closely as possible. The natural formal framework in which to characterise this procedure is clearly that of optimal control: however, macroeconometnc models are by their nature stochastic, and when they are also non-linear the solution to the deterministic control problem may differ substantially from the solution to the stochastic problem. There is therefore a need for a practical procedure which would allow optimal control techniques to be extended to stochastic non-linear models.In this paper, we propose a new technique which combines the approach of stochastic simulation with standard optimal control techniques to produce an algorithm which allows such an extension. The technique involves extending and rearranging the standard objective function to allow for the stochastic nature of the model. It then iterates between a standard optimal control algorithm using this extended objective function and stochastic simulation around the optimal control solution until convergence is achieved. We also show that such a technique may be used to derive optimal feedback rules where the minimising of the variance of a target variable around some baseline is a part of the objective function. The paper concludes with an example of the technique applied to the Bank of England's quarterly macroeconometric model.