Set Of Test Problems Research Articles

Generalized RMIL conjugate gradient method under the strong Wolfe line search with application in image processing

RMIL conjugate gradient method originally proposed by Rivaie et al. (2012) has recently gained lots of attention. In this article, we propose a generalized conjugate gradient parameter that contains both RMIL and its variant, that is, RMIL+, as special cases. We show that the search direction generated by the new method is sufficiently descent. Under standard mild conditions, we discuss the convergence analysis of the propose method. We demonstrate the numerical efficiency of the propose method on a set of unconstrained minimization benchmark test problems as well as an image restoration problem. The results of the experiment reveal that the proposed method performs better than its main competitors.

Gradient-based descent linesearch to solve interval-valued optimization problems under gH-differentiability with application to finance

In this article, gradient based descent line search scheme is proposed to solve interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational point of view. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient, without transforming it into real valued function. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search for interval optimization problem. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by our proposed scheme.

A new hybrid conjugate gradient algorithm for unconstrained optimization

It is well known that conjugate gradient methods are useful for solving large-scale unconstrained nonlinear optimization problems. In this paper, we consider combining the best features of two conjugate gradient methods. In particular, we give a new conjugate gradient method, based on the hybridization of the useful DY (Dai-Yuan), and HZ (Hager-Zhang) methods. The hybrid parameters are chosen such that the proposed method satisfies the conjugacy and sufficient descent conditions. It is shown that the new method maintains the global convergence property of the above two methods. The numerical results are described for a set of standard test problems. It is shown that the performance of the proposed method is better than that of the DY and HZ methods in most cases.

Two-stage species conservation for multimodal multi-objective optimization with local Pareto sets

Compared to general multi-objective optimization problems, multimodal multi-objective optimization problems (MMOPs) with local Pareto sets (PSs) must determine multiple global and local PSs simultaneously. Therefore, MMOPs with local PSs are challenging. To resolve this issue, this study proposes a multimodal multi-objective optimization evolutionary algorithm based on two-stage species conservation (MMOEA/TSC). MMOEA/TSC divides the evolutionary process into two stages: diversity-oriented species conservation and convergence-oriented species conservation. The former is aimed at locating promising regions in which global and local PSs may exist. To balance the distribution of solutions, a Gaussian variation strategy is used to iteratively generate diverse offspring in regions that contain the smallest number of solutions. The latter mainly focused on obtaining one PS with good convergence in each promising region. To help the solutions converge to the global and local PSs uniformly, a species stratification strategy was adopted according to the Pareto level of the well-converged solution for each species. The proposed algorithm was compared with seven state-of-the-art algorithms. For the CEC 2020 MMOP test problem set, the experimental results show that MMOEA/TSC has the capacity to find global and local PSs.

Integrated mixed-model assembly line balancing and parts feeding with supermarkets

In the automotive industry, decentralized in-house logistic areas called supermarkets are used for parts feeding to mixed-model assembly lines. In the literature, generally, the number of supermarkets and their locations are determined according to a previously balanced assembly line. As a result, supermarket locations have to be completely dependent on the line balance. There may be more than one solution that gives the same performance criterion value in the line balancing. Among these alternative solutions, it is more reasonable to use a solution that provides lower logistics costs than others. For this purpose, in this study, the mixed-model assembly line balancing problem and the supermarket location problem are considered simultaneously. Accordingly, in order to minimize the total costs of the assembly line and supermarkets, a mixed-integer mathematical model is developed, and the developed model is solved by using constraint programming (CP). In addition, a new approach based on Ant Colony Optimization (ACO) and Simulated Annealing (SA) is presented for large-sized problems. An illustrative example problem is solved using both the developed model and the proposed algorithm. The effectiveness of the proposed approach is tested through a new data set of test problems presented to the literature for a computational experiment. The experimental results show that the total costs are reduced meaningfully and a more realistic and applicable structure is achieved by the proposed approach.

A perturbation based modified BAT algorithm for priority based covering salesman problem

In this study, Bat algorithm (BA) is modified along with K-opt operation and one newly proposed perturbation approach to solve the well known covering salesman problem (CSP). Here, along with the restriction of the radial distances of the unvisited cities from the visited cities another restriction is imposed where a priority is given to some cities for the inclusion in the tour, i.e., some clusters to be created where the prioritised cities must be the visiting cities and the corresponding CSP is named as Prioritised CSP (PCSP). In the algorithm, 3-opt and 4-opt operations are used for two different purposes. The 4-opt operation is applied for generating an initial solution set of CSP for the BA and the 3-opt operation generates some perturbed solutions of a solution. A new perturbation approach is proposed for generating neighbour solutions of a potential solution where the exchange of some cities in the tour is made and is named as K-bit exchange operation. The proposed solution approach for the CSP and PCSP is named as the modified BA embedded with K-bit exchange and K-opt operation (MBAKEKO). It is a two-stage algorithm where in the first stage of the algorithm the clustering of the cities is done with respect to a fixed visiting city of each cluster in such a manner that the distances of the other cities of the cluster must lie with in the fixed covering distance of the problem and in the second stage the BA is applied to find the minimum cost Hamiltonian circuit by passing through the visiting cities of the clusters. MBAKEKO is tested with a set of benchmark test problems with significantly large sizes from the TSPLIB. To measure the performance of MBAKEKO, its results are compared with the results of different well-known approaches for CSPs available in the literature. It is observed from the comparison studies that MBAKEKO searches the minimum cost tour for any of the considered instances compared to all other well-known algorithms in the literature. It can be concluded from the numerical studies that the performance of MBAKEKO is better with respect to the state-of-the-art algorithms available in the literature.

Instance Space Analysis for Algorithm Testing: Methodology and Software Tools

Instance Space Analysis (ISA) is a recently developed methodology to (a) support objective testing of algorithms and (b) assess the diversity of test instances. Representing test instances as feature vectors, the ISA methodology extends Rice’s 1976 Algorithm Selection Problem framework to enable visualization of the entire space of possible test instances, and gain insights into how algorithm performance is affected by instance properties. Rather than reporting algorithm performance on average across a chosen set of test problems, as is standard practice, the ISA methodology offers a more nuanced understanding of the unique strengths and weaknesses of algorithms across different regions of the instance space that may otherwise be hidden on average. It also facilitates objective assessment of any bias in the chosen test instances and provides guidance about the adequacy of benchmark test suites. This article is a comprehensive tutorial on the ISA methodology that has been evolving over several years, and includes details of all algorithms and software tools that are enabling its worldwide adoption in many disciplines. A case study comparing algorithms for university timetabling is presented to illustrate the methodology and tools.

Secant penalized BFGS: a noise robust quasi-Newton method via penalizing the secant condition

In this paper, we introduce a new variant of the BFGS method designed to perform well when gradient measurements are corrupted by noise. We show that by treating the secant condition with a penalty method approach motivated by regularized least squares estimation, one can smoothly interpolate between updating the inverse Hessian approximation with the original BFGS update formula and not updating the inverse Hessian approximation. Furthermore, we find the curvature condition is smoothly relaxed as the interpolation moves towards not updating the inverse Hessian approximation, disappearing entirely when the inverse Hessian approximation is not updated. These developments allow us to develop a method we refer to as secant penalized BFGS (SP-BFGS) that allows one to relax the secant condition based on the amount of noise in the gradient measurements. SP-BFGS provides a means of incrementally updating the new inverse Hessian approximation with a controlled amount of bias towards the previous inverse Hessian approximation, which allows one to replace the overwriting nature of the original BFGS update with an averaging nature that resists the destructive effects of noise and can cope with negative curvature measurements. We discuss the theoretical properties of SP-BFGS, including convergence when minimizing strongly convex functions in the presence of uniformly bounded noise. Finally, we present extensive numerical experiments using over 30 problems from the CUTEst test problem set that demonstrate the superior performance of SP-BFGS compared to BFGS in the presence of both noisy function and gradient evaluations.

A Newton-type proximal gradient method for nonlinear multi-objective optimization problems

In this paper, a globally convergent Newton-type proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth function. The proposed method deals with unconstrained convex multi-objective optimization problems. This method is free from any kind of priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a suitable descent direction. The subproblem uses a quadratic approximation of each smooth function. An Armijo type line search is conducted to find a suitable step length. A sequence is generated using the descent direction and the step length. The global convergence of this method is justified under some mild assumptions. The proposed method is verified and compared with some existing methods using a set of test problems.

A Study on a Probabilistic Method for Designing Artificial Neural Networks for the Formation of Intelligent Technology Assemblies with High Variability

Currently, ensemble approaches based, among other things, on the use of non-network models are powerful tools for solving data analysis problems in various practical applications. An important problem in the formation of ensembles of models is ensuring the synergy of solutions by using the properties of a variety of basic individual solutions; therefore, the problem of developing an approach that ensures the maintenance of diversity in a preliminary pool of models for an ensemble is relevant for development and research. This article is devoted to the study of the possibility of using a method for the probabilistic formation of neural network structures developed by the authors. In order to form ensembles of neural networks, the influence of parameters of neural network structure generation on the quality of solving regression problems is considered. To improve the quality of the overall ensemble solution, using a flexible adjustment of the probabilistic procedure for choosing the type of activation function when filling in the layers of a neural network is proposed. In order to determine the effectiveness of this approach, a number of numerical studies on the effectiveness of using neural network ensembles on a set of generated test tasks and real datasets were conducted. The procedure of forming a common solution in ensembles of neural networks based on the application of an evolutionary method of genetic programming is also considered. This article presents the results of a numerical study that demonstrate a higher efficiency of the approach with a modified structure formation procedure compared to a basic approach of selecting the best individual neural networks from a preformed pool. These numerical studies were carried out on a set of test problems and several problems with real datasets that, in particular, describe the process of ore-thermal melting.

A Hybrid Conjugate Gradient Algorithm for Nonlinear System of Equations through Conjugacy Condition

the purpose of solving a large-scale system of nonlinear equations, a hybrid conjugate gradient algorithm is introduced in thispaper, based on the convex combination ofβFRkandβPRPkparameters. It is made possible by incorporating the conjugacy condition togetherwith the proposed conjugate gradient search direction. Furthermore, a significant property of the method is that through a non-monotone typeline search it gives a descent search direction. Under appropriate conditions, the algorithm establishes its global convergence. Finally, resultsfrom numerical tests on a set of benchmark test problems indicate that the method is more effective and robust compared to some existingmethods.

On the optimal solution of a distributionally robust multi-product newsvendor problem

This paper derives the optimal solution for a distributionally robust multi-product newsvendor problem, in which different products are produced under a capacity constraint, while only the mean and variance of the product demand are known. The problem aims to find a capacity allocation scheme to minimize the system cost of the worst-case among all possible demand distributions. When the total capacity is among certain ranges, the optimal solution has a closed-form. For other ranges, the optimal solution can be derived by solving one equation. The solution shows that there exist a number of threshold values for the capacity, below which some products are neglected (allocated with zero capacity). Besides, in the optimal solution, which products are prioritized for production is determined by the order of an index measured by the unit production cost, unit shortage cost, demand mean, and demand variance. For a special case where the cost structures for different products are identical, a closed-form solution is derived for any total capacity value. For this special case, the index order is simply determined by the coefficient of variation of each product demand. Sensitivity analysis shows that when the capacity is abundant, a larger demand variability of one product may cause a higher production quantity for this product; while if the capacity is tight, a larger demand variability cause a lower production quantity. For a set of test problems, the performance of the robust optimization solution is quite close to that of the stochastic optimization solution.

Using a Random Forest Model to Choose Optimized Group Structures

It is challenging to select an appropriate group structure for any given multigroup neutron transport problem. Many group structures were designed long ago, and the reasoning behind the creator’s choices may be unknown. In this work, we apply the simulated annealing optimization method to develop improved group structures for a set of test problems. We then use a random forest (a machine learning method) to identify which group structure will be the best for any new problem based on input characteristics, such as geometry and isotopics. Simulated annealing spans a large solution space before narrowing in on an optimal solution, avoiding local minima by jumping around. Our solution space, however, is large and inconsistent, making finding the optimal group structure infeasible. Instead, we find potentially optimal group structures, ones that yield more accurate solutions than our standard group structures, but are probably not the “best” possible. Group structures are obtained for six classes of problems, ranging from a fast 233U system to a thermal 239Pu system. These were chosen to encompass a series of critical assemblies from the International Criticality Safety Benchmark Evaluation Project (ICSBEP) handbook. These optimized group structures were used in PARTISN for a large range of ICSBEP critical assemblies and compared to the traditional Los Alamos National Laboratory group structures. Our reference solution was from 618-group PARTISN runs. The results were used to train a random forest regressor model with bagging, which was then tested on similar benchmarks. The bagging regressor model chose the best group structure from 52% to 65% of the time, and a subjectively “good” group structure up to 91% of the time.

A stable interface-preserving reinitialization equation for conservative level set method

Existing artificial compression based reinitialization approach for conservative level set method has a few drawbacks, like distortion of fluid–fluid interface and formation of unphysical fluid patches away from the interface. In this paper, a novel reinitialization approach has been presented which circumvents these limitations by reformulating the existing reinitialization equation. The new reinitialization equation by design ensures that the interface contour is kept invariant under the action of level set reinitialization. The unphysical patch formation away from the interface is also resolved here by avoiding the use of ill-conditioned contour normal vectors. Owing to the use of much simpler terms, a significant reduction in the numerical computations is achieved with the new reinitialization equation. The new reinitialization equation also enables one to choose a larger time step during the reinitialization iteration, leading to an overall reduction in computational efforts. In order to evaluate the performance of the present formulation, a set of test problems involving reinitialization of stationary level set functions is carried out first. Then, the efficacy of the proposed reinitialization approach is demonstrated using a set of standard two-dimensional scalar advection based test problems and incompressible two-phase flow problems. Finally, the ability to deal with complex mesh types is demonstrated by solving a test problem on an unstructured mesh consisting of finite volume cells having triangular and quadrilateral shapes.

Multisurrogate-Assisted Ant Colony Optimization for Expensive Optimization Problems With Continuous and Categorical Variables.

As an effective optimization tool for expensive optimization problems (EOPs), surrogate-assisted evolutionary algorithms (SAEAs) have been widely studied in recent years. However, most current SAEAs are designed for continuous/ combinatorial EOPs, which are not suitable for mixed-variable EOPs. This article focuses on one kind of mixed-variable EOP: EOPs with continuous and categorical variables (EOPCCVs). A multisurrogate-assisted ant colony optimization algorithm (MiSACO) is proposed to solve EOPCCVs. MiSACO contains two main strategies: 1) multisurrogate-assisted selection and 2) surrogate-assisted local search. In the former, the radial basis function (RBF) and least-squares boosting tree (LSBT) are employed as the surrogate models. Afterward, three selection operators (i.e., RBF-based selection, LSBT-based selection, and random selection) are devised to select three solutions from the offspring solutions generated by ACO, with the aim of coping with different types of EOPCCVs robustly and preventing the algorithm from being misled by inaccurate surrogate models. In the latter, sequence quadratic optimization, coupled with RBF, is utilized to refine the continuous variables of the best solution found so far. By combining these two strategies, MiSACO can solve EOPCCVs with limited function evaluations. Three sets of test problems and two real-world cases are used to verify the effectiveness of MiSACO. The results demonstrate that MiSACO performs well in solving EOPCCVs.

Parallel assembly lines worker assignment and balancing problem: A mathematical model and an artificial bee colony algorithm

In today’s competitive world, assembly lines are one of the most important industrial manufacturing systems, which can accommodate the massive increase in customer demands and product range, for the production industry. One possible way to increase the capacity and efficiency of assembly lines is to design a parallel assembly line system, consisting of two or more assembly lines placed next to each other. In the literature, the assembly lines are usually balanced according to tasks with fixed processing times without considering worker–task compatibility. In real-life applications, however, since the processing time of each task may vary according to the capability of each worker, the assignment of tasks to stations in the assembly line depends on the skills of the worker assigned to the related station. Considering the worker and task assignments together for each assembly line will provide a more realistic perspective. This paper, to the best of our knowledge, is the first study that considers the minimization of joint cycle time for the worker assignment and assembly line balancing problem in parallel assembly lines (PALWABP-II). Accordingly, a binary linear mathematical programming (BLP) model is developed and an artificial bee colony (ABC)-based solution approach is proposed for medium and large-sized problems. A new set of test problems is presented to the literature for a computational experiment. In addition, a summary of the literature is presented. An example problem is solved using the BLP model. In order to evaluate the effectiveness of the proposed model and approach, they are compared with the classical particle swarm optimization (PSO) algorithm and ten different priority rules. The computational results on test problems prove the effectiveness of the proposed ABC algorithm for the PALWABP-II.

Solving Bilevel Optimization Problems Using Kriging Approximations.

Bilevel optimization involves two levels of optimization, where one optimization problem is nested within the other. The structure of the problem often requires solving a large number of inner optimization problems that make these kinds of optimization problems expensive to solve. The reaction set mapping and the lower level optimal value function mapping are often used to reduce bilevel optimization problems to a single level; however, the mappings are not known a priori, and the need is to be estimated. Though there exist a few studies that rely on the estimation of these mappings, they are often applied to problems where one of these mappings has a known form, that is, piecewise linear, convex, etc. In this article, we utilize both these mappings together to solve general bilevel optimization problems without any assumptions on the structure of these mappings. Kriging approximations are created during the generations of an evolutionary algorithm, where the population members serve as the samples for creating the approximations. One of the important features of the proposed algorithm is the creation of an auxiliary optimization problem using the Kriging-based metamodel of the lower level optimal value function that solves an approximate relaxation of the bilevel optimization problem. The auxiliary problem when used for local search is able to accelerate the evolutionary algorithm toward the bilevel optimal solution. We perform experiments on two sets of test problems and a problem from the domain of control theory. Our experiments suggest that the approach is quite promising and can lead to substantial savings when solving bilevel optimization problems. The approach is able to outperform state-of-the-art methods that are available for solving bilevel problems, in particular, the savings in function evaluations for the lower level problem are substantial with the proposed approach.

A primal-dual interior-point relaxation method with global and rapidly local convergence for nonlinear programs

Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs with general equality and nonnegative constraints. In each iteration, our method approximately solves the KKT system of a parametric equality constrained mini-max subproblem, which avoids the requirement that any primal or dual iterate is an interior-point. The method has some similarities to the warmstarting interior-point methods in relaxing the interior-point requirement and is easily extended for solving problems with general inequality constraints. In particular, it has the potential to circumvent the jamming difficulty that appears with many interior-point methods for nonlinear programs and improve the ill conditioning of existing primal-dual interior-point methods as the barrier parameter is small. A new smoothing approach is introduced to develop our relaxation method and promote convergence of the method. Under suitable conditions, it is proved that our method can be globally convergent and locally quadratically convergent to the KKT point of the original problem. The preliminary numerical results on a well-posed problem for which many interior-point methods fail to find the minimizer and a set of test problems from the CUTEr collection show that our method is efficient.

A Hybrid Stochastic Deterministic Algorithm for Solving Unconstrained Optimization Problems

In this paper, a new deterministic method is proposed. This method depends on presenting (suggesting) some modifications to existing parameters of some conjugate gradient methods. The parameters of our suggested method contain a mix of deterministic and stochastic parameters. The proposed method is added to a line search algorithm to make it a globally convergent method. The convergence analysis of the method is established. The gradient vector is estimated by a finite difference approximation approach, and a new step-size h of this approach is generated randomly. In addition, a set of stochastic parameter formulas is constructed from which some solutions are generated randomly for an unconstrained problem. This stochastic technique is hybridized with the new deterministic method to obtain a new hybrid algorithm that finds an approximate solution for the global minimization problem. The performance of the suggested hybrid algorithm is tested in two sets of benchmark optimization test problems containing convex and non-convex functions. Comprehensive comparisons versus four other hybrid algorithms are listed in this study. The performance profiles are utilized to evaluate and compare the performance of the five hybrid algorithms. The numerical results show that our proposed hybrid algorithm is promising and competitive for finding the global optimum point. The comparison results between the performance of our suggested hybrid algorithm and the other four hybrid algorithms indicate that the proposed algorithm is competitive with, and in all cases superior to, the four algorithms in terms of the efficiency, reliability, and effectiveness for finding the global minimizers of non-convex functions.

Multi-objective sparrow search algorithm: A novel algorithm for solving complex multi-objective optimisation problems

This study proposes a multi-objective sparrow search algorithm (MOSSA) based on a 2k crowding-distance entropy and the optimal strategy of the positions to solve complex multi-objective optimisation problems (MOPs) using the excellent performance of the sparrow search algorithm (SSA). A fast non-dominated sorting approach is incorporated to permit SSA to solve MOPs. To maintain the evenness and spread of the Pareto solution set obtained in each run, a 2k crowding-distance entropy is proposed to measure the diversity of the solution set. Modified formulas for updating positions and additional adaptive parameters can improve the global search ability of MOSSA. The position archive of the population is introduced to realise the optimal strategy of the positions. This strategy ensures that the positions of the population in each generation are optimal, which significantly improves the performance of MOSSA. MOSSA is compared with three well-known algorithms using a set of complex unconstrained test problems and a complex constrained engineering optimisation problem. This study explores the effect of the 2k crowding-distance entropy and crowding distance on maintaining diversity. The impact of the k in 2k crowding-distance entropy on the performance of MOSSA is analysed. The experimental results demonstrate that MOSSA exhibits competitive performance in solving complex MOPs.