Set Of Test Problems Research Articles

Application of the Parabola Method in Nonconvex Optimization

We consider the Golden Section and Parabola Methods for solving univariate optimization problems. For multivariate problems, we use these methods as line search procedures in combination with well-known zero-order methods such as the coordinate descent method, the Hooke and Jeeves method, and the Rosenbrock method. A comprehensive numerical comparison of the obtained versions of zero-order methods is given in the present work. The set of test problems includes nonconvex functions with a large number of local and global optimum points. Zero-order methods combined with the Parabola method demonstrate high performance and quite frequently find the global optimum even for large problems (up to 100 variables).

A projected-search interior-point method for nonlinearly constrained optimization

AbstractThis paper concerns the formulation and analysis of a new interior-point method for constrained optimization that combines a shifted primal-dual interior-point method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. Shifts on the dual variables allow the method to be safely “warm started” from a good approximate solution and avoids the possibility of very large solutions of the associated path-following equations. The approximate Newton direction is used in conjunction with a new projected-search line-search algorithm that employs a flexible non-monotone quasi-Armijo line search for the minimization of each penalty-barrier function. Numerical results are presented for a large set of constrained optimization problems. For comparison purposes, results are also given for two primal-dual interior-point methods that do not use projection. The first is a method that shifts both the primal and dual variables. The second is a method that involves shifts on the primal variables only. The results show that the use of both primal and dual shifts in conjunction with projection gives a method that is more robust and requires significantly fewer iterations. In particular, the number of times that the search direction must be computed is substantially reduced. Results from a set of quadratic programming test problems indicate that the method is particularly well-suited to solving the quadratic programming subproblem in a sequential quadratic programming method for nonlinear optimization.

TREVR2: illuminating fast N log2N radiative transfer

ABSTRACT We present trevr2 (Tree-based REVerse Ray Tracing 2), a fast, general algorithm for computing the radiation field, suitable for both particle and mesh codes. It is designed to self-consistently evolve chemistry for zoomed-in astrophysical simulations, such as cosmological galaxies with both internal sources and prescribed background radiation, rather than large periodic volumes. Light is propagated until absorbed, with no imposed speed limit other than those due to opacity changes (e.g. ionization fronts). trevr2 searches outward from receiving gas in discrete directions set by the healpix algorithm (unlike its slower predecessor trevr), accumulating optical depth and adding the flux due to sources combined into progressively larger tree cells with distance. We demonstrate Nactivelog2N execution time with absorption and many sources. This allows multiband radiative transfer costs comparable to tree-based gravity and hydrodynamics, and the usual speed-up when active particles evolve on individual time-steps. Sources embedded in non-homogeneous absorbing material introduce systematic errors. We introduce transmission averaging instead of absorption averaging which dramatically reduces these systematic effects. We outline other ways to address systematics including an explicit complex source model. We demonstrate the overall performance of the method via a set of astrophysical test problems.

Bounds on eigenvalues of real symmetric interval matrices for αBB method in global optimization

In this paper, we investigate bounds on eigenvalues of real symmetric interval matrices. We present a method that computes bounds on eigenvalues of real symmetric interval matrices. It outperforms many methods developed in the literature and produces as sharp as possible bounds on eigenvalues of real symmetric interval matrices. The aim is to apply the proposed method to compute lower bounds on eigenvalues of a symmetric interval hessian matrix of a nonconvex function in the ?BB method and use them to produce a tighter underestimator that improves the ?BB algorithm for solving global optimization problems. In the end, we illustrate by example, the comparison of various approaches of bounding eigenvalues of real symmetric interval matrices. Moreover, a set of test problems found in the literature are solved efficiently and the performances of the proposed method are compared with those of other methods.

Investigating the performance of a surrogate-assisted nutcracker optimization algorithm on multi-objective optimization problems

This paper introduces a novel surrogate-assisted multi-objective nutcracker optimization algorithm. This algorithm is built upon the recently proposed nutcracker optimization algorithm, drawing inspiration from the behaviours observed in Clark’s nutcrackers. The algorithm is developed based on two distinct behaviours exhibited by these birds. To comprehensively evaluate the performance of the proposed algorithm, a dual-pronged approach is adopted. On the one hand, a set of artificial test problems is employed to scrutinize the algorithm's capabilities, while on the other hand, a set of real-world problems is considered to assess its practical efficacy. The results of the proposed algorithm are evaluated in comparison to existing baseline algorithms and state-of-the-art algorithms, using well-recognized performance metrics, both qualitatively and quantitatively. The obtained results provide convincing evidence of the performance of the proposed algorithm.

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

In this paper, we consider a global optimization problem where the objective function is assumed to be Lipschitz-continuous with an unknown Lipschitz constant. Building upon the recently introduced BIRECT (BIsection of RECTangles) algorithm, we propose a new diagonal partitioning and sampling scheme. Our framework, named BIRECT-V (V for vertices), combines bisection with the sampling of two points. In the initial hyper-rectangle, these points are located at 1/3 and 1 along the main diagonal. Unlike most DIRECT-type algorithms, where evaluating the objective function at vertices is not suitable for bisection, our strategy, when combined with bisection, provides more comprehensive information about the objective function. However, the creation of new sampling points may coincide with existing ones at shared vertices, resulting in additional evaluations of the objective function and increasing the number of function evaluations per iteration. To overcome this issue, we propose modifying the original optimization domain to obtain a good approximation of the global solution. Experimental investigations demonstrate that this modification positively impacts the performance of the BIRECT-V algorithm. Our proposal shows promise as a global optimization algorithm compared to the original BIRECT and two popular DIRECT-type algorithms on a set of test problems. It particularly excels at high-dimensional problems

Data-driven simulation-based decision support system for resource allocation in industry 4.0 and smart manufacturing

Data-driven simulation (DDS) is fundamental to analytical and decision-support technologies in Industry 4.0 and smart manufacturing. This study investigates the potential of DDS for resource allocation (RA) in high-mix, low-volume smart manufacturing systems with mixed automation levels. A DDS-based decision support system (DDS-DSS) is developed by incorporating two RA strategies: simulation-based bottleneck analysis (SB-BA) and simulation-based multi-objective optimization (SB-MOO). To enhance the performance of SB-MOO, a unique meta-learning mechanism featuring memory, dynamic orthogonal array, and learning rate is integrated into the NSGA-II, resulting in a modified version of the NSGA-II with meta-learning (i.e., NSGA-II-ML). The proposed DSS also benefits from a post-optimality analysis that leverages a clustering algorithm to derive actionable insights. A real-life marine engine manufacturing application study is presented to demonstrate the applicability and exhibit efficacy of the proposed DSS and NSGA-II-ML. To this aim, NSGA-II-ML was tested against the original NSGA-II and differential evolution (DE) algorithm across a set of test problems. The results revealed that NSGA-II-ML surpassed the other two in terms of the number of non-dominated solutions and hypervolume, particularly in medium and large-sized problems. Furthermore, NSGA-II-ML achieved a 24% improvement in the best throughput found in the real case problem, outperforming SB-BA, NSGA-II, and DE. The post-optimality analysis led to the extraction of valuable knowledge about the key, influencing decision variables on the throughput.

Автоматизовані засоби тестування програмних модулів розв’язування інтегральних рівнянь Вольтерри другого роду

The article is part of the methodology of automated testing of software modules for solving Volterra integral levels of the second kind. For the implementation of automated testing, the Matlab software environment was selected, which has a wide range of software testing capabilities, in particular: functions for generating data sets for testing; functions for comparing test results; functions for generating test reports, etc. For the development of automated testing tools, the Unit Testing Framework was selected, which is a component of the MATLAB Test Framework, has many ready-made methods for checking the correctness of values and the formation of statistical errors. A set of test problems has been developed for Volterra integral equations of the second kind, which are divided into different types, including linear Volterra integral equations of the second kind, which consist of a kernel with power, exponential, hyperbolic, logarithmic, trigonometric, inverse trigonometric functions and their combination. Developed testing tools used for automated quality control of software modules built on the basis of left rectangle, right rectangle, trapezoidal, and Simpson methods. The developed set of test tasks covers a wide range of possible operating conditions of software modules. The results of the testing allowed to improve the existing software modules to achieve the set conditions for their operation. The conducted research should contribute to the development of more reliable and efficient software modules for solving Volterra integral levels of the second kind. The obtained results are the basis for further research, which will be used in the following directions: development of testing methods for more complex types of Volterra integral levels of the second kind, including the level with nonlinear and non-stationary kernels.

Bilevel Optimization via Collaborations Among Lower-Level Optimization Tasks

Bilevel metaheuristics have been widely used for bilevel optimization. However, recent studies have indicated that most bilevel metaheuristics are inefficient since they perform the lower-level optimization task for each upper-level solution independently and neglect the relationship among lower-level optimization tasks. In this paper, we develop a bilevel metaheuristic with the collaborations among lower-level optimization tasks. Specifically, a population is evolved to solve the lower-level optimization tasks for all upper-level solutions collaboratively at each generation. In the population, each solution is associated with a lower-level optimization task. In such a way, all lower-level optimization tasks can be solved in a single run. To capture the individual features of different lower-level optimization tasks, we construct a lower-level search distribution for each lower-level optimization task based on all solutions in the population. In addition, an information-sharing mechanism is proposed to share good solutions among lower-level optimization tasks. Experiments on two sets of test problems and three practical applications demonstrate that our proposed algorithm performs better than other bilevel metaheuristics in comparison.

Surrogate-Assisted Differential Evolution With Adaptive Multisubspace Search for Large-Scale Expensive Optimization

Real-world industrial engineering optimization problems often have a large number of decision variables. Most existing large-scale evolutionary algorithms need a large number of function evaluations to achieve high-quality solutions. However, the function evaluations can be computationally intensive for many of these problems, particularly, which makes large-scale expensive optimization challenging. To address this challenge, surrogate-assisted evolutionary algorithms based on the divide-and-conquer strategy have been proposed and shown to be promising. Following this line of research, we propose a surrogate-assisted differential evolution algorithm with adaptive multi-subspace search for large-scale expensive optimization to take full advantage of the population and the surrogate mechanism. The proposed algorithm constructs multi-subspace based on principal component analysis and random decision variable selection, and searches adaptively in the constructed subspaces with three search strategies. The experimental results on a set of large-scale expensive test problems have demonstrated its superiority over three state-of-the-art algorithms on the optimization problems with up to 1000 decision variables.

On the Advantages of Searching Infeasible Regions in Constrained Evolutionary-Based Multi-Objective Engineering Optimization

Abstract Solving a multiple-criteria optimization problem with severe constraints remains a significant issue in multi-objective evolutionary algorithms. The problem primarily stems from the need for a suitable constraint handling technique. One potential approach is balancing the search in feasible and infeasible regions to find the Pareto front efficiently. The justification for such a strategy is that the infeasible region also provides valuable information, especially in problems with a small percentage of feasibility areas. To that end, this paper investigates the potential of the infeasibility-driven principle based on multiple constraint ranking-based techniques to solve a multi-objective problem with a small feasibility ratio. By analyzing the results from intensive experiments on a set of test problems, including the realistic multi-objective car structure design and actuator design problem, it is shown that there is a significant improvement gained in terms of convergence by utilizing the generalized version of the multiple constraint ranking techniques.

Performance Benchmarking of Different Methods to Solve Gauss-Newton Trust Region Subproblems

Summary The Gauss-Newton (GN) trust region optimization methods perform robustly but may introduce significant overhead cost when using the conventional matrix factorization method to solve the associated GN trust region subproblem (GNTRS). Solving a GNTRS involves solving a nonlinear equation using an iterative Newton-Raphson (NR) method. In each NR iteration, a symmetric linear system can be solved by different matrix factorization methods, including Cholesky decomposition (CD), eigenvalue decomposition (EVD), and singular value decomposition (SVD). Because CD fails to factorize a singular symmetric matrix, we propose solving a GNTRS using the robust EVD method. In this paper, we analyze the performances of different methods to solve a GNTRS using different matrix factorization subroutines in LAPACK with different options and settings. The cost of solving a GNTRS mainly depends on the number of observed data (m) and the number of uncertainty parameters (n). When n≤m, we recommend directly solving the original GNTRS with n variables. When n>m, we propose an indirect method that transforms the original GNTRS with n variables to a new problem with m unknowns. The proposed indirect method can significantly reduce the computational cost by dimension reduction. However, dimension reduction may introduce numerical errors, which, in turn, may result in accuracy degradation and cause failure of convergence using the popular iterative NR method. To further improve the overall performance, we introduce a numerical error indicator to terminate the iterative NR process when numerical errors become dominant. Finally, we benchmarked the performances of different approaches on a set of testing problems with different settings. Our results confirm that the GNTRS solver using the EVD method together with the modified NR method performs the best, being both robust (no failure for all testing problems) and efficient (consuming comparable CPU time to other methods).

An adaptive moments-based interface reconstruction using intersection of the cell with one half-plane, two half-planes and a circle

We present a new adaptive moment-of-fluid (A-MOF) interface reconstruction method. It uses the zeroth, first, and second moments of the fragment of material inside a cell of the mesh to construct a shape that approximates the respective material fragment. The new method requires information about the material moments only for the cell under consideration. The adaptive method chooses between shapes obtained by the intersection of the cell with one half-plane, two half-planes, or a circle. The A-MOF method allows to exactly reproduce several convex shapes: corners, filaments, and their concave cell-complements; as well as pieces of the circles and its cell-compliments. Interface reconstruction is formulated as a local (for each cell), non-linear, equality constrained optimization problem, which does not require additional communication and allows for an efficient parallel implementation.We present an extensive set of test problems, both for interface reconstruction on a single cell, and for reconstruction of a variety of shapes on the entire mesh.

A new nonmonotone spectral projected gradient algorithm for box-constrained optimization problems in [formula omitted] real matrix space with application in image clustering

Box-constrained optimization problems in the real m×n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature.

Four Points Block Method with Second Derivative for Solving First Order Ordinary Differential Equations

In this study, we devised 4-point implicit block (New 4pb) approaches that take advantage of second derivatives. The aim provides is more accurate and quicker numerical solutions to first - order ordinary differential equations (ODEs). Whereas the properties of the (New 4pb) method such as the order and zero-stability have been discussed. After that, the approaches are applied to a collection of first-order (ODEs). In addressing the set of test problems, numerical results clearly indicate that the newly proposed systems outperformed previous, well-known existing methods.

Bundle Enrichment Method for Nonsmooth Difference of Convex Programming Problems

The Bundle Enrichment Method (BEM-DC) is introduced for solving nonsmooth difference of convex (DC) programming problems. The novelty of the method consists of the dynamic management of the bundle. More specifically, a DC model, being the difference of two convex piecewise affine functions, is formulated. The (global) minimization of the model is tackled by solving a set of convex problems whose cardinality depends on the number of linearizations adopted to approximate the second DC component function. The new bundle management policy distributes the information coming from previous iterations to separately model the DC components of the objective function. Such a distribution is driven by the sign of linearization errors. If the displacement suggested by the model minimization provides no sufficient decrease of the objective function, then the temporary enrichment of the cutting plane approximation of just the first DC component function takes place until either the termination of the algorithm is certified or a sufficient decrease is achieved. The convergence of the BEM-DC method is studied, and computational results on a set of academic test problems with nonsmooth DC objective functions are provided.

A Multiform Optimization Framework for Constrained Multiobjective Optimization.

Constrained multiobjective optimization problems (CMOPs) pose great difficulties to the existing multiobjective evolutionary algorithms (MOEAs), in terms of constraint handling and the tradeoffs between diversity and convergence. The constraints divide the search space into feasible and infeasible regions. A key to solving CMOPs is how to effectively utilize the information of both feasible and infeasible solutions during the optimization process. In this article, we propose a multiform optimization framework to solve a CMOP task together with an auxiliary CMOP task in a multitask setting. The proposed framework is designed to conduct a search in different sizes of feasible space that is derived from the original CMOP task. The derived feasible space is easier to search and can provide a useful inductive bias to the search process of the original CMOP task, by leveraging the transferable knowledge shared between them, thereby helping the search to toward the Pareto optimal solutions from both the infeasible and feasible regions of the search space. The proposed framework is instantiated in three kinds of MOEAs: 1) dominance-based; 2) decomposition-based; and 3) indicator-based algorithms. Experiments on four sets of benchmark test problems demonstrate the superiority of the proposed method over four representative constraint-handling techniques. In addition, the comparison against five state-of-the-art-constrained MOEAs demonstrates that the proposed approach outperforms these contender algorithms. Finally, the proposed method is successfully applied to solve a real-world antenna array synthesis problem.

Challenging test problems for multi- and many-objective optimization

In spite of the extensive studies that have been conducted regarding the construction of multi-objective test problems, researchers have mainly focused their interests on designing complicated search spaces, disregarding, in many cases, the design of the Pareto optimal fronts of the problems. In this regard, the work related to scalable multi-objective test problems—i.e., problems that can be formulated for an arbitrary number of objectives—has been much less studied. This paper introduces a new set of continuous and box-constrained multi-objective test problems which are scalable in both the number of objectives and in the number of decision variables. Each test problem included in the proposed test suite has a peculiar Pareto front different from those observed in the existing scalable multi-objective test suites. In addition to different Pareto fronts, the proposed test suite introduces features related to the search space that place obstacles that complicate exploring Pareto optimal solutions. Such features can be easily switched on and off by the user to analyze specific mechanisms of multi-objective evolutionary algorithms (MOEAs). The components used in the proposed test suite can be used as a toolkit to construct new test instances not included in this set of problems. To illustrate the use and difficulties of the proposed test suite, some experiments are presented adopting three MOEAs using selection mechanisms based on Pareto optimality, decomposition, and a performance indicator (hypervolume).

A conservative finite volume method for incompressible two-phase flows on unstructured meshes

A robust finite volume framework for solving incompressible two-phase flow problems based on an explicit dual-time stepping based artificial compressibility formulation has been developed by Parameswaran and Mandal (2019) for structured meshes. This artificial compressibility framework is extended to unstructured meshes in the present paper. In order to further enhance the applicability to real life engineering problems, the surface tension effect is also modeled here. To deal with arbitrary unstructured mesh topology, least square and Green-Gauss integral based methods are used. The efficacy of the proposed method is demonstrated using a set of scalar advection based test problems and a set of standard incompressible two-phase flow test problems involving gravitational, viscous and surface tension forces. The numerical results show improvement in accuracy compared to the results of the structured mesh formulation.

A clustering heuristic to improve a derivative-free algorithm for nonsmooth optimization

In this paper we propose an heuristic to improve the performances of the recently proposed derivative-free method for nonsmooth optimization CS-DFN. The heuristic is based on a clustering-type technique to compute an estimate of Clarke’s generalized gradient of the objective function, obtained via calculation of the (approximate) directional derivative along a certain set of directions. A search direction is then calculated by applying a nonsmooth Newton-type approach. As such, this direction (as it is shown by the numerical experiments) is a good descent direction for the objective function. We report some numerical results and comparison with the original CS-DFN method to show the utility of the proposed improvement on a set of well-known test problems.