Test Problems Research Articles

Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling

We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution \(\mu\) , an \(\varepsilon \gt 0\) , and access to sampling oracle(s) for a hidden distribution \(\pi\) , the goal in identity testing is to distinguish whether the two distributions \(\mu\) and \(\pi\) are identical or are at least \(\varepsilon\) -far apart. When there is only access to full samples from the hidden distribution \(\pi\) , it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various “conditional” sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the \(\mathsf{Coordinate\ Oracle}\) , and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an \(n\) -dimensional visible distribution \(\mu\) , then there is an efficient identity testing algorithm for any hidden distribution \(\pi\) using \(\widetilde{O}(n/\varepsilon)\) queries to the \(\mathsf{Coordinate\ Oracle}\) . Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of \(n\) -dimensional distributions, specifically sparse antiferromagnetic Ising models over \(\{+1,-1\}^{n}\) , we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless \(\mathsf{RP}=\mathsf{NP}\) . We complement our results with a matching \(\Omega(n/\varepsilon)\) statistical lower bound for the sample complexity of identity testing in the \(\mathsf{Coordinate\ Oracle}\) model.

A spectral collocation scheme for solving nonlinear delay distributed-order fractional equations

The basic purpose of this paper is to implement a variant of the Jacobi–Gauss collocation scheme for solving nonlinear delay (in difference format) fractional differential equations of distributed-order type. At the first stage of the solving procedure, the Legendre–Gauss quadrature rule is implemented to approximate the main problem by a difference delay fractional differential equation of multi-term type. So, one can apply the Jacobi–Gauss collocation approach for localizing the resulting multi-term fractional differential equation with the difference delay factor. Moreover, some special cases of Jacobi–Gauss quadrature rules can be used for approximating the involved integrals in the aforementioned problem. By considering the Lagrange interpolating polynomials, as the numerical solutions, the solution of the main problem can be approximated by solving the associated system of nonlinear algebraic equations in terms of unknown Lagrange multipliers. Convergence analysis of the problem is investigated rigorously under some mild conditions at a spectral rate. Extensive test problems are considered to justify the theoretical analysis.

Optimal Network Pairwise Comparison

We are interested in the problem of two-sample network hypothesis testing: given two networks with the same set of nodes, we wish to test whether the underlying Bernoulli probability matrices of the two networks are the same or not. We propose Interlacing Balance Measure (IBM) as a new two-sample testing approach. We consider the Degree-Corrected Mixed-Membership (DCMM) model for undirected networks, where we allow severe degree heterogeneity, mixed-memberships, flexible sparsity levels, and weak signals. In such a broad setting, how to find a test that has a tractable limiting null and optimal testing performances is a challenging problem. We show that IBM is such a test: in a broad DCMM setting with only mild regularity conditions, IBM has N ( 0 , 1 ) as the limiting null and achieves the optimal phase transition. While the above is for undirected networks, IBM is a unified approach and is directly implementable for directed networks. For a broad directed-DCMM (extension of DCMM for directed networks) setting, we show that IBM has N ( 0 , 1 / 2 ) as the limiting null and continues to achieve the optimal phase transition. We have also applied IBM to the Enron email network and a gene co-expression network, with interesting results.

Impact of north-west Bay of Bengal bathymetry along the off-coast of Digha on the propagation of shallow water waves using finite difference method

Numerical simulation of shallow water equations has been carried out using the Crank-Nicolson scheme of Finite Difference Method (FDM) to study the interaction of waves with different forms of bathymetry especially that of the off-coast of Digha, a tourist spot situated in the north-west coast of Bay of Bengal. MATLAB code has been developed for the simulation. One-dimensional (1-D) linear shallow water model has been considered here. Three different bathymetry profile are chosen for the investigation: (i) flat bathymetry (with or without seamount/basin); (ii) piecewise linear approximation to the real bathymetry of the off-coast of Digha; (iii) real bathymetry of the off-coast of Digha. Real bathymetry data has been obtained from the gridded data set of GEBCO (General Bathymetric Chart of the Oceans) published in 2022. Validation of the present numerical scheme has been performed by considering test problems with results existing in the literature. Comparative study of the effect of the bathymetries, on the propagation of shallow water waves, has been performed and demonstrated with figures and tables. Results of wave-bathymetry interaction obtained here are believed to be new with respect to the present study area and will definitely give some insight into the phenomena in this region.

Analysis on failure test and evaluation about refueling receptacle of onboard hydrogen system

ABSTRACT As the key component of the onboard hydrogen system, the service life and failure problem of the refueling receptacle easily leads to the hydrogen leakage accident which give rise to the suffocation, the frostbite, the burns, and the impact injuries. The typical hydrogen leakage accidents include the accidental leakage, the ignition, the deflagration, and the explosion. Therefore, the failure test and evaluation problem about the refueling receptacle is focused on. In this paper, the failure test and evaluation about the refueling receptacle covering the vibration resistance performance, the temperature resistance performance, and the durability was studied. First, the failure mechanisms of the refueling receptacle involving the vibration, the temperature, and the durability based on the vehicle operating conditions were clarified. Second, according to the test experience and the vehicle operating condition, the test methods about the failure phenomenon of the refueling receptacle were established, respectively. Finally, the failure test results of the refueling receptacle were analyzed. The failure of the refueling receptacle may caused by the aging and failure of sealing element, the metal creep aging and material damage, the decline in the contact stress at the low-temperature connection, the loose connection, and the internal structural ectopia.

The Association between Pressure Pain Thresholds, Conditioned Pain Modulation, Clinical Status, and Sleep Quality in Fibromyalgia Patients: A Clinical Trial Secondary Analysis.

Background: Fibromyalgia (FM) is a complex multidimensional disorder primarily characterized by chronic widespread pain, significantly affecting patients' quality of life. FM is associated with some clinical signs found with quantitative sensory testing (QST), sleep disturbance, or psychological problems. This study aims to explore the associations between pressure pain thresholds (PPTs), conditioned pain modulation (CPM), clinical status, and sleep quality in FM patients, offering insights for better clinical management and assessment tools. Methods: This secondary analysis utilized data from a clinical trial involving 129 FM patients. Various assessments, including the Fibromyalgia Impact Questionnaire (FIQ), Pain Catastrophizing Scale (PCS), State-Trait Anxiety Inventory (STAI), and Jenkins Sleep Scale (JSS), were employed to evaluate the clinical and psychological status and sleep quality. PPTs and CPM were measured to understand their relationship with clinical parameters. Results: Our findings revealed that PPTs and CPM are not significantly associated with the clinical status or sleep quality of FM patients. Instead, pain catastrophizing and anxiety state showed a stronger correlation with the impact of fibromyalgia and sleep disturbances. These results highlight the importance of psychological and cognitive factors in managing FM. Conclusions: The study suggests that while PPTs and CPM may not be reliable biomarkers for clinical status in FM, the use of comprehensive assessments including FIQ, PCS, STAI, and JSS can provide a more accurate evaluation of patients' condition. These tools are cost-effective, can be self-administered, and facilitate a holistic approach to FM management, emphasizing the need for personalized treatment plans.

A two-phase sequential algorithm for global optimization of the standard quadratic programming problem

AbstractWe introduce a new sequential algorithm for the Standard Quadratic Programming Problem (StQP), which exploits a formulation of StQP as a Linear Program with Linear Complementarity Constraints (LPLCC). The algorithm is finite and guarantees at least in theory a $$\delta $$ δ -approximate global minimum for an arbitrary small $$\delta $$ δ , which is a global minimum in practice. The sequential algorithm has two phases. In Phase 1, Stationary Points (SP) with strictly decreasing objective function values are computed. Phase 2 is designed for giving a certificate of global optimality for the last SP computed in Phase 1. Two different Nonlinear Programming Formulations for LPLCC are proposed for each one of these phases, which are solved by efficient enumerative algorithms. New procedures for computing a lower bound for StQP are also proposed, which are easy to implement and give tight bounds in general. Computational experiments with a number of test problems from known sources indicate that the two-phase sequential algorithm is, in general, efficient in practice. Furthermore, the algorithm seems to be an efficient way to study the copositivity of a matrix by exploiting an StQP with this matrix.

A rational‐Chebyshev projection method for nonlinear eigenvalue problems

AbstractThis article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function under consideration is approximated by a matrix polynomial in . The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

Deep reinforcement learning as multiobjective optimization benchmarks: Problem formulation and performance assessment

The successful deployment of Deep learning in several challenging tasks has been translated into complex control problems from different domains through Deep Reinforcement Learning (DRL). Although DRL has been extensively formulated and solved as single-objective problems, nearly all real-world RL problems often feature two or more conflicting objectives, where the goal is to obtain a high-quality and diverse set of optimal policies for different objective preferences. Consequently, the development of Multi-Objective Deep Reinforcement Learning (MODRL) algorithms has gained a lot of traction in the literature. Generally, Evolutionary Algorithms (EAs) have been demonstrated to be scalable alternatives to the classical DRL paradigms when formulated as an optimization problem. Hence it is reasonable to employ Multi-objective Evolutionary Algorithms (MOEAs) to handle MODRL tasks. However, there are several factors constraining the progress of research along this line: first, there is a lack of a general problem formulation of MODRL tasks from an optimization perspective; second, there exist several challenges in performing benchmark assessments of MOEAs for MODRL problems. To overcome these limitations: (i) we present a formulation of MODRL tasks as general multi-objective optimization problems and analyze their complex characteristics from an optimization perspective; (ii) we present an end-to-end framework, termed DRLXBench, to generate MODRL benchmark test problems for seamless running of MOEAs (iii) we propose a test suite comprising of 12 MODRL problems with different characteristics such as many-objectives, degenerated Pareto fronts, concave and convex optimization problems, etc. (iv) Finally, we present and discuss baseline results on the proposed test problems using seven representative MOEAs.

The weighted Euclidean one-center problem in $${\mathbb {R}}^n$$

AbstractGiven a finite set of distinct points in $${\mathbb {R}}^n$$ R n and a positive weight for each point, primal and dual algorithms are developed for finding the Euclidean ball of minimum radius so that the weighted Euclidean distance between the center of the ball and each point is less than or equal to the radius of the ball. Each algorithm is based on a directional search method in which the search path at each iteration is either a ray or a two-dimensional circular arc in $${\mathbb {R}}^n$$ R n . At each iteration, a search path is constructed by intersecting bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional spheres. Each search path preserves complementary slackness and primal (dual) feasibility for the primal (dual) algorithm. The step size along each search path is determined explicitly. Test problems up to 1000 dimensions and 10,000 points were solved to optimality by both primal and dual algorithms. Computational results also show that these algorithms outperform several open-source SOCP codes.

A New Variant of the Conjugate Descent Method for Solving Unconstrained Optimization Problems and Applications

Unconstrained optimization problems have a long history in computational mathematics and have been identified as being among the crucial problems in the fields of applied sciences, engineering, and management sciences. In this paper, a new variant of the conjugate descent method for solving unconstrained optimization problems is introduced. The proposed algorithm can be seen as a modification of the popular conjugate descent (CD) algorithm of Fletcher. The algorithm of the proposed method is well-defined, and the sequence of the directions of search is shown to be sufficiently descending. The convergence result of the proposed method is discussed under the common standard conditions. The proposed algorithm together with some existing ones in the literature is implemented to solve a collection of benchmark test problems. Numerical experiments conducted show the performance of the proposed method is very encouraging. Furthermore, an additional efficiency evaluation is carried out on problems arising from signal processing and it works well.

Arbitrary order positivity preserving finite-volume schemes for 2D elliptic problems

The positivity preservation is very important in most applications solving elliptic problems. Many schemes preserving positivity has been proposed but are at most second-order convergent. Besides, in general, high-order schemes do not preserve positivity. In the present paper, we propose an arbitrary-order positivity preserving method for elliptic problems in 2D. We show how to adapt our method to the case of a discontinuous and/or tensor-valued diffusion coefficient, while keeping the expected order of convergence. We assess the new scheme on several test problems.

Sequential testing in batches with resource constraints

This paper studies a problem of determining the state of a system through costly tests of its components, where components can be tested simultaneously in batches to exploit economies of scale. This problem is a generalization of the classical sequential testing problem and it has applications in various settings, including machine maintenance, disease diagnosis, and new product development. We prove that the problem is strongly NP-hard, model it as a mixed-integer programming formulation, and we also propose a dynamic program for it. Additionally, we design a tabu search and a hybrid solution method that combines a tabu search metaheuristic and a proximity search matheuristic. Based on extensive computational experiments, we find that the dynamic program can solve instances with up to 25 components within a 15-minute time limit and 16 GB of RAM. With respect to larger instances, the proposed metaheuristic and hybrid method are better than a greedy heuristic.

Hypervolume-Based Cooperative Coevolution With Two Reference Points for Multiobjective Optimization

An important issue in hypervolume-based evolutionary multi-objective optimization (EMO) algorithms is the specification of a reference point for hypervolume calculation. However, its appropriate specification has not been carefully studied in the literature. Some recent studies have pointed out the importance and difficulty of the reference point specification. Its appropriate specification depends on problem characteristics such as the Pareto front shape and the number of objectives. In this paper, the difficulty of the reference point specification in hypervolume-based EMO algorithms is circumvented by using two reference points. Instead of using only a single reference point, we propose a new hypervolume-based EMO algorithm that can effectively utilize two reference points cooperatively. Experimental results show that the proposed algorithm has good and robust performance on a wide range of test problems. In comparison to hypervolume-based EMO algorithms with only a single reference point, the proposed algorithm can find a wider and more uniformly distributed solution set. On a recently proposed real-world problem suite, the proposed algorithm shows competitive performance in comparison to state-of-the-art algorithms.

Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample

Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample

Micro Multiobjective Evolutionary Algorithm With Piecewise Strategy for Embedded-Processor-Based Industrial Optimization.

In some industrial applications, it is required to do off-line multiobjective optimization in embedded systems. Due to their limited computing and memory capability, embedded processor may not be able to run conventional multiobjective optimization evolutionary algorithms (MOEAs). This article proposes a micro MOEA with piecewise strategy (μ MOEA) for industrial optimization in embedded processor. μ MOEA introduces an improved piecewise strategy based on the MOEA/D framework, which serially optimizes subclusters to be compatible with embedded processor under limited computing power. For the purpose of further enhancing μ MOEA, a dynamic and flexible weight vector update trigger mechanism is proposed, so that the algorithm can save and utilize the computing resources of the embedded processor as much as possible. Abundant artificial test problems are carrying out to test the performance of μ MOEA. Through various experiments, it can be found that μ MOEA has outstanding performance in ZDT, DTLZ, SMOP, and MaF problems. Last and most importantly, μ MOEA is successfully applied to two specific application scenarios of industrial optimization on embedded processor for simulation, such as two different types of semi-autogenous grinding optimization problems and micro-grid energy optimization problem, which prove the feasibility of applying MOEA to embedded processor.

THE IMPACT OF CHINA’S CARBON PLEDGE ON THE COUNTRY’S HYDROPOWER PRODUCTION: EVIDENCE FROM THE STOCK MARKET

China’s national climate goals to “peak” its carbon emissions before 2030 and to achieve “carbon neutrality” before 2060 are considered to exert a considerable impact on the country’s production of hydropower, one of the major clean energy sources. Although the existing literature provides a large number of studies on the impacts of China’s “dual carbon” climate goals, only very few studies discuss their impact on the hydropower industry. This paper selects Chinese hydropower companies listed on the country’s stock market to examine. To overcome the traditional problem of the Chow test, the Quandt–Andrews test is used to identify the structural changes in the stock market performance data of those selected companies. The analysis provides essential evidence that shows the considerable impact of China’s announcement of “dual carbon” climate goals on the country’s hydropower industry.

Selective clustered tourist trip design problem with time windows under intuitionistic fuzzy score and exponential travel times

Creating travel plans is a difficult task for tourists. Hence, this is due to planning trips for a selection of Points of Interest (POIs) that consider several constraints and preferences to identify POIs and optimize the attractiveness of routes in general. The literature addresses the problem as the Tourist Trip Design Problem (TTDP). This paper considers a particular variant of TTDP, where POIs are grouped into clusters corresponding to various destinations. Additionally, the visit to each POI must occur within a specific time interval. Minimum and maximum restrictions exist on how many POIs from each cluster may be visited on the same itinerary. Moreover, we proposed the Tourist Trip Design Problem with Clustered POIs under intuitionistic fuzzy scores and exponential travel times in uncertainty and randomness. This problem is finding a feasible route that maximizes the total score collected. The route must begin and end at a specified initial location, and its duration is limited to a specific maximum value. A mathematical model is proposed for this problem. Test problems are generated and solved by the proposed model. The solution to the test problems is obtained by using different parameters. The results show that the proposed methodology is an approach that allows tourists to get other travel plans and see how well these travel plans meet their demands and preferences.

Going Beyond the MHD Approximation: Physics-based Numerical Solution of the CGL Equations

We present a new numerical model for solving the Chew–Goldberger–Low system of equations describing a bi-Maxwellian plasma in a magnetic field. Heliospheric and geospace environments are often observed to be in an anisotropic state with distinctly different parallel and perpendicular pressure components. The Chew–Goldberger–Low (CGL) system represents the simplest leading order correction to the common isotropic MHD model that still allows the incorporation of the latter’s most desirable features. However, the CGL system presents several numerical challenges: the system is not in conservation form, the source terms are stiff, and unlike MHD, it is prone to a loss of hyperbolicity if the parallel and perpendicular pressures become too different. The usual cure is to bring the parallel and perpendicular pressures closer to one another, but that has usually been done in an ad hoc manner. We present a physics-informed method of pressure relaxation based on the idea of pitch-angle scattering that keeps the numerical system hyperbolic and naturally leads to zero anisotropy in the limit of very large plasma beta. Numerical codes based on the CGL equations can, therefore, be made to function robustly for any magnetic field strength, including the limit where the magnetic field approaches zero. The capabilities of our new algorithm are demonstrated using several stringent test problems that provide a comparison of the CGL equations in the weakly and strongly collisional limits. This includes a test problem that mimics the interaction of a shock with a magnetospheric environment in 2D.

Adaptive rank-based tests for high dimensional mean problems

The Wilcoxon signed-rank test and the Wilcoxon–Mann–Whitney test are commonly employed in one sample and two sample mean tests for one-dimensional hypothesis problems. For high-dimensional mean test problems, we calculate the asymptotic distribution of the maximum of rank statistics for each variable and suggest a max-type test. This max-type test is then merged with a sum-type test, based on their asymptotic independence offered by stationary and strong mixing assumptions. Our numerical studies reveal that this combined test demonstrates robustness and superiority over other methods, especially for heavy-tailed distributions.