Non-separable Variables Research Articles

A decomposition framework based on memorized binary search for large-scale optimization problems

Cooperative co-evolution (CC) is an evolutionary framework for dealing with large-scale optimization problems. The divide-and-conquer strategy is widely used in CC. The large-scale problem is decomposed into multiple smaller and easier to optimize subcomponents to reduce the complexity and improve the optimization performance. However, CC typically requires appropriate decomposition methods and numerous functional evaluations. To address this problem, this study proposes a new decomposition framework known as hierarchical differential grouping (HDG). Hierarchy 1 is used to identify the separable and nonseparable variables. The core of the HDG is Hierarchy 2, where a memorized binary search is used to group nonseparable variables into multiple subcomponents. Hierarchy 3 merges the variables with indirect interactions. Finally, Hierarchy 4 implements sequential decomposition for the larger subcomponents. Furthermore, this study theoretically analyzes the computational resources consumed by HDG to decompose large-scale problems. The experimental results demonstrate that HDG outperforms other state-of-the-art differential grouping methods in terms of both the decomposition accuracy and computational complexity. HDG combined with the covariance matrix adaptive evolution strategy can be competitive on multiple benchmark functions.

Merged Differential Grouping for Large-Scale Global Optimization

The divide-and-conquer strategy has been widely used in cooperative co-evolutionary algorithms to deal with large-scale global optimization problems, where a target problem is decomposed into a set of lower-dimensional and tractable subproblems to reduce the problem complexity. However, such a strategy usually demands a large number of function evaluations to obtain an accurate variable grouping. To address this issue, a merged differential grouping (MDG) method is proposed in this article based on the subset–subset interaction and binary search. In the proposed method, each variable is first identified as either a separable variable or a nonseparable variable. Afterward, all separable variables are put into the same subset, and the nonseparable variables are divided into multiple subsets using a binary-tree-based iterative merging method. With the proposed algorithm, the computational complexity of interaction detection is reduced to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\max \{n,n_{ns}\times \log _{2} k\})$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{ns}(\leq n)$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k( &lt; n)$ </tex-math></inline-formula> indicate the numbers of decision variables, nonseparable variables, and subsets of nonseparable variables, respectively. The experimental results on benchmark problems show that MDG is very competitive with the other state-of-the-art methods in terms of efficiency and accuracy of problem decomposition.

Parameter identification for lithium batteries: Model variable-coupling analysis and a novel cooperatively coevolving identification algorithm

In the operational control of renewable energy system, the efficient parameter identification for lithium battery is of great importance. In this study, the parameter identification of lithium battery is modelled as a large-scale global optimization problem with thousands of dimensionalities. In addition, the developed identification model is proved to be a partial-separable problem by comprehensively analysing its variable-coupling relationships, and the detailed proof is also provided. In order to overcome the high-dimensional characteristic of the developed model, a novel algorithm namely incomplete multi-context cooperatively coevolving PSO (IMCCPSO) is developed, in which some efficient algorithmic mechanisms are proposed: On one hand, the non-separable variables are grouped together with each of the separable-variable components, and the context vectors for separable and non-separable variables are discriminatively reconstituted for balancing the local and global exploration; On the other hand, the coevolving efficiency index is proposed for selecting the group-size values and coevolving rules dynamically and adaptively. Experimental results show that the developed methodology can effectively identify the parameters of the evaluated lithium battery bank under typical load profiles, and the developed IMCCPSO also outperforms the compared state-of-the-art algorithms on identification accuracy and robustness.

Frequency domain analysis of pre-stressed elastomeric vibration isolators

Two types of elastomeric vibration isolators used for equipment vibration isolation in aerospace vehicles are considered for the present study. These isolators are constructed using elastomers mounted in steel encasings. These isolators are initially deformed statically and dynamic loads are applied on the deformed configuration. To capture the static deformation, equivalent static load corresponding to its load rating and specified displacements are created. Static deformation is computed using Finite Element methods with four node axi-symmetric element which include the geometric non-linear effect for steel and with standard Yeoh hyper-elastic material model for elastomers (Muhammed and Zu, 2012) [1]. Yeoh material constants are derived from uni-axial tension test data of the elastomer specimen. These isolators are subjected to harmonic and random excitations in the pre-deformed state. For numerical analysis, elastomeric constants at dynamic conditions are obtained as complex function of frequency using Dynamic Mechanical Analyzer (DMA) for a range of frequencies. The standard material model of Yeoh is modified incorporating frequency dependant material characteristics and damping in the range of frequencies of interest. A multiplicative non-separable variables law is derived for Yeoh material model to include the effect of static pre-stress, based on the methodology given in literature (Nashif et al., 1985; Beda et al., 2014) [2,3]. The modifications of Yeoh model suitable for frequency domain analysis is the novelty in the present study. In the analysis, while dynamic loads are applied, the configuration is updated considering initial static loading. The frequency response of the isolators is computed using material properties evaluated at progressive dynamic strains until a match in natural frequency is observed. Appropriate damping corrections are then incorporated to match the test observed transmissibility. Then updated material properties are used to compute the random response which showed good agreement with results of experiments, validating the approach taken for the development of this model.

Tech-eco Efficiency Evaluation of hydrogen production industry under carbon dioxide Emissions regulation in China

This study introduces the concept of non-separable variables, and conducts a super-efficiency slack-based measure (SBM) model considered undesirable output variable to evaluate the economic and technical efficiencies of 15 hydrogen production methods in China. The results show that the average technical efficiency and scale efficiency values of hydrogen production industry are 0.438 and 0.352, respectively, which are relevantly low. The average pure technical efficiency value is 1.090. When the strong carbon constraint is introduced, the average technical efficiency and pure technical efficiency values of hydrogen production industry are increased by about 12.3% and 22.2%, respectively, but the average scale efficiency is reduced by 1.7%. The coal-to-hydrogen production with carbon capture, utilization and storage (CCUS) technology produces the highest technical efficiency, and the technical efficiency value and pure technical efficiency value are 1.435 and 1.679, respectively. The scale efficiency value of hydrogen production from chlor-alkali is 0.951, which is the most effective scale among all measured hydrogen production processes. The methanol-to-hydrogen production also performs great. However, hydrogen production by water electrolysis does not have the advantages of economic nor technical efficiencies. The results of the input-output slack of 15 hydrogen production methods show that the excessive production inputs and carbon dioxide emission are the main reasons for the poor efficiency evaluation results of most hydrogen production methods. The CO2 emission regulation reduces the redundancy of the cost, comprehensive energy consumption and CO2 emissions of 13 hydrogen production processes by more than four times, which affected the technical efficiency and the scale efficiency of hydrogen production industry. It indicates that CO2 emission regulation has improved the economic and technical efficiency of hydrogen production processes in China.

Potential distribution of polar optical phonons and their electron–phonon coupling properties of a wurtizite GaN-based trianglar quantum dot

Polar optical phonon modes of a wurtzite nitride trianglar quantum dot (QD) are deduced and analyzed by using the dielectric continuum model. Based on a method of nonseparable variables, the Laplace equation of electristatic potential of the structure is solved, and the exact and analytical phonon states of exact confined (EC) modes are derived. It is found that the frequency of EC modes in the nitride trianglar QD systems can take both the longitudinal optical phonons in [Formula: see text]-direction [Formula: see text] and the longitudinal optical phonons in [Formula: see text]-plane [Formula: see text], which is obviously different from the cases of EC modes in wurtzite GaN quantum wells and quantum wires due to their different confined dimensions. The polarization eigenvectors and their orthogonality relations for the EC phonon modes are derived, which could form a set of orthogonal and completed basis vectors. By using the orthogonality relations of polarization eigenvector and field quantization method, the free phonon field of EC modes and the corresponding Fröhlich electron–phonon interaction Hamiltonian are obtained. Numerical calculations on a wurtztie GaN trianglar QD are performed. The spatial distributions of phonon potentials of EC phonons both in [Formula: see text]-plane and in [Formula: see text]-direction as well as the electron–phonon coupling strength are plotted and discussed. It is observed that the potential of these EC phonon modes is strictly confined within the trianglar QD ranges. The dependence of symmetries and electron–phonon coupling strength on the quantum numbers [Formula: see text] and [Formula: see text] in [Formula: see text]-plane, and the quantum number [Formula: see text] in [Formula: see text]-direction are analyzed in detail. The relationship of the total number of peaks and troughs with these quantum numbers are also discussed and induced.

Generalized Method of Finite Integral Transforms in Static Problems for Anisotropic Prisms

A new approach to solving three-dimensional elliptic linear boundary-value problems with nonseparable variables is developed using the ideas of methods of finite integral transforms. It consists in setting up a coupled system of three integral transforms with three pairs of independent variables of the domain, from which the transforms and kernels are determined. The approach is used to solve static problems for anisotropic prisms with elastic properties of low order of symmetry and arbitrary conditions on the faces. The approach is tested and the deformation of specific bodies of this class is analyzed.

Cooperative Coevolution with Formula-Based Variable Grouping for Large-Scale Global Optimization.

For a large-scale global optimization (LSGO) problem, divide-and-conquer is usually considered an effective strategy to decompose the problem into smaller subproblems, each of which can then be solved individually. Among these decomposition methods, variable grouping is shown to be promising in recent years. Existing variable grouping methods usually assume the problem to be black-box (i.e., assuming that an analytical model of the objective function is unknown), and they attempt to learn appropriate variable grouping that would allow for a better decomposition of the problem. In such cases, these variable grouping methods do not make a direct use of the formula of the objective function. However, it can be argued that many real-world problems are white-box problems, that is, the formulas of objective functions are often known a priori. These formulas of the objective functions provide rich information which can then be used to design an effective variable group method. In this article, a formula-based grouping strategy (FBG) for white-box problems is first proposed. It groups variables directly via the formula of an objective function which usually consists of a finite number of operations (i.e., four arithmetic operations "", "", "", "" and composite operations of basic elementary functions). In FBG, the operations are classified into two classes: one resulting in nonseparable variables, and the other resulting in separable variables. In FBG, variables can be automatically grouped into a suitable number of non-interacting subcomponents, with variables in each subcomponent being interdependent. FBG can easily be applied to any white-box problem and can be integrated into a cooperative coevolution framework. Based on FBG, a novel cooperative coevolution algorithm with formula-based variable grouping (so-called CCF) is proposed in this article for decomposing a large-scale white-box problem into several smaller subproblems and optimizing them respectively. To further enhance the efficiency of CCF, a new local search scheme is designed to improve the solution quality. To verify the efficiency of CCF, experiments are conducted on the standard LSGO benchmark suites of CEC'2008, CEC'2010, CEC'2013, and a real-world problem. Our results suggest that the performance of CCF is very competitive when compared with those of the state-of-the-art LSGO algorithms.

Cooperation coevolution with fast interdependency identification for large scale optimization

Cooperative coevolution (CC) provides a powerful divide-and-conquer architecture for large scale global optimization (LSGO). However, its performance relies highly on decomposition. To make near-optimal decomposition, most developed decomposition strategies either cannot obtain the correct interdependency information or require a lot of fitness evaluations (FEs) in the identification. To alleviate the limitations in previous works, in this paper we propose a fast interdependency identification (FII) algorithm for CC in LSGO. The proposed algorithm firstly identifies separable and nonseparable variables efficiently. Then, the interdependency information of nonseparable variables is further investigated. To make near-optimal decomposition for CC, our algorithm avoids the necessity of obtaining the full interdependency information of nonseparable variables. Therefore, a significant number of FEs can be saved. Extensive experiments have been conducted on two suites of LSGO benchmark functions with up to 2000 variables. FII correctly identified the interdependency information on most benchmark functions with much fewer FEs in comparison with three state-of-the-art algorithms. Furthermore, combined with CC and coupled with a differential evolution variant serving as the optimizer, FII has shown its promising performance in LSGO.

Loaded rubber-like materials subjected to small-amplitude vibrations

This paper proposes a constitutive law and a method for characterizing highly preloaded viscoelastic materials subjected to linear (small-amplitude) vibrations. A multiplicative non-separable variables law has been suggested to model the behavior that depends on both stretch and time/frequency. This approach allows splitting the intricate combined test performed simultaneously on both stretch and frequency, generally in a limited experimental domain up to 100 Hz, into two independent tests. Thus, on one hand, the dynamic complex modulus dependent on frequency alone is evaluated on the basis of vibration tests in a large experimental domain up to 100 kHz. On the other hand, energetic parameters are determined from a quasi-static hyperelastic tensile test. The complex modulus, dependent on both stretch and frequency, is then deduced from the results acquired from uncoupled investigations. This work shows that, in extension, the elastic modulus increases with increasing stretch, and the loss factor decreases with increasing stretch; while, in compression, around the material undeformed state, the modulus increases as the stretch increases till a certain value of compression stretch (upturn point depending on material characteristics), and then the modulus decreases as the stretch increases. Globally, preload rigidifies materials but reduces their damping property. These results closely match a well-known observation in solid mechanics.

Separability Detection Cooperative Particle Swarm Optimizer based on Covariance Matrix Adaptation

The particle swarm optimizer (PSO) is a population- based optimization technique that can be widely utilized to many applications. The cooperative particle swarm optimization (CPSO) applies cooperative behavior to improve the PSO on finding the global optimum in a high-dimensional space. This is achieved by employing multiple swarms to partition the search space. However, independent changes made by different swarms on correlated variables will deteriorate the performance of the algorithm. This paper proposes a separability detection approach based on covariance matrix adaptation to find non-separable variables so that they can previously be placed into the same swarm to address the difficulty that the original CPSO encounters.

An asymptotic solution of the Schrödinger equation for the elliptic wire in the magnetic field

An asymptotic solution of the Schrödinger equation with non-separable variables is obtained for a particle confined to an infinite elliptic cylinder potential well under an applied uniform longitudinal magnetic field. Using the standard-problem method, dimension-quantized eigenvalues have been calculated when the magnetic length is large enough in comparison with the half of the distance between the boundary ellipse focuses. In semi-classical approximation, the confined electron (hole) states are divided into the boundary states (BS), ring states (RS), hyperbolic caustic states (HCS) and harmonic oscillator states (HOS). For large angular momentum quantum numbers and small radial quantum numbers, the BS and RS are grouped into the ‘whispering gallery’ mode. They associate with particles moving along the wire cross section boundary. The motion is limited from the wire core by the elliptic caustic. Consisting of the HCS and HOS, the ‘jumping ball’ modes correspond to the states of particle moving along a wire diameter when the angular momentum quantum number is much less than the radial quantum number. In this case, the motion is restricted by the hyperbolic caustics and two boundary ellipse arcs. For excited hole states in a Bi wire, the energy spectrum and space probability distribution are analyzed.

Eigenstates of incommensurate systems

Quantization of non-separable variables and quasiperiodic systems is considered. A Bloch electron in a magnetic field, diamagnetic edge states of open orbits, an elliptic shell in a magnetic field and a tunneling time response to an impulse are discussed. New irregular quantum oscillations, branching phase transitions and local space-time tunneling resonances are predicted.

Kinetic analysis of TG data: study of calcium oxalate monohydrate dehydration with a non-separable variables model

A study of the kinetic behaviour of calcium oxalate dehydration using TG data is reported. The discrimination among non-separable variables models was carried out in a differential form. It was found that a two step model namely nucleation, E 1 = 168 kJ mol −1, and reaction at the cylinder interface, E 2 = 70 kJ mol −1 is required to descibe the reaction course. Good agreement between experimental and calculated α- T curves is observed.

Kinetic equations of mechanistic type with nonseparable variables for catalyst deactivation by coke. Models and data analysis methods

Kinetic equations of mechanistic type with nonseparable variables for catalyst deactivation by coke. Models and data analysis methods