An Overview of Sequential Approximation in Topology Optimization of Continuum Structure

This paper offers an extensive overview of the utilization of sequential approximate optimization approaches in the context of numerically simulated large-scale continuum structures.These structures,commonly encountered in engineering applications,often involve complex objective and constraint functions that cannot be readily expressed as explicit functions of the design variables.As a result,sequential approximation techniques have emerged as the preferred strategy for addressing a wide array of topology optimization challenges.Over the past several decades,topology optimization methods have been advanced remarkably and successfully applied to solve engineering problems incorporating diverse physical backgrounds.In comparison to the large-scale equation solution,sensitivity analysis,graphics post-processing,etc.,the progress of the sequential approximation functions and their corresponding optimizersmake sluggish progress.Researchers,particularly novices,pay special attention to their difficulties with a particular problem.Thus,this paper provides an overview of sequential approximation functions,related literature on topology optimization methods,and their applications.Starting from optimality criteria and sequential linear programming,the other sequential approximate optimizations are introduced by employing Taylor expansion and intervening variables.In addition,recent advancements have led to the emergence of approaches such as Augmented Lagrange,sequential approximate integer,and non-gradient approximation are also introduced.By highlighting real-world applications and case studies,the paper not only demonstrates the practical relevance of these methods but also underscores the need for continued exploration in this area.Furthermore,to provide a comprehensive overview,this paper offers several novel developments that aim to illuminate potential directions for future research.

Cyclic Solution and Optimal Approximation of the Quaternion Stein Equation

In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .

Accuracy analysis of the optimal separable approximation method of one-way wave migration

An approximation for the one-way wave operator takes the form of separated space and wave-number variables and makes it possible to use the FFT, which results in a great improvement in the computational efficiency. From the function approximation perspective, the OSA method shares the same separable approximation format to the one-way wave operator as other separable approximation methods but it is the only global function approximation among these methods. This leads to a difference in the phase error curve, impulse response, and migration result from other separable approximation methods. The difference is that the OSA method has higher accuracy, and the sensitivity to the velocity variation declines with increasing order.

Computation of Peak Output for Inputs Restricted in L_2 and L_∞ Norms Using Finite Difference Schemes and Convex Optimization

Control systems designed by the principle of matching gives rise to problems of evaluating the peak output. This paper proposes a practical method for computing the peak output of linear time-invariant and non-anticipative systems for a class of possible sets that are characterized with many bounding conditions on the two- and/or the infinity-norms of the inputs and their derivatives. The original infinite-dimensional convex optimization problem is approximated as a large-scale convex programme defined in a Euclidean space, which are associated with sparse matrices and thus can be solved efficiently in practice. The numerical results show that the method performs satisfactorily, and that using a possible set with many bounding conditions can help to reduce the design conservatism and thereby yield a better match.

3D wavefield extrapolation with optimal separable approximation

An accurate and wide-angle one-way propagator for wavefield extrapolation is an important topic for research on wave-equation prestack depth migration in the presence of large and rapid velocity variations. Based on the optimal separable approximation presented in this paper, the mixed domain algorithm with forward and inverse Fourier transforms is used to construct the 3D one-way wavefield extrapolation operator. This operator separates variables in the wavenumber and spatial domains. The phase shift operation is implemented in the wavenumber domain while the time delay for lateral velocity variation is corrected in the spatial domain. The impulse responses of the one-way wave operator show that the numeric computation is consistent with the theoretical value for each velocity, revealing that the operator constructed with the optimal separable approximation can be applied to lateral velocity variations for the case of small steps. Imaging results of the SEG/EAGE model and field data indicate that the new method can be used to image complex structure.

On the Finite-dimensional Optimal Approximations to the Infinite dimensional Linear Operators

A method of approaching to the infinite dimensional linear operators by the finite dimensional operators is discussed. It is shown that,for every infinite dimensional operator A and every natural number n, there exists an n dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n dimensional optimal approximations to be unique is obtained.

The Offset-Domain Prestack Depth Migration with Optimal Separable Approximation

The offset-domain prestack depth migration with optimal separable approximation, based on the double square root equation, is used to image complex media with large and rapid velocity variations. The method downward continues the source and the receiver wavefields simultaneously. The mixed domain algorithm with forward Fourier and inverse Fourier transform is used to construct the double square root equation wavefield extrapolation operator. This operator separates variables in the wave number domain and variables in the space domain. The phase operation is implemented in the wave number domain, whereas the time delay for lateral velocity variation is corrected in the space domain. The migration algorithm is efficient since the seismic data are not computed shot by shot. The data set test of the Marmousi model indicates that the offset-domain migration provides a satisfied seismic migration section on which complex geologic structures are imaged in media with large and rapid lateral velocity variations.

A Novel Approach for Optimum Conjunctive Use Management of Groundwater and Surface Water Resources under Uncertainty

Uncertainty in determining optimum conjunctive water use lies not only on variability of hydrological cycle and climate but also on lack of adequate data and perfect knowledge about groundwater-surface water system interactions, errors in historic data and inherent variability of system parameters both in space and time. Simulation-optimization models are used for conjunctive water use management under uncertain conditions. However, direct application of such approach whereby all realizations are considered at every-iteration of the optimization process leads to a highly computational time-consuming optimization problem as the number of realizations increases. Hence, this study proposes a novel approach—a Retrospective Optimization Approximation (ROA) approach. In this approach, a simulation model was used to determine aquifer system responses (draw-downs) which were assembled as response matrices and incorporated in the optimization model (procedure) as coefficients in the constraints. The sample optimization sub-problems generated, were solved and analyzed through ROA-Active-Set procedure implemented under MATLAB code. The ROA-Active Set procedure solves and evaluates a sequence of sample path optimization sub-problems in an increasing number of realizations. The methodology was applied to a real-world conjunctive water use management problem found in Great Letaba River basin, South Africa. In the River basin, surface water source contributes 87% of the existing un-optimized total conjunctive water use withdrawal rate (6512.04 m3/day) and the remaining 13% is contributed by groundwater source. Through ROA approach, results indicate that the optimum percentages contribution of the surface and subsurface sources to the total water demand are 58% and 42% respectively. This implies that the existing percentage contribution can be increased or reduced by ±29% that is groundwater source can be increased by 29% while the surface water source contribution can be reduced by 29%. This reveals that the existing conjunctive water use practice is unsustainable wherein surface water system is overstressed while groundwater system is under-utilized. Through k-means sampling technique ROA-Active Set procedure was able to attain a converged maximum expected total optimum conjunctive water use withdrawal rate of 4.35 × 104 m3/day within a relatively few numbers of iterations (6 to 8 iterations) in about 2.30 Hrs. In conclusion, results demonstrated that ROA approach is capable of managing real-world regional aquifers sustainable conjunctive water use practice under hydro-geological uncertainty conditions.

THE GENERALIZED REFLEXIVE SOLUTION FOR A CLASS OF MATRIX EQUATIONS (AX-B,XC=D)

In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.

A novel Lagrangian relaxation level approach for scheduling steelmaking-refining-continuous casting production

A Lagrangian relaxation(LR) approach was presented which is with machine capacity relaxation and operation precedence relaxation for solving a flexible job shop(FJS) scheduling problem from the steelmaking-refining-continuous casting process. Unlike the full optimization of LR problems in traditional LR approaches, the machine capacity relaxation is optimized asymptotically, while the precedence relaxation is optimized approximately due to the NP-hard nature of its LR problem. Because the standard subgradient algorithm(SSA) cannot solve the Lagrangian dual(LD) problem within the partial optimization of LR problem, an effective deflected-conditional approximate subgradient level algorithm(DCASLA) was developed, named as Lagrangian relaxation level approach. The efficiency of the DCASLA is enhanced by a deflected-conditional epsilon-subgradient to weaken the possible zigzagging phenomena. Computational results and comparisons show that the proposed methods improve significantly the efficiency of the LR approach and the DCASLA adopting capacity relaxation strategy performs best among eight methods in terms of solution quality and running time.

The Quantum Approximate Algorithm for Solving Traveling Salesman Problem

The Quantum Approximate Optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization problems.It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians.To fit this framework,one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians.In this paper,for the well-known NP-hard Traveling Salesman Problem(TSP),we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian.Moreover,we map edges(routes)connecting each pair of cities to qubits,which decreases the search space significantly in comparison to other approaches.As a result,our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach.We argue the formalization approach presented in this paper would lead to a generalized framework for finding,in the context of QAOA,high-quality approximate solutions to NP optimization problems.

MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL SPACES

Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries consisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-dimensional spaces. It offers a method to process signals in arbitrary dimensions.

A quadratic programming method for optimal degree reduction of Bézier curves with G^1-continuity

This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.

Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices

In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.

The Reflexive Solutions of the Matrix Equations (AX,XB^H )= (C,D^H )

The matrix equations (AX, XBH)=(C, DH) have been widely used in structural design, parameter identification, linear optimal control, and so on. But few researches studied the reflexive solutions. A new approach for the reflexive solutions to the matrix equations was proposed. By applying the canonical correlation decomposition (CCD) of matrix pairs, the necessary and sufficient conditions for the existence and the general expression for the reflexive solutions of the matrix equations (AX, XBH)=(C, DH) were established. In addition, by using the methods of space decomposition, the expression of the optimal approximation solution to a given matrix was derived.

Proximal Point-Like Method for Updating Simultaneously Mass and Stiffness Matrices of Finite Element Model

The problem of correcting simultaneously mass and stiffness matrices of finite element model of undamped structural systems using vibration tests is considered in this paper.The desired matrix properties,including satisfaction of the characteristic equation,symmetry,positive semidefiniteness and sparsity,are imposed as side constraints to form the optimal matrix pencil approximation problem.Using partial Lagrangian multipliers,we transform the nonlinearly constrained optimization problem into an equivalent matrix linear variational inequality,develop a proximal point-like method for solving the matrix linear variational inequality,and analyze its global convergence.Numerical results are included to illustrate the performance and application of the proposed method.

Updating Mass and Stiffness Matrices Using Eigenstructure Assignment Methods

A novel numerical method is presented to update mass and stiffness matrices simultaneously with measured vibration data by means of the combined acceleration and displacement output feedback.By the method,the required displacement and acceleration output feedback gain matrices are determined,and thus the optimal approximation mass matrix and stiffness matrix which satisfy the required orthogonality relation and eigenvalue equation are found.The proposed method is computationally efficient and the updated mass and stiffness matrices are also symmetric and have the compact expressions.The numerical example shows that the proposed method is reliable and attractive.

Least-square Solutions of Inverse Problems for Anti-symmetric and Skew-symmetric Matrices

The least-square solutions of inverse problem for anti-symmetric and skew-symmetric matrices are studied. In addition, the problem of using anti-symmetric and skew-symmetric matrices to construct the optimal approximation to a given matrix is discussed, the necessary and sufficient conditions for the problem are derived, and the expression of the solution is provided. A numerical example is given to show the effectiveness of the proposed method.

Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations

Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E, CXD = F. If we choose the initial iterative matrix X0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.

Approximate Optimality Conditions for Composite Convex Optimization Problems

The purpose of this paper is to study the approximate optimality condition for composite convex optimization problems with a cone-convex system in locally convex spaces,where all functions involved are not necessarily lower semicontinuous.By using the properties of the epigraph of conjugate functions,we introduce a new regularity condition and give its equivalent characterizations.Under this new regularity condition,we derive necessary and sufficient optimality conditions ofε-optimal solutions for the composite convex optimization problem.As applications of our results,we derive approximate optimality conditions to cone-convex optimization problems.Our results extend or cover many known results in the literature.