Function Minimization Problem Research Articles

Dual Objective Multidimensional Parameter Optimization of Cam Curve Based on Improved Ant Colony Algorithm

By applying improved ant colony algorithm, this paper analyzed multi-objective design problem in optimal design of cam mechanism profile curve, which were then transformed into TSP problem, that is, multi-objective function minimization problem into TSP shortest path searching problem. For the multiple complex variables and target weight parameters in design, a concept of multidimensional space node in ant colony optimization path was proposed, converting multi-objective function with weight coefficient into a multidimensional space nodes (city).At the same time, improved ant colony algorithm, combined with genetic algorithm, was used to avoid optimization calculation falling into a locally optimal solution. Exemplified by dual objective function in cam mechanism profile curve design, this paper solved optimal value problem of dual objective optimization design involving the biggest fullness coefficient and minimum abrasion quantity of variable weight coefficient under the three-dimensional parameters n, m and w.

Deformation field correction to preserve topology for image registration

In this paper, the author addresses the issue of designing a theoretically well-motivated and computationally efficient method ensuring topology preservation on image-registration-related deformation fields. The model is motivated by a mathematical characterization of topology preservation for a deformation field mapping two subsets of Z 2 , namely, positivity of the four approximations to the Jacobian determinant of the deformation on a square patch. The first step of the proposed algorithm thus consists in correcting the gradient vector field of the deformation at the discrete level in order to fulfill this positivity condition. Once this step is achieved, it thus remains to reconstruct the deformation field, given its full set of discrete gradient vectors. The author propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is introduced and its well-posedness is studied: existence/uniqueness of the solution, characterization of the solution, convergence of the method when the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to handle large deformations (contrary to classical methods) and the interest of having decomposed the problem into smaller ones are provided.

SIMULATION-BASED OPTIMIZATION BY NEW STOCHASTIC APPROXIMATION ALGORITHM

This paper proposes one new stochastic approximation algorithm for solving simulation-based optimization problems. It employs a weighted combination of two independent current noisy gradient measurements as the iterative direction. It can be regarded as a stochastic approximation algorithm with a special matrix step size. The almost sure convergence and the asymptotic rate of convergence of the new algorithm are established. Our numerical experiments show that it outperforms the classical Robbins–Monro (RM) algorithm and several other existing algorithms for one noisy nonlinear function minimization problem, several unconstrained optimization problems and one typical simulation-based optimization problem, i.e., (s, S)-inventory problem.

A projection gradient method for computing ground state of spin-2 Bose–Einstein condensates

In this paper, a projection gradient method is presented for computing ground state of spin-2 Bose–Einstein condensates (BEC). We first propose the general projection gradient method for solving energy functional minimization problem under multiple constraints, in which the energy functional takes real functions as independent variables. We next extend the method to solve a similar problem, where the energy functional now takes complex functions as independent variables. We finally employ the method into finding the ground state of spin-2 BEC. The key of our method is: by constructing continuous gradient flows (CGFs), the ground state of spin-2 BEC can be computed as the steady state solution of such CGFs. We discretized the CGFs by a conservative finite difference method along with a proper way to deal with the nonlinear terms. We show that the numerical discretization is normalization and magnetization conservative and energy diminishing. Numerical results of the ground state and their energy of spin-2 BEC are reported to demonstrate the effectiveness of the numerical method.

MIMO radar 3D imaging based on combined amplitude and total variation cost function with sequential order one negative exponential form.

In inverse synthetic aperture radar (ISAR) imaging, a target is usually regarded as consist of a few strong (specular) scatterers and the distribution of these strong scatterers is sparse in the imaging volume. In this paper, we propose to incorporate the sparse signal recovery method in 3D multiple-input multiple-output radar imaging algorithm. Sequential order one negative exponential (SOONE) function, which forms homotopy between 1 and 0 norms, is proposed to measure the sparsity. Gradient projection is used to solve a constrained nonconvex SOONE function minimization problem and recover the sparse signal. However, while the gradient projection method is computationally simple, it is not robust when a matrix in the algorithm is ill conditioned. We thus further propose using diagonal loading and singular value decomposition methods to improve the robustness of the algorithm. In order to handle targets with large flat surfaces, a combined amplitude and total-variation objective function is also proposed to regularize the shapes of the flat surfaces. Simulation results show that the proposed gradient projection of SOONE function method is better than orthogonal matching pursuit, CoSaMp, l1-magic, Bayesian method with Laplace prior, smoothed l0 method, and l1-ls in high SNR cases for recovery of ± 1 random spikes sparse signal. The quality of the simulated 3D images and real data ISAR images obtained using the new method is better than that of the conventional correlation method and minimum l2 norm method, and competitive to the aforementioned sparse signal recovery algorithms.

Attraction area of minima in quadratic binary optimization

The paper deals with the problem of quadratic functional minimization in the space of binary variables. We analyze the efficiency of the random search procedure used in binary minimization and show that the radius of the attraction area of a minimum directly depends on its depth: the deeper the minimum, the greater its radius of attraction. Thus, the probability of finding a minimum during random search grows exponentially with depth of the minimum.

A Relax Inexact Accelerated Proximal Gradient Method for the Constrained Minimization Problem of Maximum Eigenvalue Functions

For constrained minimization problem of maximum eigenvalue functions, since the objective function is nonsmooth, we can use the approximate inexact accelerated proximal gradient (AIAPG) method (Wang et al., 2013) to solve its smooth approximation minimization problem. When we take the functiong(X)=δΩ(X) (Ω∶={X∈Sn:F(X)=b,X⪰0})in the problemmin{λmax(X)+g(X):X∈Sn}, whereλmax(X)is the maximum eigenvalue function,g(X)is a proper lower semicontinuous convex function (possibly nonsmooth) andδΩ(X)denotes the indicator function. But the approximate minimizer generated by AIAPG method must be contained inΩotherwise the method will be invalid. In this paper, we will consider the case where the approximate minimizer cannot be guaranteed inΩ. Thus we will propose two different strategies, respectively, constructing the feasible solution and designing a new method named relax inexact accelerated proximal gradient (RIAPG) method. It is worth mentioning that one advantage when compared to the former is that the latter strategy can overcome the drawback. The drawback is that the required conditions are too strict. Furthermore, the RIAPG method inherits the global iteration complexity and attractive computational advantage of AIAPG method.

A variational-difference numerical method for designing progressive-addition lenses

We propose a variational-difference method for designing the optical free form surface of progressive-addition lenses (PALs). The PAL, which has a front surface with three important zones including the far-view, near-view and intermediate zones, is often used to remedy presbyopia by distributing optical powers of the three zones progressively and smoothly. The problem for designing PALs could be viewed as a functional minimization problem. Compared with the existing literature which solved the problem by the B-spline finite element method, the essence of the proposed variational-difference numerical method lies in minimizing the functional directly by finite difference method and/or numerical quadratures rather than in approximating the solution of the corresponding Euler–Lagrange equation to the functional. It is very easily understood and implemented by optical engineers, and the numerical results indicate that it can produce satisfactory designs for optical engineers in several seconds. We believe that our method can be a powerful candidate tool for designing various specifications of PALs.

Sparse Recovery Based on Squarewave-Like Sum Function Minimization

For the problem of sparse recovery, an approximation algorithm, termed Squarewave-Like Sum Function Minimization (abr. SLSFM), is proposed. Firstly, norm is appro-ximated by squarewave-like sum function. Then sparse recovery is cast as Squarewave-Like Sum Function Minimization problem, which is solved by a fast iterative method. Finally, simulation results show that our algorithm has some advantages over most existing methods: it is more robust to noise and needs less measurement.

A New Image Demising Method Based on Partial Differential Equations

In the paper, we discuss the image demising models, based on partial differential equations. It is through the use of the concept of variations in the calculus of the objective function minimization problem, defines the image processing tasks. The results show that the model expands 2d thermal diffusion equation. Therefore, it is easy to get solution is to use a simple iterative process.

A bidirectional B-spline finite point method for the analysis of piezoelectric laminated composite plates and its application in material parameter identification

The bidirectional B-spline finite point method (BB-sFPM) can be used to construct a direction problem solver with good accuracy and little computational cost. To improve the performance of a piezoelectric structural system, we propose a parameter identification process with BB-sFPM for the analysis of a laminated composite plate with piezoelectric layers bonded to its upper and lower surfaces. The displacement and electric potential fields in the proposed method are approximated by using B-spline interpolation functions in x and y directions. And the displacement and stress solutions of the BB-sFPM are derived for a piezoelectric laminated composite plate (PLCP). The performance of the proposed method is assessed with some numerical examples available in published literature. The numerical results show that the proposed method has very good capability in the analysis of a thin-to-moderately thick PLCP. As an application, the displacements obtained by the BB-sFPM are applied to the material parameter identification of the PLCP. The parameter identification problem is formulated as an objective function minimization problem, defined as a square sum of differences between the measured displacement and the computed displacement by the BB-sFPM. The minimization problem is solved by using Levenberg–Marquardt method in which the sensitivity calculation is based on the differentiation of the governing equations of the BB-sFPM. Numerical examples are given to illustrate the validity of the proposed algorithm for the material parameter identification of the PLCP.

Type design for bilayer textile materials under low temperature: Modeling, numerical algorithm and simulation

Based on the model of steady-state heat and moisture transfer through textiles, an inverse problem of type design (IPTD) for bilayer textile materials under low temperature is put forward. According to the idea of regularization method, the IPTD for bilayer textile materials can be formulated into a function minimization problem. By means of the finite difference algorithm for nonlinear ordinary differential equation and the direct search method of one-dimensional minimization problems, an iterative algorithm for the regularized solution of the IPTD is constructed. By analyzing the results of numerical simulation in different environmental conditions, some conclusions are obtained: Hooke–Jevees’s direct search method can effectively solve the IPTD for bilayer textile materials, at the same time, numerical simulation shows the effectiveness of the algorithm and the validity of the proposed inverse problem.

最大流方法在图像去噪和分割中的研究进展

近年来，基于能量泛函的图像去噪和分割模型得到广泛的关注，解决这类模型的有力工具之一是最大流方法。本文分别介绍离散的和连续的最大流方法，包括能量泛函最小化问题转化为最大流问题的基本步骤，以及相应最大流问题的求解方法，并展望最大流方法的发展前景。 In recent years, image denoising and segmentation model based on energy functional received widely atten- tion. Max-flow method is one of the most powerful tools to solve this kind of model. Discrete and continuous max-flow methods are introduced, they both include the basic steps in which energy functional minimization problem transformed to max-flow problem, and the solutions of the corresponding max-flow problem are also reviewed. In addition, the de- velopment of max-flow method are also discussed.

Hybrid Dynamical Evolutionary Algorithm and Time Complexity Analysis

The dynamical evolutionary algorithm (DEA) is a new evolutionary algorithm based on the theory of statistical mechanics, however, DEA converges slowly and often converge at local optima for some function optimization problems. In this paper, a hybrid dynamical evolutionary algorithm (HDEA) with multi-parent crossover and differential evolution mutation is proposed for accelerating convergence velocity and easily escaping suboptimal solutions. Moreover, the population of HDEA is initialized by chaos. In order to confirm the effectiveness of our algorithm, HDEA is applied to solve the typical numerical function minimization problems. The computational complexity of HDEA is analyzed, and the experimental results show that HDEA outperforms the DEA in the aspect of convergence velocity and precision, even the two algorithms have the similar time complexity.

The Bethe Permanent of a Nonnegative Matrix

It has recently been observed that the permanent of a non-negative square matrix, i.e., of a square matrix containing only non-negative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudo-codewords and permanent-based kernels.

Equivalence of convex minimization problems over base polytopes

This paper considers convex optimization problems over base polytopes of polymatroids. We show that the decomposition algorithm for the separable convex function minimization problems helps us derive simple sufficient conditions for the rationality of optimal solutions, and leads us to some useful properties, including the equivalence of the lexicographically optimal base problem, introduced by Fujishige, and the submodular utility allocation market problem, introduced by Jain and Vazirani. In addition, we deal with a class of non-separable convex objective functions. Moreover, we describe an algorithm for the lexicographically optimal base problem, which is a variant of the Fujishige–Wolfe algorithm.

Optimal prediction of human postural response under anterior–posterior platform tilting

Previous studies have suggested that human beings movements can be related to the problem of cost function minimization. But at the present time it is not clear that which objective function(s) and constraints are used by central nervous system (CNS) to produce optimal reactions under perturbations. Present study has been done experimentally and by numerical simulations to explore the stability constraints which should be used in combination with energy based cost function (weighted minimum torque) to estimate the motor planning criterion is used by CNS for disturbance rejections. The influence of three stability criterions (ZMP, extrapolated center of mass and a vertical force criterion) in combination with minimum torque model on the optimal trajectory formation is investigated. First, the response of 10 male healthy persons to platform oscillation was recorded by motion analysis system and the hip, knee and ankle angular trajectories were derived from recorded data. Second, the dynamic simulation of a four-segment, three actuated degrees of freedom mechanical model of the human body was performed using predictive dynamic method which leads to an optimization problem. The simulated trajectories were then compared to the experimental data. With comparison between experimental results, the weighting coefficients of the objective function were found to achieve best estimation. It was seen that the minimum torque objective function with weighting coefficients gives trajectories that are mostly matched with experimental observation. Moreover, the results showed that between stability criterions, the ZMP predictions are near to experimental results. Although by using vertical force criterion some nearness to experimental results are lost (in comparison with ZMP criterion) but a secured flat-foot posture for the model is obtained which this posture is more applicable than others in humanoid implementations.

A short time existence/uniqueness result for a nonlocal topology-preserving segmentation model

Motivated by a prior applied work of Vese and the second author dedicated to segmentation under topological constraints, we derive a slightly modified model phrased as a functional minimization problem, and propose to study it from a theoretical viewpoint. The mathematical model leads to a second order nonlinear PDE with a singularity at Du=0 and containing a nonlocal term. A suitable setting is thus the one of the viscosity solution theory and, in this framework, we establish a short time existence/uniqueness result as well as a Lipschitz regularity result for the solution.

Assessing the performance of improved harmony search algorithm (IHSA) for the optimization of unconstrained functions using Taguchi experimental design

This paper describes the meta-heuristic improved harmony search algorithm (IHSA) and analyzes the performance of IHSA in solving unconstrained function minimization problems. The most important challenge to this algorithm is setting the parameters for various optimization problems. Performance of IHSA is quite sensitive to initial settings. Using Taguchi’s experimental design to arrange IHSA parameters, the performance of IHSA was analyzed to find global optima of Rosenbrock and Wood-Colville function. Key words: Improved harmony search algorithm, Taguchi experimental design, performance of IHSA, unconstrained function.

Assessing the performance of improved harmony search algorithm (IHSA) for the optimization of unconstrained functions using Taguchi experimental design

This paper describes the meta-heuristic improved harmony search algorithm (IHSA) and analyzes the performance of IHSA in solving unconstrained function minimization problems. The most important challenge to this algorithm is setting the parameters for various optimization problems. Performance of IHSA is quite sensitive to initial settings. Using Taguchi’s experimental design to arrange IHSA parameters, the performance of IHSA was analyzed to find global optima of Rosenbrock and Wood-Colville function. Key words: Improved harmony search algorithm, Taguchi experimental design, performance of IHSA, unconstrained function.