Calculation of elements of machine-building structures on vibration strength

A method and an automated system for calculating structural elements for vibration strength have been developed. The calculation algorithms are based on a new method of forming Rayleigh-type functionals and minimizing them by the coordinate descent method. The use of the coordinate descent method avoids the problems associated with the formation, storage and operation of global matrices of stiffness and mass. This makes it possible to solve large-scale problems using only the operational memory of the PC. The developed approach allows to solve problems on own and forced fluctuations. The software is formed on a modular basis, which allows you to improve and expand this package of applications. The software has been tested on a large number of test and application tasks. The proposed calculation method and the developed automated system were implemented in engineering practice.

A block coordinate descent method for sensor network localization

The problem of sensor network localization (SNL) can be formulated as a semidefinite programming problem with a rank constraint. We propose a new method for solving such SNL problems. We factorize a semidefinite matrix with the rank constraint into a product of two matrices via the Burer–Monteiro factorization. Then, we add the difference of the two matrices, with a penalty parameter, to the objective function, thereby reformulating SNL as an unconstrained multiconvex optimization problem, to which we apply the block coordinate descent method. In this paper, we also provide theoretical analyses of the proposed method and show that each subproblem that is solved sequentially by the block coordinate descent method can also be solved analytically, with the sequence generated by our proposed algorithm converging to a stationary point of the objective function. We also give a range of the penalty parameter for which the two matrices used in the factorization agree at any accumulation point. Numerical experiments confirm that the proposed method does inherit the rank constraint and that it estimates sensor positions faster than other methods without sacrificing the estimation accuracy, especially when the measured distances contain errors.

Adaptation of energy methods to automated calculation of mobile machines frame constructions

The paper shows that the adaptation of energy methods to automated calculation of mobile machines frame constructions consists of developing a single algorithm applicable to different construction schemes. The calculation outset still remains the idea of getting a function of potential energy of deformation as a function with unknown inner power factors. Search for local function minimum of potential power of deformation has been based on the function’s discrete grid-surface. We managed to reach tactical flexibility of coordinate descent method in an attempt to continue approaching local minimum in cases of a dead end situation by changing the discrete course. The paper suggests extending the implemented algorithm from 3-D surface dealing only with two power factors, to n-D one with many unknown values.

THE OPTIMIZATION OF PLACING ELECTRONIC COMPONENTS BY A MODIFIED ALTERNATING-VARIABLE DESCENT METHOD

The problem of optimal placement of elements of electrical and electronic circuits is considered. The minimum weighted connection length is selected as the criterion. A computational method is proposed that is a modification of the coordinate descent method and one of the variants of the General approach based on pair permutations. The scheme is defined by the connection matrix. We consider a fixed set of element positions and a distance matrix based on an orthogonal metric. This problem is a variant of the General mathematical model, called the quadratic assignment problem. Geometric restriction of the problem – no more than one element can be placed in one cell. It is stated that approaches based on paired and similar permutations are economical, and the method of the penalty function leads to” ditching ” and is ineffective. A modified coordinate descent method is described, which is a variant of the pair permutation method, in which pairs are selected based on the coordinate descent method. In the proposed version of the coordinate descent method, two coordinates are changed simultaneously at one stage of calculations (and not one, as in the usual optimization method). one of the coordinates is used for the usual trial step, and the other is used for correction, returning to the acceptable area. Next, the value of the target function is calculated at the found point and compared with the previously reached value. If the value has improved, the found point becomes the new starting point. Otherwise, a step is made on a different coordinate with simultaneous correction of the vector of item position numbers (return to the allowed area). The experience of using the modified method in solving the problem of placing EVA elements has shown its significant advantages in comparison with other known methods, for example, the genetic algorithm, as well as the method of penalty functions. An example of calculations using the proposed method is considered. The connection matrix was set analytically. First, the initial approximation was searched by the Monte Carlo method (10,000 iterations), after which the local optimum was calculated using a modified method of coordinate descent in the permutation space without repetitions (a limit of 100 iterations was set). The initial value of the coordinate step is equal to the size of the permutation, then at each iteration it was reduced by 1 to the minimum possible value of 1. The advantage of this method is that there is no penalty function. The search is performed automatically in the permutation space without repetitions. Computational experiments have shown high computational qualities of the proposed method.

An algorithm combining coordinate descent and split Bregman iterative for fractional image denoising model

The split Bregman algorithm and the coordinate descent method are efficient tools for solving optimization problems, which have been proven to be effective for the total variation model. We propose an algorithm for fractional total variation model in this paper, and employ the coordinate descent method to decompose the fractional-order minimization problem into scalar sub-problems, then solve the sub-problem by using split Bregman algorithm. Numerical results are presented in the end to demonstrate the superiority of the proposed algorithm.

On Gradient Descent and Co-ordinate Descent methods and its variants.

This research is focused on Unconstrained Optimization problems. Among a number of methods that can be used to solve Unconstrained Optimization problems we have worked on Gradient and Coordinate Descent methods. Step size plays an important role for optimization. Here we have performed numerical experiment with Gradient and Coordinate Descent method for several step size choices. Comparison between different variants of Gradient and Coordinate Descent methods and their efficiency are demonstrated by implementing in loss functions minimization problem.