On Subgradient Methods with Polyak’s Step and Space Transformation

Introduction. Minimization of ravine convex functions, both smooth and non-smooth, arises in many problems of planning, control, stability analysis of dynamic systems, artificial intelligence, and machine learning. Therefore, the development of new and improvement of existing methods is an important task, taking into account the fact that more and more frequently the functions to be minimized depend on a large number of variables. Special attention should be paid to the methods using the operation of linear transformation of space, which allow to improve the properties of the objective function and significantly accelerate its minimization. Known methods of this type, in particular, ellipsoid methods and r-algorithms, require recalculation of the transformation matrix at each iteration. Therefore, the development of methods with a one-time transformation of space, which have a lower cost of iteration, is definitely an urgent task. The purpose of the article is to review modifications of subgradient method with Polyak’s step that use a scalar parameter m(1 and one-time space transformation operation. Supplement the justification of the convergence and convergence rate of the modifications described. Give recommendations regarding determination of parameter m and space transformation matrix B. Results. Subgradient method with Polyak’s step in transformed space and parameter m>1 is an effective method for minimizing smooth and non-smooth convex functions that have a ravine structure. Determination of an appropriate value of the parameter m(1 and space transformation matrix B allows to significantly accelerate this method and use it for minimization functions that depend on a large number of variables. Conclusions. The development of fast methods for minimization of non-smooth convex functions of many variables with a ravine structure makes it possible to effectively solve modern problems of artificial intelligence, in particular, the problems of machine learning, image recognition, big data analysis, etc

Using the Ellipsoid Method to Study Relationships in Medical Data

Deterioration of moral and psychological state on the background of a full–scale war is observed in many social groups. Timely detection of various types of pre–depressive states and appropriate therapy is a critically important task nowadays. In addition, an equally important task is to identify relationships between physical and psychological indicators of health. Establishing such regularities will allow detecting anxious states, avoiding direct profile testing of a patient. The article is devoted to construction of a mathematical apparatus for predicting psychological conclusions based on cardiological data. For this, a linear regression model and the ellipsoid method are used to determine its parameters with a criterion based on least moduli method (LMM), a feature selection procedure and a metric for assessing consistency of a data set. Material of the article is presented in 5 sections. Section 1 describes the ellipsoid method for finding parameters of linear regression with the least moduli method as a criterion in the power of p. Problem dimensions that can be successfully solved using the ellipsoid method on modern computers are indicated. The 2nd Section is devoted to Octave program emlmp, which implements the ellipsoid method, and the results of two computational experiments with its use. The obtained results demonstrate robustness of the obtained solutions when parameter p values are close to one. The 3rd Section describes mechanism of variable selection for the best prediction of psychological state of patients based on cardiological data. Variable selection was carried out using the Python Sequential Feature Selector procedure for predicting two psychological indicators – Beck's anxiety scale and psychologist's formalized conclusion. The 4th Section contains the results of a computational experiment using the emlmp program with LMM and least square method (LSM) criteria for predicting a psychologist’s formalized conclusion based on 84 selected patients and 22 parameters. Obtained solutions and forecasts for comparing criteria based on LMM and LSM are given. In the 5th Section, a metric for evaluation consistency of a data set is proposed, which allows to evaluate consistency for each parameter separately. A linear connection was found between 4 psychological parameters and the maximum accuracy of regression models with optimal number of parameters in specified models