This book is devoted to finite-dimensional problems of non-convex non-smooth optimization and numerical methods for their solution. The problem of nonconvexity is studied in the book on two main models of nonconvex dependencies: these are the so-called generalized differentiable functions and locally Lipschitz functions. Non-smooth functions naturally arise in various applications. In addition, they often appear in the theory of extremal problems itself due to the operations of taking the maximum and minimum, decomposition techniques, exact non-smooth penalties, and duality. The considered models of nonconvexity are quite general and cover the majority of practically important optimization problems; they clearly show all the difficulties of non-convex optimization. The method of studying the generalized differentiable functions is that for these functions a generalization of the concept of gradient is introduced, a calculus is constructed, and various properties of nonconvex problems are studied in terms of generalized gradients. As for numerical methods, it is possible to extend the theory and algorithms of subgradient descent of convex optimization to problems with generalized differentiable functions. Methods for solving Lipschitz problems are characterized by the fact that the original functions are approximated by smoothed ones and iterative minimization procedures are applied to them. With this approach, it is possible to approximate the gradients of smoothed functions by stochastic finite differences and thus to construct methods without calculating gradients. A similar approach can be justified in generalized differentiable and Lipschitz stochastic programming. In these cases, various generalizations of the classical stochastic approximation and stochastic quasi-gradient method are obtained for solving constrained nonconvex nonsmooth stochastic programming problems.

In the theory of recursive functions, a recursively enumerable (but not recursive) set K = {x |x ∈ Wx } is obtained by Cantor’s diagonal method. Based on Diophantine sets, K is expressed by some polynomial that has positive roots. On the contrary, the set K = {x |x ∉ Wx } is not recursively enumerable. None of the computable functions can enumerate all the elements of the set K . As a result of the productivity of the set K , a parameter exists for which the polynomial has no positive roots. However, it is impossible to prove their absence since this parameter does not belong to any recursively enumerable set. Keywords: diagonal method, Diophantine set, recursively enumerable sets.

The article considers extremum economic and mathematical dynamic optimization problems of distributing investments between the economy sectors of the country. Leontief’s “input–output” and Solow’s models were used to construct extremum problems where the presented gross domestic product is maximized under the limited investment volume conditions. The constructed mathematical programming problem is proved to belong to the class of smooth and convex extremum problems. By using the conditional gradient method, the optimal distribution of investments for the Ukrainian economy was calculated.

The authors analyze Diophantine sets and show that all recursively enumerable sets are Diophantine. Based on the classical results from the theory of recursive functions, a simple version of the theorem on the incompleteness of arithmetic is provided: there is a polynomial that has no positive integer solutions, and for which it is impossible to prove the absence of positive roots.

An analysis of the undecidability of Diophantine equations showed that problems of recognition of the properties of the NP class are decidable, i.e., a non-deterministic algorithm or exhaustive search at the problem input gives a positive or negative answer. For polynomial Diophantine equations, such a non-deterministic algorithm does not exist. A simple version of Gödel’s theorem on the incompleteness of arithmetic follows from the undecidability of Diophantine equations.

The authors consider the application of Bayesian recognition procedures on Markov chain models to inflammatory processes in gliomas. Parameters of protein structures of blood plasma in gliomas, metastases, and brain concussion obtained with the help of a laser spectrograph are analyzed. A comparative analysis of the results of recognition on the basis of protein structures in relation to the parameters of surface plasmon resonance and modified erythrocyte sedimentation rate in gliomas is carried out.

For every disease, there is a certain set of genes whose mutations increase the risk of illness development. DNA sequencing of sick and healthy individuals results in the determination of genes related to certain diseases. Efficient procedures are described in order to determine point mutations in gene sequences of the examined patients. The optimal Bayesian procedure is used to determine risk groups for certain diseases, including the ones that underlie COVID-19.

Introduction. In the group of risk at people with COVID-19 there are persons with the such chronic diseases: heart-vessel system; respiratory system; endocrine system; oncologic diseases; immune-deficit states; patients with kidney insufficiency. For every disease there is the concrete set of genes the mutations of which multiply the risk of development of illness. Determination of DNA of sick and healthy people resulted in determination of the genes, related to the diseases which arise up at COVID-19. At persons having by had COVID-19 with the certain disease, with the high stake of probability took place points mutations in certain genes. These people can be brought in a teaching sampling «sick», in a class «healthy» persons are brought in with the negative result of PCR. Purpose of the article. On the basis of teaching selections to develop the effective methods of determination of groups of risks of diseases which COVID-19 accompanies. Results. We consider that genes in a left table column are signs for Bayesian procedure. Work of procedure is executed on the basis of count of amount of mutations or their absence in the teaching selections of classes «sick» and «healthy». We correlate the explored person in that class «sick» and «healthy», for which result of procedure higher. Conclusions. Determination of DNA of sick and healthy people resulted in determination of the genes related to the concrete diseases, including with the diseases which arise up at COVID-19. It is shown that the presence of points mutations in the genes of DNA of man results in the certain disease. On the basis of Bayesian procedure of recognition it is possible effectively to determine the groups of risks of diseases which COVID-19 accompanies. Keywords: determination of DNA, the points mutations, Bayesian procedure of recognition.

The use of Bayesian recognition procedure for surface plasmon resonance with the addition of verapamil hydrochloride and ketamine to the blood in the analysis of neurosurgical tumor pathologies significantly improved recognition compared to pure blood samples. An analysis of the difference in such pathologies due to the method of surface plasmon resonance with the addition of ketamine made it possible to improve the results of recognition of pathologies as compared with the method of the modified erythrocyte sedimentation rate.

DNA symmetry is used to generate optimal genetic codes whose noise immunity with respect to polarity of amino acids in case of mutations in nucleotides is much greater than immunity of standard codes. Noise immunity of optimal symmetric and non-symmetric genetic codes is analyzed. Databases of genetic diseases are used to show that optimal symmetric code for which symmetry holds in every second case keeps polarity of amino acids at mutations in first and second nucleotides of codon as compared with standard code.

Application of Bayesian recognition procedures to erythrocyte sedimentation rate in brain gliomas has allowed detecting inflammatory processes in a human body. The analysis of results of the recognition methods based on tree network methods, Markov chains, and nearest neighbor algorithm has shown that Bayesian procedure was the most efficient.

Fundamental relations and symmetry rules of the genetic information organization in DNA were studied. DNA symmetry was used to construct an optimal symmetric code with respect to amino acid polarity, with noise immunity much higher than that of a standard genetic code. It is well known that various diseases are associated with pointwise mutations of nucleotides in genes. Bayesian procedures allow for use of the standard and symmetric codes for genetic diseases diagnosis. Markov model of higher orders with hidden states was used to build simple algorithms for gene fragment prediction.

A promising computer approach to recognition of hematologic diseases is substantiated. Due to highly efficient Bayesian procedures, computer search is used to find combinations of indicators that have the highest recognition quality. Such method allows conducting fast diagnostics without performing it completely.

The code symmetric with respect to polarity of amino acids for the case of mutations in nucleotides is constructed using DNA symmetry. Standard code is compared with randomly generated codes. The noise immunity of genetic code against amino acid polarity is analyzed. Databases of genetic diseases are used to show that symmetric code corrects violation of polarity in mutations in most cases.

The paper considers two methods for processing sets of logical regularities of classes (LRC) found by training samples analysis. The first approach is based on the of logical descriptions of classes. As a result of solving the problem of linear discrete optimization, the shortest logic description of each class is found. Each training object satisfies at least to one LRC of found irreducible subset of logical regularities. The second approach is based on the clustering of the set of LRC and selecting standards of derived clusters. The clustering problem is reduced to the clustering of representations of LRC set. Here each LRC is represented in the form of binary vector with different informative weight. A modification of the known method of variance criterion minimization for the case where the objects have different information weights is proposed. We present the results of illustrative experiments.

The authors consider the problem of reconstruction of hidden state sequences for mixture distributions with constituents described by the generalization of high-order Markov chains and hidden Markov models. A new algorithm to solve the problem using dynamic programming is proposed, as well as its modifications to eliminate recursion and reduce search. The results are applied to the problem of gene fragment recognition in plants.

The EM algorithm is considered for the problem of separation of distribution mixtures described by Markov chains, together with the related weighted likelihood maximization problem. Auxiliary algorithms are proposed to select the initial approximation and optimal number of mixture components, as well as a method to approximate the distribution mixture with given data using support vector machines. The results are applied to gene fragment classification.

The noise immunity of genetic codes with nucleotide mutations is analyzed. The universal code is compared with randomly generated codes. The noise immunity of genetic code against polarity, hydrophobicity, and helix propensity is analyzed. A genetic algorithm for the optimization of noise immunity of a code is described.

Algorithmic compositions in the form of expert mixtures with exclusive competence zones are considered in order to increase the quality of classification of gene fragments with the help of models based on Markov chains.