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The regularization problem of degenerated nonlinear system of equations/inequalities in Banach spaces is considered. The proposed approach is based on the explicit parametrization of input data and on the utilization of the multi-valued mapping techniques. Two regularization methods are suggested: the extended minimization and the regularized partial penalty function one. According to the first method an ill-posed problem should be replaced by a search of the minimal norm element in the join of solution sets of the family of problems that are equivalent to the initial one with respect to input data accuracy. The second one is based on separation from the initial system some regular part. Suggested approach is illustrated on some examples (analytical and numerical), including a singular system of ordinary differential and finite equations.

The inverse problem of the market demand's theory is constructing a collective utility function via a trade statistics consisting of a finite set of pairs ``prices-quantities''. The main computational problem here is the solution of the Afriat's inequalities system, which determines the values of the utility function and the Lagrange multiplier on the trade statistics data, which are ``Afriat's numbers''. This inverse problem is ill-posed one because of multiplicity of inequalities system's solutions and also because of their possible inconsistency and instability. A regularization method for this problem is proposed, based on the relaxation of the Afriat's system yielding local Hausdorf continuity of its solution set, and on the use of characteristics of analytical index numbers determined via Afriat's numbers. These characteristics formalized by choice criteria are: optimism, pessimism, objectivity. The results of constructing analytical index numbers for real trade statistics of Ulyanovsk region are presented.

The article develops methods for constructing economic (analytical) indexes in the framework of the holistic theory of market demand, built in recent years. By this, the economic indexes presented in the world literature within the framework of the theory of individual demand and, accordingly, related to households, acquire practical value.The introduction provides a brief overview of the main problems of modern indexology and the implementation of an economic approach dating back to the classical work of 1924 by the Soviet statistician A.A. Konüs. The properties of the most well-known «formula» indexes of Laspeyres, Paasche, and Fischer with respect to the fulfillment of the Fisher test criteria are described. These indexes play an important role in the methods proposed by the authors for constructing analytical indexes, which are determined through the function of consumer expenditures. The latter is determined by a utility function that rationalizes trade statistics. The rationalizing utility function is constructed ambiguously, and the corresponding task should be specified. Methods for its solution are proposed, developed within a non-parametric demand analysis of Afriat-Varian. The core of this analysis is the system of linear Afriat’s inequalities that determine the values of the utility function and marginal utility corresponding to statistical demand. This system can be inconsistent and unstable with respect to variations of non-exact demand statistics. In the case of compatibility, inequalities have many solutions, and the choice of different solutions of inequalities gives different values of analytical indexes. The authors suggest three types of tasks for the stable solution of Afriat’s inequalities, which define indexes with characteristics of optimism (low price indexes and high quantity indexes), pessimism (vice versa) and objectivity.Therefore, the problem of increasing the objectivity of consumer demand indexes receives a theoretically justified toolbox methods for calculating analytical market demand indexes that take into account, in contrast to formula indices, consumer preferences.

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