Modern algorithm of linear and quadratic programming in optimization and problems of deformation of structures of variable structure in contact conditions

Abstract

Various mathematical problems, in which the goal of finding the extremum of a functional is set, belong to the problems of mathematical programming and optimization problems. Practically directed problems of finding the optimal solution are extremely numerous in economics, management, technology, and others. They are related to increasing production efficiency, reducing resource costs, improving design solutions and technological processes, reducing mass, dimensions, etc. Among them, an important role is given to the methods of limiting the maximum stresses caused by external loads. Solving such problems begins with mathematical formalization. Constructive, economic or technological indicators are chosen as variation parameters. The search for the best solution is reduced to the selection of a set of parameters that provide a stationary value of the objective function. Extreme problems of practical orientation contain equality-inequality constraints in mathematical models. In improving the technical characteristics of machines, a significant role belongs to engineering and technical workers, who find optimal options at the design stage. At the same time, an essential element of the design process is the modeling of the determining processes in structures, taking into account the main influencing factors and behavior scenarios. Optimization is an important area of applied mathematics that provides effective tools for such modeling. Universal Algorithm is proposed in work [3] – a numerous scheme for solving quadratic programming (QP) problems for calculating the optimal point of a wide range of applied problems. At the same time, the linear programming (LP) problem is considered as a partial case of the (QP) problem. That is, in the universal algorithm for setting 2 optimization problems, they are formalized in a single and convenient form of symmetric matrix dependence, which makes it possible to build a single effective algorithm based on matrix algebra operations. In particular, it allows you to consider the practical tasks of calculating VAT in constructions of a variable structure consisting of separate parts connected by one-way connections. The main goal of this work is to analyze the behavior of the algorithm when increasing the number of constraints of the inequality type, to refine the computational scheme, and to formulate conclusions. Two model problems are considered as examples of the algorithm. This is a classic "transport" problem of LP and the behavior of a model of a bridge structure with one-way connections in cables under variations of wind loads. The number of ropes has been increased to 20, and the limits of one-way connections to 40.