# Properties of the Quadratic Transformation of Dual Variables
###### Abstract
The organization of RSM methods is to find the descent direction that enables us to go beyond a particular neighborhood of the current extremum [40; 41; 42; 43], i.e., they find a descent direction that is gradient-consistent with all gradients in the neighborhood of the current extremum approximation. The narrower the CPD, the lower the convergence rate and the higher the computational cost of finding the RSM optimum [40; 41; 42; 43]. Thus, with the modulus method of transforming dual variables, the dual function is non-smooth, and its degree of degeneracy increases as it approaches the extremum. This negatively affects the convergence rate of the minimization method and leads to a decrease in the accuracy of the resulting solution. In this case, it becomes necessary to involve complex subgradient methods of accelerated convergence for solving the problem. Considering the noted shortcomings of the modulus transformation, it is important to search for a transformation of dual variables without gradient discontinuities. Our contribution is as follows. We propose a quadratic transformation of dual variables for the optimization problems with the number of constraints much less than the number of variables. The quadratic transformation method has proved successful in the problem of imposing constraints on the pollutant emissions of enterprises [33], which aroused interest in studying the properties of the function transformed in this way. In this work, we prove that in the case of a quadratic transformation, the gradient of the newly formed function is Lipschitz. In addition to subgradient methods, the noted property enables us to exploit methods of unconstrained minimization of smooth functions in solving the problem. As a result of a quadratic method compared to a modulus one, this is an increase in the convergence speed of the minimization method and the accuracy of the resulting solution when using well-studied and proven fast-converging methods for minimizing smooth functions. We conducted a computer experiment for practical analysis of the methods for transforming dual variables. Models of convex programming problems similar in properties to those used in the software package for environmental monitoring of atmospheric air (ERA-AIR) [33] were taken as tests. Based on the extremum conditions, convex programming problems with two-sided constraints were generated with a number of variables up to 10,000 and a number of linear constraints up to 1000. The number of active two-sided and linear constraints varied. To solve test problems, we used the multi-step subgradient method [43], subgradient methods with a change in the space metric [40; 41; 42], the BFGS (Broyden-Fletcher-Goldfarb-Shanno) quasi-Newtonian method [44; 45; 46; 47], and the conjugate gradient method [34]. In the case of a quadratic transformation, the number of iterations required to solve the problem was reduced and the accuracy of its solution was increased. Problems with a quadratic transformation of the dual function variables were also successfully solved by rapidly converging methods for minimizing smooth functions: the conjugate gradient method [34] and the quasi-Newtonian BFGS method [44; 45; 46; 47]. The noted transformations of dual variables were used in the program module for calculating the MPE, which is the part of the software package for environmental monitoring of atmospheric air, ERA-AIR. In related literature, the authors present different types of models for this applied problem: regression models [48; 49], the particle swarm optimization model [50], the fuzzy c-means model [50], and a nonlinear programming model [51; 52; 53]. The rest of the paper is organized as follows. In Section 2, we formulate the dual problem for a convex programming problem with a strongly convex objective function and a summary of the results of the duality theory necessary for further research.