A Bfgs Method for the Problem of Building S-Shaped Curve

Abstract

We consider the application of the BFGS method (Broyden, Fletcher, Goldfarb, Shanno) and its projective version L-BFGS-B for minimization of nonlinear function, which corresponds to finding solutions to a system of five nonlinear equations. Among them three equations are integral and depend on unknown parameters of integrands and unknown upper bounds for definite integrals. This system represents the problem of building an S-shaped curve in natural parameterization, which passes through the two given points with given tangents inclination angles and provides a given tangent inclination angle at an intermediate point with a given abscissa. The tangent inclination angle at the point with known abscissa is used to control the inflection point of the S-shaped curve, by which a fragment of the external contour of the Frankl nozzle is modeled.The material of the article is presented in three sections. Section 1 describes a system of five nonlinear equations and analyses its properties related to the existence of many solutions. In section 2 the optimization problem with a non-smooth objective function and linear constraints is presented, the global minima of which correspond to solutions of a system of nonlinear equations. There also is described the application of the modification of r-algorithm to find global minima of the optimization problem.Section 3 describes the application of BFGS and L-BFGS-B methods to find local minima of smooth functions corresponding to finding solutions of a system of nonlinear equations. We formulate two optimization problems with smooth objective functions and two-sided constraints on variables and describe the results of computational experiments for finding solution of the system of nonlinear equations, presented in Section 1, by S-shaped curve. It is shown that BFGS and L-BFGS-B methods are effective if the starting point is selected in such a neighborhood of the minimum point, where the minimized function is approximated fairly accurately by a convex quadratic function.Manuscript received 15.06.2020