The Use of Certain Equations of Mathematical Physics in Optimization of a Function on a Set. I

The results of application of potential theory to optimization are used to extend the use of (Helmholtz) diffusion and diffraction equations for optimization of their solutions (x, ) with respect to both x, and . If the aim function is modified such that the optimal point does not change, then the function (x, ) is convex in (x, for small . The possibility of using heat conductivity equation with a simple boundary layer for global optimization is investigated. A method is designed for making the solution U(x,t) of such equations to have a positive-definite matrix of second mixed derivatives with respect to x for any x in the optimization domain and any small t < 0 (the point is remote from the extremum) or a negative-definite matrix in x (the point is close to the extremum). For the functions (x, ) and U(x,t) having these properties, the gradient and the Newton–Kantorovich methods are used in the first and second stages of optimization, respectively.