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 ϕ(x, ω) with respect to both x, and ω. If the aim function is modified such that the optimal point does not change, then the function ϕ(x, ω) is convex in (x, ω for small ω. 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 ϕ(x, ω) 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.