The constructive techniques for the study of equations with deviating arguments described below are based on a specific approximation of original problems within the class of computable functions and operators [2].
This optimization problem is within the class of MDS problems, with several alternative tools available to find a local minimum [9, 10].
The procedure for the constructive study of the problem (8) consists of the following steps: approximation of the problem (8) within the class of computable functions and operators, study of the principal boundary value problem (9), analysis of its solvability.
approximation of the problem (8) within the class of computable functions and operators, study of the principal boundary value problem (9), analysis of its solvability.
If the input spectrum is discretized (either by considering a Riemmann approximation, or by restricting it to the span of a finite dimensional linear space), this problem falls within the class of semi-infinite convex programs.
Theorem 2.3 Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied: (H ) 4 2 < σ − < σ + < 2 p + p + − 1, p + ⩾ 2. Then the nonnegative solution of problem (1.1) is unique within the class of all nonnegative weak solutions.
Theorem 2.2 Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied: (H ) 3 σ + ( p + − 1 ) p + ⩽ σ − < σ + < 2, 1 < p − < p + ⩽ 1 + 2. Then the nonnegative bounded solution of problem (1.1) is unique within the class of all nonnegative bounded weak solutions.
This problem is well-known to fall within the class of chance-constrained optimization problems, which are nonconvex in most cases.
First, are the solutions that the computer provides close enough to the actual (but unavailable) solutions to be useful?…this is a purely mathematical question and falls within the class of problems we have just mentioned.
This class of optimization problems is called the class of pseudoinvex programs.
