A decision process over variables and their number: Two-phase optimality conditions

The paper considers the following unconstrained optimization problem: minimize f_n(x_n, n) = g_n(x_n) + h(n), where g_n(x_n) = α?_{i = 0}^n?1 ∫ _xi^{x_i = 1} [ ∫ _x^{x_i + 1}b (t) d(t)]d x, over both the variables x₁, x₂, …, x_n?1, and their number n, when x₀, x_n ≡ X are known boundary conditions, and h (n) is increasing, unbounded and weakly convex. Examples that demonstrate the practical usefulness of this decision process are given, and a two-phase approach for solving it, namely, minimize first over the arguments for a fixed n, and then over n, is suggested. Simple conditions, for a general g_n(x_n), under which phase II can be carried out by a finite search are stated, and then the question: “Under what circumstances does the search for the optimal n reduce to a simple condition?” is posed. Kuhn-Tucker type two-phase optimality conditions that answer this question are established, and moreover, specifications on the dominating single variable function b (x) for which the conditions hold are set forth.