A study on concave optimization via canonical dual function

Abstract

a b s t r a c tIn this study we find a global minimizer of a concave function over a sphere. By introducing a differential equation, we obtain the invariant characteristics for a given optimization problem by constructing a canonical dual function. We present two theorems concerning the global optimality of an extrema of the optimization problem.

Solution to an Optimal Control Problem via Canonical Dual Method

“…In this section we present a differential flow to deal with the global optimization, which is used to find the optimal control expressed by the costate in the next section. Here we use the method in our another paper (see [9]).…”

“…when P(u) is a nonconvex quadratic function, the problem (9) can be solved completely by the canonical dual transformation [6][7][8]. In [9], the global concave optimization over a sphere is solved by use of a differential system with the canonical dual function. Because the Pontryagin principle is a necessary condition for a control to be optimal, it is not sufficient for obtaining an optimal control to solve only the optimization (9).…”

“…In [9], the global concave optimization over a sphere is solved by use of a differential system with the canonical dual function. Because the Pontryagin principle is a necessary condition for a control to be optimal, it is not sufficient for obtaining an optimal control to solve only the optimization (9). In this paper, combing the method given in [6,9] with the Pontryagin principle, we solve problem (1)-( 2) which has nonconvex integrand on the control variable in the cost functional and present the optimal control expressed by the costate via canonical dual variables.…”

On the paper "A study on concave optimization via canonical dual function"

“…In [2] one says: "The primary goal of this paper is to study the global minimizers for the following concave optimization problem (primal problem (P ) in short).…”