Application of Collaborative Optimization to Optimal Control Problems

Abstract

F ROM the view point of practical computation, multidisciplinary design optimization (MDO) can be considered as methods for solving complex optimization problems. We think that MDO is, in some sense, a bridge between the conventional optimization algorithms and complex applications. There are two main strategies used in these MDO methods, which are approximation and decomposition. Although these two strategies are not mutually exclusive, they are applicable to problems with different properties, respectively. Some reported flight-vehicle configuration shape-optimizationdesign problems, integrated with complex analysis models (e.g., computational fluid dynamics or computational structural mechanics), have a small number of design variables and constraints [1,2]. Methods using approximation, such as surrogate-based methods, are applicable to these problems in which the original complex analysis models are replaced by corresponding approximate and relatively simple models, such as radial basis function and Kriging models. The optimization computation is then performed based on these approximate models. On the other hand, a flight-vehicle trajectory optimization-design problemwith constraints of differential equations, also called optimal control, can be viewed as an infinite-dimensional extension of a common nonlinear optimization problem [3], which is a practical solution that is to convert the infinite-dimensional problem into a finite-dimensional problem. Several conversion methods, for example, direct shooting, multiple direct shooting, collocation, and pseudospectral, have been developed [3,4]. In some cases, the conversion will result in a very high-dimensional nonlinear optimization problem with a large number of design variables and constraints [5]. Some optimizers for large-scale optimizations, such as SNOPT [6], have been presented. In this Note, an alternative solution using collaborative optimization (CO), an MDO method with decomposition strategy, is discussed. Compared with approximation, decomposition ismore applicable to this kind of large-scale problem in which the original large-scale problem is decomposed into several reduced subproblems. That is to say, the original complex computation task of one optimization is decomposed into several relatively small computation tasks of several optimizations. Although the computational difficulties in CO have not been solved ideally, the decomposition strategy is a natural and potential way for solving optimization problems with a large number of design variables and constraints. As far as we know, the solution to this kind of large-scale optimization problem converted from optimal control problems by using decomposedmethod has not been reported, which is the main motivation of the work in this Note. The organization of this Note is as follows: in Sec. II, we briefly review the collocation method for converting an optimal control problem into a nonlinear optimization problem, and then the discussions on the decomposition of the converted problem usingCO are presented in Sec. III. In Sec. IV, a numerical test case illustrates the discussions in Secs. II and III. The conclusions are stated in Sec. V.