This paper proposes a new bi-level multidisciplinary design optimization (MDO) architecture (or MDO formulation) with variants and proves their convergence properties. MDO architectures are reformulations of an original MDO problem into one or combined optimization problems. Distributed architectures define several optimization problems, and can be exploited in industrial contexts to map the separation of responsibilities, models, and computational resources in the design of sub-systems. Also, they allow to use only some disciplinary blocks of the overall MDO problem Jacobian, and do not require the complete coupled adjoint, which is rarely available in an industrial context. These advantages come at the price of weaker convergence properties compared to monolithic architectures. The Institut de Recherche Technologique (IRT) Saint Exupéry’s bi-level (BL) architecture, which falls within this category, is derived from the Multidisciplinary Feasible (MDF) architecture and is highly reliant on a block decomposition of the lower optimization problem, in which blocks are optimized in parallel. In order to enhance its convergence capability, a novel bi-level architecture is proposed (BL-BCD-MDF), which solves the lower optimization problem using the well-known Block Coordinate Descent (BCD) algorithm. The primary findings of this study are twofold. First, under the hypothesis that the lower optimization problem always has a unique solution, the BL-BCD-MDF and MDF optimization problems admit the same set of solutions. Next, the BL-BCD-MDF architecture is proved to be locally convergent by considering two additional stronger hypotheses: the Unique Block Minimizer (UBM) and the First Order Separability (FOS) properties. Variants are proposed to enhance flexibility in assumptions and the decomposition. To the authors’ knowledge, this is the first convergence proof of a distributed MDF bi-level MDO architecture in the MDO literature. The architectures are benchmarked on standard MDO problems (Sellar and Sobieski’s Supersonic Business Jet). Improved continuity and robustness of the system functions compared to the IRT bi-level architecture are demonstrated, as well as a superior scalability with respect to disciplinary design variables than MDF when the full Jacobian is not available.
We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the cone. We show that this is a weaker notion when compared to other similar ones and we explore several links with the existing literature. We subsequently demonstrate that this concept furnishes the properties required to obtain various generalizations of important results and techniques, including conic cancellation rules and the Rådström embedding procedure.
In this paper, we study a class of interval variational inequalities involving generalized set-valued vector fields (GIVIs) on Hadamard manifolds. The concepts of LU-pseudomonotone and gH-hemicontinuous operators are proposed. By employing a KKM-type lemma adapted to Hadamard manifolds, we prove existence theorems for GIVIs under appropriate conditions. In addition, we define the gH-subdifferential for convex interval-valued functions and investigate its fundamental properties in the locally Lipschitz case. We further establish the equivalence between GIVIs and nonsmooth convex interval optimization problems. Our results extend classical variational inequality theory from Euclidean spaces to Hadamard manifolds and, even in the Euclidean setting, yield new existence and equivalence results for interval variational inequalities and nonsmooth interval optimization problems.
