Efficiency for multiobjective multidisciplinary optimization problems with quasiseparable subproblems

Multidisciplinary optimization (MDO) has proved to be a useful tool for engineering design problems. Multiobjective optimization has been introduced to strengthen MDO techniques and deal with non-comparable and conflicting design objectives. A large majority of papers on multiobjective MDO have been applied in nature. This paper develops theory of multiobjective MDO and examines relationships between efficient solutions of a quasi-separable multiobjective multidisciplinary optimization problem and efficient solutions of its separable counterpart. Equivalence of the original and separable problems in the context of the Kuhn-Tucker constraint qualification and efficiency conditions are proved. Two decomposition approaches are proposed and offer a possibility of finding efficient solutions of the original problem by only finding efficient solutions of the subproblems. The presented results are related to algorithms published in the engineering literature on multiobjective MDO.     

A TWO-LEVEL DECOMPOSITION METHOD FOR DESIGN OPTIMIZATION

In this paper, a two-level decomposition method for design optimization is proposed which is an extension of the model coordination methods. The method couples the global monotonicity analysis of the first-level subproblem(s) with an optimization method (single-level method) or the second-level problem. Three classes of problems are considered where in the first-level they have: (1) one subproblem with one local variable, (2) several subproblems with one local variable, and (3) several subproblems with several local variables. Some test results have been presented which shows the improved performance of the proposed approach over a conventional single-level optimization method.     

Decomposition of multidisciplinary optimization problems: formulations and application to a simplified wing design

Several formulations for solving multidisciplinary design optimization (MDO) problems are presented and applied to a test case. Two bi-level hierarchical decomposition approaches are compared with two classical single-level approaches without decomposition of the optimization problem. A methodology to decompose MDO problems and a new formulation based on this decomposition are proposed. The problem considered here for validation of the different formulations involves the shape and structural optimization of a conceptual wing model. The efficiency of the design strategies are compared on the basis of optimization results.     

Co-evolutionary algorithm: An efficient approach for bilevel programming problems

The bilevel programming problem involves two optimization problems, which is hierarchical, strongly NP-hard and very challenging for most existing optimization approaches. An efficient universal co-evolutionary algorithm is developed in this article to deal with various bilevel programming problems. In the proposed algorithm, evolutionary algorithms are used to explore the leader's and the follower's decision-making spaces interactively. Unlike other existing approaches, in the suggested procedure the follower's problem is solved in two phases. First, an evolutionary algorithm is run for a few generations to obtain an approximation of lower level solutions. In the second phase, from all approximate solutions obtained above, only a small number of good points are selected and evolved again by a newly designed multi-criteria evolutionary algorithm. The technique refines some candidate solutions and can efficiently reduce the computational cost of obtaining feasible solutions. Proof-of-principle experiments demonstrate the efficiency of the proposed approach.     

USE OF COUNTERPROPAGATION NEURAL NETWORKS TO ENHANCE THE CONCURRENT SUBSPACE OPTIMIZATION STRATEGY

The paper explores the use of artificial neural networks in a concurrent optimization strategy that derives from a decomposition based approach to design of large-scale engineering systems. These problems are characterized by complex couplings that render parametric design methods inappropriate as solution tools. Decomposition methods reduce the large dimensionality problem into a sequence of smaller, more tractable optimization problems, each with a smaller set of design variables and constraints. The decomposed subproblems are rarely decoupled completely, and design changes in one subproblem have a profound influence on changes in another subproblem. Essential components of decomposition based design methods are strategies to identify a topology for problem decomposition, and to develop coordination strategies which account for couplings among the decomposed problems. The paper examines the effectiveness of artificial neural networks as a tool to both account for the coupling, and to develop methods to coordinate the solution in the different subproblems to a converged optimal design     

Multidisciplinary design optimization based on independent subspaces

Optimization has been successfully applied to systems with a single discipline. Since many disciplines are involved in a coupled fashion in modern engineering, multidisciplinary design optimization (MDO) technology has been developed. MDO algorithms are designed to solve the coupled aspects generated from the interdisciplinary relationship. In a general MDO algorithm, a large design problem is decomposed into smaller ones which can be easily solved. Although various methods have been proposed for MDO, research is still in the early stage. This study proposes a new MDO method which is named MDO based on independent subspaces (MDOIS). Many real engineering problems consist of physically separate components and they can be independently designed. The inter‐relationship occurs through coupled physics. MDOIS is developed for such problems. In MDOIS, a large system is decomposed into small subsystems. The coupled aspects are solved via system analysis which solves the coupled physics. The algorithm is mathematically validated by showing that the solution satisfies the Karush–Kuhn–Tucker condition. Copyright © 2005 John Wiley & Sons, Ltd.     

Riemannian optimization and multidisciplinary design optimization

Riemannian Optimization (RO) generalizes standard optimization methods from Euclidean spaces to Riemannian manifolds. Multidisciplinary Design Optimization (MDO) problems exist on Riemannian manifolds, and with the differential geometry framework which we have previously developed, we can now apply RO techniques to MDO. Here, we provide background theory and a literature review for RO and give the necessary formulae to implement the Steepest Descent Method (SDM), Newton’s Method (NM), and the Conjugate Gradient Method (CGM), in Riemannian form, on MDO problems. We then compare the performance of the Riemannian and Euclidean SDM, NM, and CGM algorithms on several test problems (including a satellite design problem from the MDO literature); we use a calculated step size, line search, and geodesic search in our comparisons. With the framework’s induced metric, the RO algorithms are generally not as effective as their Euclidean counterparts, and line search is consistently better than geodesic search. In our post-experimental analysis, we also show how the optimization trajectories for the Riemannian SDM and CGM relate to design coupling and thereby provide some explanation for the observed optimization behaviour. This work is only a first step in applying RO to MDO, however, and the use of quasi-Newton methods and different metrics should be explored in future research.     

A move coordination method for alternative loads in structural optimization

A decomposition technique for alternative loading conditions in the integrated optimal structural design is developed. The method, called the move coordination, consists of partitioning the large structural optimization problem into a set of smaller coupled subproblems. In each subproblem only one loading condition is considered and the subproblems are solved in a parallel cyclic way. The coupling among the subproblems is accomplished through the introduction of coordinating constraints between each subproblem. These constraints ensure that the final design is the same and feasible for all subproblems. The method developed is illustrated by two examples of member sizing of truss structures using the integrated optimal design formulation and geometric programming. The method presents the advantage of reducing the size of the optimization problem as well as the computer processing time. The method is also suitable for implementation on computers using parallel processing.     

On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem

We study the ‘classical’ topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms. Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non‐linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form. For the special case when only a single linear constraint on volume is present, we note that application of the ever‐popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor η proposed by Bendsøe, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor η=0.5 is identical to the use of SAO with reciprocal intervening variables. Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two‐point exponential approximation we have used, using gradient information at the previously visited point. Copyright © 2007 John Wiley & Sons, Ltd.     

Decoupled approach to multidisciplinary design optimization under uncertainty

We propose solution methods for multidisciplinary design optimization (MDO) under uncertainty. This is a class of stochastic optimization problems that engineers are often faced with in a realistic design process of complex systems. Our approach integrates solution methods for reliability-based design optimization (RBDO) with solution methods for deterministic MDO problems. The integration is enabled by the use of a deterministic equivalent formulation and the first order Taylor’s approximation in these RBDO methods. We discuss three specific combinations: the RBDO methods with the multidisciplinary feasibility method, the all-at-once method, and the individual disciplinary feasibility method. Numerical examples are provided to demonstrate the procedure. Anukal Chiralaksanakul is currently a full-time lecturer in the Graduate School of Business Administration at National Institute of Development Administration (NIDA), Bangkok, Thailand.     

Non‐convex dual forms based on exponential intervening variables,with application to weight minimization

We study the weight minimization problem in a dual setting. We propose new dual formulations for non‐linear multipoint approximations with diagonal approximate Hessian matrices, which derive from separable series expansions in terms of exponential intervening variables. These, generally, nonconvex approximations are formulated in terms of intervening variables with negative exponents, and are therefore applicable to the solution of the weight minimization problem in a sequential approximate optimization (SAO) framework. Problems in structural optimization are traditionally solved using SAO algorithms, like the method of moving asymptotes, which require the approximate subproblems to be strictly convex. Hence, during solution, the nonconvex problems are approximated using convex functions, and this process may in general be inefficient. We argue, based on Falk's definition of the dual, that it is possible to base the dual formulation on nonconvex approximations. To this end we reintroduce a nonconvex approach to the weight minimization problem originally due to Fleury, and we explore certain convex and nonconvex forms for subproblems derived from the exponential approximations by the application of various methods of mixed variables. We show in each case that the dual is well defined for the form concerned, which may consequently be of use to the future code developers. Copyright © 2009 John Wiley & Sons, Ltd.     

A bilevel game theoretic approach to optimum design of flywheels

Multiobjective optimization problems arise frequently in mechanical design. One approach to solving these types of problems is to use a game theoretic formulation. This article illustrates the application of a bilevel, leader–follower model for solving an optimum design problem. In particular, the optimization problem is modelled as a Stackelberg game. The partitioning of variables between the leader and follower problem is discussed and a variable partitioning metric is introduced to compare various variable partitions. A computational procedure based on variable updating using sensitivity information is developed for exchanging information between the follower and leader problems. The proposed approach is illustrated through the design of a flywheel. The two objective functions used for the design problem include maximizing the kinetic energy stored in the flywheel while simultaneously minimizing the manufacturing cost.     

Benchmarking multidisciplinary design optimization algorithms

A comparison of algorithms for multidisciplinary design optimization (MDO) is performed with the aid of a new software framework. This framework, pyMDO, was developed in Python and is shown to be an excellent platform for comparing the performance of the various MDO methods. pyMDO eliminates the need for reformulation when solving a given problem using different MDO methods: once a problem has been described, it can automatically be cast into any method. In addition, the modular design of pyMDO allows rapid development and benchmarking of new methods. Results generated from this study provide a strong foundation for identifying the performance trends of various methods with several types of problems.     

Convex multilevel decomposition algorithms for non-monotone problems

A convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for a model structure with non-monotone interface or boundary conditions. First the non-monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non-linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non-convex min-min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,¹ is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non-monotone stick-slip frictional effects illustrate the theory.     

On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic search tree decomposition for balancing assembly lines by parallel search

We consider the simple assembly line balancing problem of type 1 (SALB-1). Although several branch-and-bound algorithms have been developed for optimally solving the SALB-1, until now to the best of our knowledge no parallel approach has been presented for this classical problem. In this paper we propose a parallel algorithm for solving the SALB-1. The parallel approach is based on a competitive serial algorithm. The parallel approach benefits from the substitution of a powerful but memory-intensive dominance concept through solution characteristics. Using the master-slave principle the master dynamically generates subproblems which are, upon request, assigned to the slaves for evaluation. The subproblems depend on previously assigned and on previously evaluated subproblems. The characteristics allow each slave to decide on potential dominances without requiring information from another slave or the master. The algorithm has been coded in GNU C using the Parallel Virtual Machine to allow process communication. Experimental studies have been performed on a simulated cluster of personal computers. Benchmark sets from the literature served as a testbed. The computational results are extremely promising. It seems that the SALB-1 is an ideal application of parallel search.     

Product decomposition strategy for optimization of supply chain planning

Optimization of large-scale supply chain planning models requires the application of decomposition strategies to reduce the computational expense. Two major options are to use either spatial or temporal Lagrangean decomposition. In this paper, to further reduce the computational expense a novel decomposition scheme by products is presented. The decomposition is based on a reformulation of knapsack constraints in the problem. The new approach allows for simultaneous decomposition by products and by time periods, enabling the generation of a large number of subproblems, that can be solved by using parallel computing. The case study shows that the proposed product decomposition exhibits similar performance as the temporal decomposition, and that selecting different orders of products and aggregating the linking constraints can improve the efficiency of the algorithm.     

A study on evolutionary multi-objective optimization for flow geometry design

In this paper, we investigate three recently proposed multi-objective optimization algorithms with respect to their application to a design-optimization task in fluid dynamics. The usual approach to render optimization problems is to accumulate multiple objectives into one objective by a linear combination and optimize the resulting single-objective problem. This has severe drawbacks such that full information about design alternatives will not become visible. The multi-objective optimization algorithms NSGA-II, SPEA2 and Femo are successfully applied to a demanding shape optimizing problem in fluid dynamics. The algorithm performance will be compared on the basis of the results obtained.     

An object‐oriented programming approach to the Lagrangian FEM analysis of large inelastic deformations and metal‐forming processes

The general deformation problem with material and geometric non‐linearities is typically divided into a number of subproblems including the kinematic, the constitutive, and the contact/friction subproblems. These problems are introduced for algorithmic purposes; however, each of them represents distinct physical aspects of the deformation process. For each of these subproblems, several well‐established mathematical and numerical models based on the finite element method have been proposed for their solution. Recent developments in software engineering and in the field of object‐oriented C++ programming have made it possible to model physical processes and mechanisms more expressively than ever before. In particular, the various subproblems and computational models in a large inelastic deformation analysis can be implemented using appropriate hierarchies of classes that accurately represent their underlying physical, mathematical and/or geometric structures. This paper addresses such issues and demonstrates that an approach to deformation processing using classes, inheritance and virtual functions allows a very fast and robust implementation and testing of various physical processes and computational algorithms. Here, specific ideas are provided for the development of an object‐oriented C++ programming approach to the FEM analysis of large inelastic deformations. It is shown that the maintainability, generality, expandability, and code re‐usability of such FEM codes are highly improved. Finally, the efficiency and accuracy of an object‐oriented programming approach to the analysis of large inelastic deformations are investigated using a number of benchmark metal‐forming examples. Copyright © 1999 John Wiley & Sons, Ltd.     

Combining stochastic programming and optimal control to decompose multistage stochastic optimization problems

The paper suggests a possible cooperation between stochastic programming and optimal control for the solution of multistage stochastic optimization problems. We propose a decomposition approach for a class of multistage stochastic programming problems in arborescent form (i.e. formulated with implicit non-anticipativity constraints on a scenario tree). The objective function of the problem can be either linear or nonlinear, while we require that the constraints are linear and involve only variables from two adjacent periods (current and lag 1). The approach is built on the following steps. First, reformulate the stochastic programming problem into an optimal control one. Second, apply a discrete version of Pontryagin maximum principle to obtain optimality conditions. Third, discuss and rearrange these conditions to obtain a decomposition that acts both at a time stage level and at a nodal level. To obtain the solution of the original problem we aggregate the solutions of subproblems through an enhanced mean valued fixed point iterative scheme.