Editing Multidisciplinary design optimization (section)

==History==
Traditionally engineering has normally been performed by teams, each with expertise in a specific discipline, such as aerodynamics or structures. Each team would use its members' experience and judgement to develop a workable design, usually sequentially. For example, the aerodynamics experts would outline the shape of the body, and the structural experts would be expected to fit their design within the shape specified. The goals of the teams were generally performance-related, such as maximum speed, minimum [[drag (physics)|drag]], or minimum structural weight.

Between 1970 and 1990, two major developments in the aircraft industry changed the approach of aircraft design engineers to their design problems. The first was [[computer-aided design]], which allowed designers to quickly modify and analyse their designs. The second was changes in the procurement policy of most [[airline]]s and military organizations, particularly the [[military of the United States]], from a performance-centred approach to one that emphasized [[Product lifecycle management|lifecycle]] cost issues. This led to an increased concentration on economic factors and the attributes known as the "[[ilities]]" including [[manufacturability]], [[reliability (engineering)|reliability]], [[maintainability]], etc.

Since 1990, the techniques have expanded to other industries. Globalization has resulted in more distributed, decentralized design teams. The high-performance [[personal computer]] has largely replaced the centralized [[supercomputer]] and the [[Internet]] and [[local area network]]s have facilitated sharing of design information. Disciplinary design software in many disciplines (such as [[OptiStruct]] or [[NASTRAN]], a [[finite element analysis]] program for structural design) have become very mature. In addition, many optimization algorithms, in particular the population-based algorithms, have advanced significantly.

=== Origins in structural optimization ===
Whereas optimization methods are nearly as old as [[calculus]], dating back to [[Isaac Newton]], [[Leonhard Euler]], [[Daniel Bernoulli]], and [[Joseph Louis Lagrange]], who used them to solve problems such as the shape of the [[catenary]] curve, numerical optimization reached prominence in the digital age. Its systematic application to structural design dates to its advocacy by Schmit in 1960.<ref>{{cite book |author-last=Vanderplaats |author-first=G.N. |chapter=Numerical Optimization Techniques |year=1987 |title=Computer Aided Optimal Design: Structural and Mechanical Systems |editor-last=Mota Soares |editor-first=C.A. |series=NATO ASI Series (Series F: Computer and Systems Sciences) |volume=27 |pages=197–239 |publisher=Springer |location=Berlin |doi=10.1007/978-3-642-83051-8_5 |isbn=978-3-642-83053-2 |quote=The first formal statement of nonlinear programming (numerical optimization) applied to structural design was offered by Schmit in 1960.}}</ref><ref>{{cite journal |last=Schmit |first=L.A. |title=Structural Design by Systematic Synthesis |journal=Proceedings, 2nd Conference on Electronic Computations |publisher=ASCE |location=New York |pages=105–122 |date=1960}}</ref> The success of structural optimization in the 1970s motivated the emergence of multidisciplinary design optimization (MDO) in the 1980s. Jaroslaw Sobieski championed decomposition methods specifically designed for MDO applications.<ref name="martins2013">{{cite journal|url=http://arc.aiaa.org/doi/full/10.2514/1.J051895|title=Multidisciplinary design optimization: A Survey of architectures|last1=Martins|first1=Joaquim R. R. A.|last2=Lambe|first2= Andrew B. |journal=AIAA Journal |date=2013|volume=51 |issue=9 |pages=2049–2075 |doi=10.2514/1.J051895|bibcode=2013AIAAJ..51.2049M |language=en|citeseerx=10.1.1.669.7076}}</ref> The following synopsis focuses on optimization methods for MDO. First, the popular gradient-based methods used by the early structural optimization and MDO community are reviewed. Then those methods developed in the last dozen years are summarized.

=== Gradient-based methods ===
There were two schools of structural optimization practitioners using [[gradient]]-based methods during the 1960s and 1970s: optimality criteria and [[Mathematical optimization|mathematical programming]]. The optimality criteria school derived recursive formulas based on the [[Karush–Kuhn–Tucker conditions|Karush–Kuhn–Tucker (KKT) necessary conditions]] for an optimal design. The KKT conditions were applied to classes of structural problems such as minimum weight design with constraints on stresses, displacements, buckling, or frequencies [Rozvany, Berke, Venkayya, Khot, et al.] to derive resizing expressions particular to each class. The mathematical programming school employed classical gradient-based methods to structural optimization problems. The method of usable feasible directions, Rosen's gradient projection (generalized reduce gradient) method, sequential unconstrained minimization techniques, sequential linear programming and eventually sequential quadratic programming methods were common choices. Schittkowski et al. reviewed the methods current by the early 1990s.

The gradient methods unique to the MDO community derive from the combination of optimality criteria with math programming, first recognized in the seminal work of Fleury and Schmit who constructed a framework of approximation concepts for structural optimization. They recognized that optimality criteria were so successful for stress and displacement constraints, because that approach amounted to solving the dual problem for [[Lagrange multipliers]] using linear [[Taylor series]] approximations in the reciprocal design space. In combination with other techniques to improve efficiency, such as constraint deletion, regionalization, and design variable linking, they succeeded in uniting the work of both schools. This approximation concepts based approach forms the basis of the optimization modules in modern structural design software.

Approximations for structural optimization were initiated by the reciprocal approximation Schmit and Miura for stress and displacement response functions. Other intermediate variables were employed for plates. Combining linear and reciprocal variables, Starnes and Haftka developed a conservative approximation to improve buckling approximations. Fadel chose an appropriate intermediate design variable for each function based on a gradient matching condition for the previous point. Vanderplaats initiated a second generation of high quality approximations when he developed the force approximation as an intermediate response approximation to improve the approximation of stress constraints. Canfield developed a [[Rayleigh quotient]] approximation to improve the accuracy of eigenvalue approximations. Barthelemy and Haftka published a comprehensive review of approximations in 1993.

=== Non-gradient-based methods ===
In recent years, non-gradient-based evolutionary methods including [[genetic algorithm]]s, [[simulated annealing]], and [[Ant colony optimization algorithms|ant colony algorithms]] came into existence. At present, many researchers are striving to arrive at a consensus regarding the best modes and methods for complex problems like impact damage, dynamic failure, and [[Real-time analyzer|real-time analyses]]. For this purpose, researchers often employ multiobjective and multicriteria design methods.

=== Recent MDO methods ===
MDO practitioners have investigated [[Optimization (mathematics)|optimization]] methods in several broad areas in the last dozen years. These include decomposition methods, [[approximation]] methods, [[evolutionary algorithm]]s, [[memetic algorithm]]s, [[response surface methodology]], reliability-based optimization, and [[multi-objective optimization]] approaches.

The exploration of decomposition methods has continued in the last dozen years with the development and comparison of a number of approaches, classified variously as hierarchic and non hierarchic, or collaborative and non collaborative. 
Approximation methods spanned a diverse set of approaches, including the development of approximations based on [[surrogate model]]s (often referred to as metamodels), variable fidelity models, and trust region management strategies. The development of multipoint approximations blurred the distinction with response surface methods. Some of the most popular methods include [[Kriging]] and the [[moving least squares]] method.

[[Response surface methodology]], developed extensively by the statistical community, received much attention in the MDO community in the last dozen years. A driving force for their use has been the development of massively parallel systems for high performance computing, which are naturally suited to distributing the function evaluations from multiple disciplines that are required for the construction of response surfaces. Distributed processing is particularly suited to the design process of complex systems in which analysis of different disciplines may be accomplished naturally on different computing platforms and even by different teams.

Evolutionary methods led the way in the exploration of non-gradient methods for MDO applications. They also have benefited from the availability of massively parallel high performance computers, since they inherently require many more function evaluations than gradient-based methods. Their primary benefit lies in their ability to handle discrete design variables and the potential to find globally optimal solutions.

Reliability-based optimization (RBO) is a growing area of interest in MDO. Like response surface methods and evolutionary algorithms, RBO benefits from parallel computation, because the numeric integration to calculate the probability of failure requires many function evaluations. One of the first approaches employed approximation concepts to integrate the probability of failure. The classical first-order reliability method (FORM) and second-order reliability method (SORM) are still popular. Professor Ramana Grandhi used appropriate normalized variables about the most probable point of failure, found by a two-point adaptive nonlinear approximation to improve the accuracy and efficiency. [[Southwest Research Institute]] has figured prominently in the development of RBO, implementing state-of-the-art reliability methods in commercial software. RBO has reached sufficient maturity to appear in commercial structural analysis programs like Altair's [[OptiStruct|Optistruct]] and MSC's [[Nastran]].

Utility-based probability maximization was developed in response to some logical concerns (e.g., Blau's Dilemma) with reliability-based design optimization.<ref>{{cite journal |last1=Bordley |first1=Robert F. |last2=Pollock |first2=Steven M. |date=September 2009 |title=A Decision Analytic Approach to Reliability-Based Design Optimization |journal=Operations Research |volume=57 |number=5 |pages=1262–1270|doi=10.1287/opre.1080.0661 }}</ref> This approach focuses on maximizing the joint probability of both the objective function exceeding some value and of all the constraints being satisfied. When there is no objective function, utility-based probability maximization reduces to a probability-maximization problem. When there are no uncertainties in the constraints, it reduces to a constrained utility-maximization problem. (This second equivalence arises because the utility of a function can always be written as the probability of that function exceeding some random variable). Because it changes the constrained optimization problem associated with reliability-based optimization into an unconstrained optimization problem, it often leads to computationally more tractable problem formulations.

In the marketing field there is a huge literature about optimal design for multiattribute products and services, based on experimental analysis to estimate models of consumers' utility functions. These methods are known as [[Conjoint analysis|Conjoint Analysis]]. Respondents are presented with alternative products, measuring preferences about the alternatives using a variety of scales and the utility function is estimated with different methods (varying from regression and surface response methods to choice models). The best design is formulated after estimating the model. The experimental design is usually optimized to minimize the variance of the estimators. These methods are widely used in practice.