# Design optimization

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**Design optimization** is an engineering design methodology using a mathematical formulation of a design problem to support selection of the optimal design among many alternatives. Design optimization involves the following stages: [1] [2]

1. [Variables](/source/Variable_(mathematics)): Describe the design alternatives

1. Objective: Elected functional combination of variables (to be maximized or minimized)

1. Constraints: Combination of Variables expressed as equalities or inequalities that must be satisfied for any acceptable design alternative

1. Feasibility: Values for set of variables that satisfies all constraints and minimizes/maximizes Objective.

## Design optimization problem

Main article: [Optimization problem](/source/Optimization_problem)

The formal mathematical ([standard form](/source/Canonical_form)) statement of the design optimization problem is [3]

minimize f ( x ) s u b j e c t t o h i ( x ) = 0 , i = 1 , … , m 1 g j ( x ) ≤ 0 , j = 1 , … , m 2 and x ∈ X ⊆ R n {\displaystyle {\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h_{i}(x)=0,\quad i=1,\dots ,m_{1}\\&&&g_{j}(x)\leq 0,\quad j=1,\dots ,m_{2}\\&\operatorname {and} &&x\in X\subseteq R^{n}\end{aligned}}}

where

- x {\displaystyle x} is a vector of *n* real-valued design variables x 1 , x 2 , . . . , x n {\displaystyle x_{1},x_{2},...,x_{n}}

- f ( x ) {\displaystyle f(x)} is the **objective function**

- h i ( x ) {\displaystyle h_{i}(x)} are m 1 {\displaystyle m_{1}} **equality constraints**

- g j ( x ) {\displaystyle g_{j}(x)} are m 2 {\displaystyle m_{2}} **inequality constraints**

- X {\displaystyle X} is a set constraint that includes additional restrictions on x {\displaystyle x} besides those implied by the equality and inequality constraints.

The problem formulation stated above is a convention called the *negative null form*, since all constraint function are expressed as equalities and negative inequalities with zero on the right-hand side. This convention is used so that numerical algorithms developed to solve design optimization problems can assume a standard expression of the mathematical problem.

We can introduce the vector-valued functions

h = ( h 1 , h 2 , … , h m 1 ) and g = ( g 1 , g 2 , … , g m 2 ) {\displaystyle {\begin{aligned}&&&{h=(h_{1},h_{2},\dots ,h_{m1})}\\\operatorname {and} \\&&&{g=(g_{1},g_{2},\dots ,g_{m2})}\end{aligned}}}

to rewrite the above statement in the compact expression

minimize f ( x ) s u b j e c t t o h ( x ) = 0 , g ( x ) ≤ 0 , x ∈ X ⊆ R n {\displaystyle {\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h(x)=0,\quad g(x)\leq 0,\quad x\in X\subseteq R^{n}\\\end{aligned}}}

We call h , g {\displaystyle h,g} the *set* or *system of* (*functional*) *constraints* and X {\displaystyle X} the *set constraint*.

## Application

Design optimization applies the methods of [mathematical optimization](/source/Mathematical_optimization) to design problem formulations and it is sometimes used interchangeably with the term [engineering optimization](/source/Engineering_optimization). When the objective function *f* is a [vector](/source/Vector-valued_function) rather than a [scalar](/source/Scalar_field), the problem becomes a [multi-objective optimization](/source/Multi-objective_optimization) one. If the design optimization problem has more than one mathematical solutions the methods of [global optimization](/source/Global_optimization) are used to identified the global optimum.

**Optimization Checklist** [2]

- Problem Identification

- Initial Problem Statement

- Analysis Models

- Optimal Design Model

- Model Transformation

- Local Iterative Techniques

- Global Verification

- Final Review

A detailed and rigorous description of the stages and practical applications with examples can be found in the book [Principles of Optimal Design](http://www.cambridge.org/gh/academic/subjects/engineering/control-systems-and-optimization/principles-optimal-design-modeling-and-computation-3rd-edition?format=HB&isbn=9781107132672#C3r6r6aLRe2bUoef.97).

Practical design optimization problems are typically solved numerically and many [optimization software](https://en.wikipedia.org/wiki/Category:Mathematical_optimization_software) exist in academic and commercial forms.[4] There are several domain-specific applications of design optimization posing their own specific challenges in formulating and solving the resulting problems; these include, [shape optimization](/source/Shape_optimization), [wing-shape optimization](/source/Wing-shape_optimization), [topology optimization](/source/Topology_optimization), [architectural design optimization](/source/Architectural_design_optimization), [power optimization](/source/Power_optimization_(EDA)). Several books, articles and journal publications are listed below for reference.

One modern application of design optimization is structural design optimization (SDO) is in building and construction sector. SDO emphasizes automating and optimizing structural designs and dimensions to satisfy a variety of performance objectives. These advancements aim to optimize the configuration and dimensions of structures to optimize augmenting strength, minimize material usage, reduce costs, enhance energy efficiency, improve sustainability, and optimize several other performance criteria. Concurrently, structural design automation endeavors to streamline the design process, mitigate human errors, and enhance productivity through computer-based tools and optimization algorithms. Prominent practices and technologies in this domain include the parametric design, generative design, building information modelling (BIM) technology, machine learning (ML), and artificial intelligence (AI), as well as integrating finite element analysis (FEA) with simulation tools.[5][6]

## Journals

- [Journal of Engineering for Industry](http://manufacturingscience.asmedigitalcollection.asme.org/issue.aspx?journalid=125&issueid=27495)

- [Journal of Mechanical Design](http://mechanicaldesign.asmedigitalcollection.asme.org/journal.aspx)

- [Journal of Mechanisms, Transmissions, and Automation in Design](http://mechanicaldesign.asmedigitalcollection.asme.org/issue.aspx?journalid=126&issueid=28068)

- [Design Science](https://www.cambridge.org/core/journals/design-science)

- [Engineering Optimization](https://www.tandfonline.com/toc/geno20/current)

- [Journal of Engineering Design](https://www.tandfonline.com/toc/cjen20/current)

- [Computer-Aided Design](https://www.journals.elsevier.com/computer-aided-design/)

- [Journal of Optimization Theory and Applications](https://www.springer.com/mathematics/journal/10957)

- [Structural and Multidisciplinary Optimization](/source/Structural_and_Multidisciplinary_Optimization)

- [Journal of Product Innovation Management](/source/Journal_of_Product_Innovation_Management)

- [International Journal of Research in Marketing](/source/International_Journal_of_Research_in_Marketing)

## See also

- [Design Decisions Wiki (DDWiki)](https://wiki.ece.cmu.edu/ddl/index.php/Main_Page) : Established by the Design Decisions Laboratory at Carnegie Mellon University in 2006 as a central resource for sharing information and tools to analyze and support decision-making

## References

1. **[^](#cite_ref-edo2021_1-0)** Martins, Joaquim R. R. A.; Ning, Andrew (2021-10-01). [*Engineering Design Optimization*](http://flowlab.groups.et.byu.net/mdobook.pdf) (PDF). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-1108833417](https://en.wikipedia.org/wiki/Special:BookSources/978-1108833417).

1. ^ [***a***](#cite_ref-pyp2017_2-0) [***b***](#cite_ref-pyp2017_2-1) Papalambros, Panos Y.; Wilde, Douglass J. (2017-01-31). [*Principles of Optimal Design: Modeling and Computation*](http://principlesofoptimaldesign.org/). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [9781316867457](https://en.wikipedia.org/wiki/Special:BookSources/9781316867457).

1. **[^](#cite_ref-3)** Boyd, Stephen; Boyd, Stephen P.; California), Stephen (Stanford University Boyd; Vandenberghe, Lieven; Angeles), Lieven (University of California Vandenberghe, Los (2004-03-08). [*Convex Optimization*](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf) (PDF). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [9780521833783](https://en.wikipedia.org/wiki/Special:BookSources/9780521833783).{{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list))

1. **[^](#cite_ref-4)** [Messac, Achille](/source/Achille_Messac) (2015-03-19). [*Optimization in Practice with MATLAB®: For Engineering Students and Professionals*](https://books.google.com/books?id=wy5hBwAAQBAJ&q=design+optimization+with+matlab+achille+messac&pg=PR17). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [9781316381373](https://en.wikipedia.org/wiki/Special:BookSources/9781316381373).

1. **[^](#cite_ref-5)** Towards BIM-Based Sustainable Structural Design Optimization: A Systematic Review and Industry Perspective. Sustainability 2023, 15, 15117. [https://doi.org/10.3390/su152015117](https://doi.org/10.3390/su152015117)

1. **[^](#cite_ref-6)** Manmatharasan, Piragash; Bitsuamlak, Girma; Grolinger, Katarina (April 2025). ["AI-driven design optimization for sustainable buildings: A systematic review"](https://linkinghub.elsevier.com/retrieve/pii/S0378778825001707). *Energy and Buildings*. **332** 115440. [doi](/source/Doi_(identifier)):[10.1016/j.enbuild.2025.115440](https://doi.org/10.1016%2Fj.enbuild.2025.115440).

## Further reading

- Rutherford., Aris, ([2016], ©1961). *The optimal design of chemical reactors : a study in dynamic programming*. Saint Louis: Academic Press/Elsevier Science. [ISBN](/source/ISBN_(identifier)) [9781483221434](https://en.wikipedia.org/wiki/Special:BookSources/9781483221434). [OCLC](/source/OCLC) 952932441

- Jerome., Bracken, ([1968]). *Selected applications of nonlinear programming*. McCormick, Garth P.,. New York,: Wiley. [ISBN](/source/ISBN_(identifier)) [0471094404](https://en.wikipedia.org/wiki/Special:BookSources/0471094404). [OCLC](/source/OCLC) 174465

- L., Fox, Richard ([1971]). *Optimization methods for engineering design*. Reading, Mass.,: Addison-Wesley Pub. Co. [ISBN](/source/ISBN_(identifier)) [0201020785](https://en.wikipedia.org/wiki/Special:BookSources/0201020785). [OCLC](/source/OCLC) 150744

- Johnson, Ray C. Mechanical Design Synthesis With Optimization Applications. New York: Van Nostrand Reinhold Co, 1971.

- 1905-, Zener, Clarence, ([1971]). *Engineering design by geometric programming*. New York,: Wiley-Interscience. [ISBN](/source/ISBN_(identifier)) [0471982008](https://en.wikipedia.org/wiki/Special:BookSources/0471982008). [OCLC](/source/OCLC) 197022

- H., Mickle, Marlin ([1972]). *Optimization in systems engineering*. Sze, T. W., 1921-2017,. Scranton,: Intext Educational Publishers. [ISBN](/source/ISBN_(identifier)) [0700224076](https://en.wikipedia.org/wiki/Special:BookSources/0700224076). [OCLC](/source/OCLC) 340906.

- Optimization and design; [papers]. Avriel, M.,, Rijckaert, M. J.,, Wilde, Douglass J.,, NATO Science Committee., Katholieke Universiteit te Leuven (1970- ). Englewood Cliffs, N.J.,: Prentice-Hall. [1973]. [ISBN](/source/ISBN_(identifier)) [0136380158](https://en.wikipedia.org/wiki/Special:BookSources/0136380158). OCLC 618414.

- J., Wilde, Douglass (1978). *Globally optimal design*. New York: Wiley. [ISBN](/source/ISBN_(identifier)) [0471038989](https://en.wikipedia.org/wiki/Special:BookSources/0471038989). [OCLC](/source/OCLC) 3707693.

- J., Haug, Edward (1979). *Applied optimal design : mechanical and structural systems*. Arora, Jasbir S.,. New York: Wiley. [ISBN](/source/ISBN_(identifier)) [047104170X](https://en.wikipedia.org/wiki/Special:BookSources/047104170X). [OCLC](/source/OCLC) 4775674.

- Uri., Kirsch, (1981). *Optimum structural design : concepts, methods, and applications*. New York: McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [0070348448](https://en.wikipedia.org/wiki/Special:BookSources/0070348448). [OCLC](/source/OCLC) 6735289.

- Uri., Kirsch, (1993). *Structural optimization : fundamentals and applications*. Berlin: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [3540559191](https://en.wikipedia.org/wiki/Special:BookSources/3540559191). [OCLC](/source/OCLC) 27676129.

- *Structural optimization : recent developments and applications*. Lev, Ovadia E., American Society of Civil Engineers. Structural Division., American Society of Civil Engineers. Structural Division. Committee on Electronic Computation. Committee on Optimization. New York, N.Y.: ASCE. 1981. [ISBN](/source/ISBN_(identifier)) [0872622819](https://en.wikipedia.org/wiki/Special:BookSources/0872622819). [OCLC](/source/OCLC) 8182361.

- *Foundations of structural optimization : a unified approach*. Morris, A. J. Chichester [West Sussex]: Wiley. 1982. [ISBN](/source/ISBN_(identifier)) [0471102008](https://en.wikipedia.org/wiki/Special:BookSources/0471102008). [OCLC](/source/OCLC) 8031383.

- N., Siddall, James (1982). *Optimal engineering design : principles and applications*. New York: M. Dekker. [ISBN](/source/ISBN_(identifier)) [0824716337](https://en.wikipedia.org/wiki/Special:BookSources/0824716337). [OCLC](/source/OCLC) 8389250.

- 1944-, Ravindran, A., (2006). *Engineering optimization : methods and applications*. Reklaitis, G. V., 1942-, Ragsdell, K. M. (2nd ed.). Hoboken, N.J.: John Wiley & Sons. [ISBN](/source/ISBN_(identifier)) [0471558141](https://en.wikipedia.org/wiki/Special:BookSources/0471558141). [OCLC](/source/OCLC) 61463772.

- N.,, Vanderplaats, Garret (1984). *Numerical optimization techniques for engineering design : with applications*. New York: McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [0070669643](https://en.wikipedia.org/wiki/Special:BookSources/0070669643). [OCLC](/source/OCLC) 9785595.

- T., Haftka, Raphael (1990). *Elements of Structural Optimization*. Gürdal, Zafer., Kamat, Manohar P. (Second rev. edition ed.). Dordrecht: Springer Netherlands. [ISBN](/source/ISBN_(identifier)) [9789401578622](https://en.wikipedia.org/wiki/Special:BookSources/9789401578622). [OCLC](/source/OCLC) 851381183.

- S., Arora, Jasbir (2011). *Introduction to optimum design* (3rd ed.). Boston, MA: Academic Press. [ISBN](/source/ISBN_(identifier)) [9780123813756](https://en.wikipedia.org/wiki/Special:BookSources/9780123813756). [OCLC](/source/OCLC) 760173076.

- S.,, Janna, William. *Design of fluid thermal systems* (SI edition; fourth edition ed.). Stamford, Connecticut. [ISBN](/source/ISBN_(identifier)) [9781285859651](https://en.wikipedia.org/wiki/Special:BookSources/9781285859651). [OCLC](/source/OCLC) 881509017.

- *Structural optimization : status and promise*. Kamat, Manohar P. Washington, DC: American Institute of Aeronautics and Astronautics. 1993. [ISBN](/source/ISBN_(identifier)) [156347056X](https://en.wikipedia.org/wiki/Special:BookSources/156347056X). [OCLC](/source/OCLC) 27918651.

- *Mathematical programming for industrial engineers*. Avriel, M., Golany, B. New York: Marcel Dekker. 1996. [ISBN](/source/ISBN_(identifier)) [0824796209](https://en.wikipedia.org/wiki/Special:BookSources/0824796209). [OCLC](/source/OCLC) 34474279.

- Hans., Eschenauer, (1997). *Applied structural mechanics : fundamentals of elasticity, load-bearing structures, structural optimization : including exercises*. Olhoff, Niels., Schnell, W. Berlin: Springer. [ISBN](/source/ISBN_(identifier)) [3540612327](https://en.wikipedia.org/wiki/Special:BookSources/3540612327). [OCLC](/source/OCLC) 35184040.

- 1956-, Belegundu, Ashok D., (2011). *Optimization concepts and applications in engineering*. Chandrupatla, Tirupathi R., 1944- (2nd ed.). New York: Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [9781139037808](https://en.wikipedia.org/wiki/Special:BookSources/9781139037808). [OCLC](/source/OCLC) 746750296.

- Okechi., Onwubiko, Chinyere (2000). *Introduction to engineering design optimization*. Upper Saddle River, NJ: Prentice-Hall. [ISBN](/source/ISBN_(identifier)) [0201476738](https://en.wikipedia.org/wiki/Special:BookSources/0201476738). [OCLC](/source/OCLC) 41368373.

- *Optimization in action : proceedings of the Conference on Optimization in Action held at the University of Bristol in January 1975*. Dixon, L. C. W. (Laurence Charles Ward), 1935-, Institute of Mathematics and Its Applications. London: Academic Press. 1976. [ISBN](/source/ISBN_(identifier)) [0122185501](https://en.wikipedia.org/wiki/Special:BookSources/0122185501). [OCLC](/source/OCLC) 2715969.

- P., Williams, H. (2013). *Model building in mathematical programming* (5th ed.). Chichester, West Sussex: Wiley. [ISBN](/source/ISBN_(identifier)) [9781118506189](https://en.wikipedia.org/wiki/Special:BookSources/9781118506189). [OCLC](/source/OCLC) 810039791.

- *Integrated design of multiscale, multifunctional materials and products*. McDowell, David L., 1956-. Oxford: Butterworth-Heinemann. 2010. [ISBN](/source/ISBN_(identifier)) [9781856176620](https://en.wikipedia.org/wiki/Special:BookSources/9781856176620). [OCLC](/source/OCLC) 610001448.

- M.,, Dede, Ercan. *Multiphysics simulation : electromechanical system applications and optimization*. Lee, Jaewook,, Nomura, Tsuyoshi,. London. [ISBN](/source/ISBN_(identifier)) [9781447156406](https://en.wikipedia.org/wiki/Special:BookSources/9781447156406). [OCLC](/source/OCLC) 881071474.

- 1962-, Liu, G. P. (Guo Ping), (2001). *Multiobjective optimisation and control*. Yang, Jian-Bo, 1961-, Whidborne, J. F. (James Ferris), 1960-. Baldock, Hertfordshire: Research Studies Press. [ISBN](/source/ISBN_(identifier)) [0585491941](https://en.wikipedia.org/wiki/Special:BookSources/0585491941). [OCLC](/source/OCLC) 54380075.

### Structural Topology Optimization

- Bendsøe, Martin Philip; Kikuchi, Noboru (1988-11-01). "Generating optimal topologies in structural design using a homogenization method". *Computer Methods in Applied Mechanics and Engineering*. **71** (2): 197–224. [doi](/source/Doi_(identifier)):[10.1016/0045-7825(88)90086-2](https://doi.org/10.1016%2F0045-7825%2888%2990086-2). [hdl](/source/Hdl_(identifier)):[2027.42/27079](https://hdl.handle.net/2027.42%2F27079). [ISSN](/source/ISSN_(identifier)) [0045-7825](https://search.worldcat.org/issn/0045-7825).

- Bendsøe, Martin P. (1995). *Optimization of structural topology, shape, and material*. Springer. [ISBN](/source/ISBN_(identifier)) [3540590579](https://en.wikipedia.org/wiki/Special:BookSources/3540590579).

- Behrooz, Hassani (1999). *Homogenization and Structural Topology Optimization : Theory, Practice and Software*. Springer. [ISBN](/source/ISBN_(identifier)) [9781447108917](https://en.wikipedia.org/wiki/Special:BookSources/9781447108917). [OCLC](/source/OCLC_(identifier)) [853262659](https://search.worldcat.org/oclc/853262659).

- Bendsøe, Martin P.; Sigmund, O. (2013). [*Topology optimization : theory, methods, and applications*](https://books.google.com/books?id=ZCjsCAAAQBAJ&pg=PR1) (2nd ed.). Springer. [ISBN](/source/ISBN_(identifier)) [9783662050866](https://en.wikipedia.org/wiki/Special:BookSources/9783662050866). [OCLC](/source/OCLC_(identifier)) [50448149](https://search.worldcat.org/oclc/50448149).

- Rozvany, G.I.N.; Lewiński, T., eds. (2014). [*Topology optimization in structural and continuum mechanics*](https://books.google.com/books?id=B8HHBAAAQBAJ&pg=PP1). Springer. [ISBN](/source/ISBN_(identifier)) [9783709116432](https://en.wikipedia.org/wiki/Special:BookSources/9783709116432). [OCLC](/source/OCLC_(identifier)) [859524179](https://search.worldcat.org/oclc/859524179).

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