Design optimization

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]:

Variables: Describe the design alternatives
Objective: Elected functional combination of variables (to be maximized or minimized)
Constraints: Combination of Variables expressed as equalities or inequalities that must be satisfied for any acceptable design alternative
Feasibility: Values for set of variables that satisfies all constraints and minimizes/maximizes Objective.

Design optimization problem

The formal mathematical (standard form) statement of the design optimization problem is ^[2]

${\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h_{i}(x)=0,\quad i=1,\dots ,m_{1}\\&&&g_{j}(x)\leq 0,\quad j=1,\dots ,m_{2}\\&\operatorname {and} &&x\in X\subseteq R^{n}\end{aligned}}$

where

$x$ is a vector of n real-valued design variables $x_{1},x_{2},...,x_{n}$
$f(x)$ is the objective function
$h_{i}(x)$ are $m_{1}$ equality constraints
$g_{j}(x)$ are $m_{2}$ inequality constraints
$X$ is a set constraint that includes additional restrictions on $x$ 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

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

to rewrite the above statement in the compact expression

${\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h(x)=0,\quad g(x)\leq 0,\quad x\in X\subseteq R^{n}\\\end{aligned}}$

We call $h,g$ the set or system of (functional) constraints and $X$ the set constraint.

Application

Design optimization applies the methods of mathematical optimization to design problem formulations and it is sometimes used interchangeably with the term engineering optimization. When the objective function f is a vector rather than a scalar, the problem becomes a multi-objective optimization one. If the design optimization problem has more than one mathematical solutions the methods of global optimization are used to identified the global optimum.

Optimization Checklist

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.

Practical design optimization problems are typically solved numerically and many optimization software exist in academic and commercial forms^[3]. 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, wing-shape optimization, topology optimization, architectural design optimization, power optimization.

References

^ Papalambros, Panos Y.; Wilde, Douglass J. (2017-01-31). Principles of Optimal Design: Modeling and Computation. Cambridge University Press. ISBN 9781316867457.
^ Boyd, Stephen; Boyd, Stephen P.; California), Stephen (Stanford University Boyd; Vandenberghe, Lieven; Angeles), Lieven (University of California Vandenberghe, Los (2004-03-08). Convex Optimization (PDF). Cambridge University Press. ISBN 9780521833783.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Messac, Achille (2015-03-19). Optimization in Practice with MATLAB®: For Engineering Students and Professionals. Cambridge University Press. ISBN 9781316381373.

Design optimization