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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
# selection of a set of variables to describe the design alternatives;
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The formal mathematical statement of the design optimization problem is
<math>\begin{align}
&{\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{align}</math>
where
* <math>x</math> is a vector of ''n'' real-valued design variables <math>x_1, x_2, ..., x_n</math>
* <math>f(x)</math> is the '''objective function'''
* <math>h_i(x)</math> are <math>m_1</math>'''equality constraints'''
* <math>g_j(x)</math> are <math>m_2</math> '''inequality constraints'''
* <math>X</math> is a set constraint that includes additional restrictions on <math>x</math> besides those implied by the equality and inequality constraints.
We can introduce the vector-valued functions
▲where '''x''' is a vector of ''n'' real-valued design variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>, f''('''x''') is the objective function, ''h<sub>i</sub>''('''x''') are ''m''<sub>1</sub> equality constraints, ''g<sub>i</sub>''('''x''') are ''m''<sub>2</sub> inequality constraints, and 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.
<math>\begin{align}
&&&{h = (h_1,h_2,\dots,h_{m1})}\\
\operatorname{and}\\
&&&{g = (g_1, g_2,\dots, g_{m2})}
\end{align}
to rewrite the above statement in the compact expression
<math>\begin{align}
We call '''h''', '''g''' the ''set'' or ''system of'' (''functional'') ''constraints'' and X the ''set constraint''. ▼
&{\operatorname{minimize}}& & f(x) \\
&\operatorname{subject\;to}
& &h(x) = 0,\quad g(x) \leq 0,\quad x \in X \subseteq R^n\\
\end{align}</math>
▲We call
== 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. Practical design optimization problems are typically solved numerically and many [[optimization software]] exist in academic and commercial forms. 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 (EDA)|power optimization]].
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