The formal mathematical (
standard form) statement of the design optimization problem is \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} 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{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 \begin{align} &{\operatorname{minimize}}& & f(x) \\ &\operatorname{subject\;to} & &h(x) = 0,\quad g(x) \leq 0,\quad x \in X \subseteq R^n\\ \end{align} We call h, g the
set or
system of (
functional)
constraints and X the
set constraint. == Application ==