## Optimization Theory Overview

Optimization techniques are used to find a set of design parameters, *x* = {*x*_{1},*x*_{2},...,*x*_{n}},
that can in some way be defined as optimal. In a simple case this
might be the minimization or maximization of some system characteristic
that is dependent on* x*. In a more advanced formulation
the objective function, *f*(*x*),
to be minimized or maximized, might be subject to constraints in the
form of equality constraints, *G*_{i}(*x*) = 0 (* i* = 1,...,*m*_{e});
inequality constraints, *G*_{i}(* x*) ≤ 0 (*i* = *m*_{e} + 1,...,*m*); and/or
parameter bounds, *x*_{l}, *x*_{u}.

A General Problem (GP) description is stated as

$$\underset{x}{\mathrm{min}}f(x),$$ | **(2-1)** |

$$\begin{array}{cc}{G}_{i}(x)=0& i=1,\mathrm{...},{m}_{e},\\ {G}_{i}(x)\le 0& i={m}_{e}+1,\mathrm{...},m,\end{array}$$

where *x* is the vector of length *n* design
parameters, *f*(*x*) is the objective
function, which returns a scalar value, and the vector function *G*(*x*)
returns a vector of length *m* containing the values
of the equality and inequality constraints evaluated at *x*.

An efficient and accurate solution to this problem depends not
only on the size of the problem in terms of the number of constraints
and design variables but also on characteristics of the objective
function and constraints. When both the objective function and the
constraints are linear functions of the design variable, the problem
is known as a Linear Programming (LP)
problem. Quadratic
Programming (QP) concerns the minimization or maximization of a quadratic
objective function that is linearly constrained. For both the LP and
QP problems, reliable solution procedures are readily available. More
difficult to solve is the Nonlinear Programming (NP) problem in which the objective
function and constraints can be nonlinear functions of the design
variables. A solution of the NP problem generally requires an iterative
procedure to establish a direction of search at each major iteration.
This is usually achieved by the solution of an LP, a QP, or an unconstrained
subproblem.