Back to Continuous Optimization

** Constrained optimization problems** consider the problem of optimizing an objective function subject to constraints on the variables. In general terms,

\[ \begin{array}{lllll}

\mbox{minimize} & f(x) & & & \\

\mbox{subject to} & c_i(x) & = & 0 & \forall i \in \mathcal{E} \\

& c_i(x) & \leq & 0 & \forall i \in \mathcal{I}

\end{array}

\]

where \(f\) and the functions \(c_i(x) \,\) are all smooth, real-valued functions on a subset of \(R^n \,\) and \(\mathcal{E}\) and \(\mathcal{I}\) are index sets for equality and inequality constraints, respectively. The

*feasible set*is the set of points \(x\) that satisfy the constraints.

Constrained optimization covers a large number of subfields, including many important special cases for which specialized algorithms are available.

*Bound Constrained Optimization*: the only constraints are lower and upper bounds on the variables*Linear Programming*: the objective function \(f\) and all of the constraints \(c_i\) are linear functions*Quadratic Programming*: the objective function \(f\) is quadratic and the constraints \(c_i\) are linear functions*Semidefinite Programming*: the objective function \(f\) is linear and the feasible set is the intersection of the cone of positive semidefinite matrices with an affine space*Nonlinear Programming*: at least some of the constraints \(c_i\) are nonlinear functions*Semi-infinite Programming*: there is an infinite number of variables or an infinite number of constraints (but not both)