$\newcommand{\RR}{{\mathbb R}}
\newcommand{\RRm}{\RR^m}
\newcommand{\RRmE}{\RR^{m_E}}
\newcommand{\RRmI}{\RR^{m_I}}
\newcommand{\RRmS}{\RR^{m_S}}
\newcommand{\RRn}{\RR^n}$
SQPlab can solve a smooth nonlinear optimization problem possibly
having bound contraints on the $n$ variables $x$, nonlinear inequality
constraints, and nonlinear equality constraints. The problem is
supposed to be written in the form:
$$
\begin{cases}
\min\,f(x)\\
l_B\leq x \leq u_B\\
l_I\leq c_I(x)\leq u_I\\
c_E(x)= 0\\
c_S(x)=0.
\end{cases}
$$
The constraint functions are $c_I: \RRn\to \RRmI$, $c_E: \RRn\to \RRmE$,
and $c_S: \RRn\to \RRmS$, which means that there are $m_I$ ($\geq0$)
two-side nonlinear inequality constraints, $m_E$
($\geq0$) nonlinear equality constraints, and $m_S$
($\geq0$) nonlinear state constraints. Lower and upper
bounds can be infinite, but must verify $l_B< u_B$ and $l_I< u_I$. If
the lower and upper bounds of a particular inequality constraint are
both infinite, the constraint is ignored. The functions $f$, $c_I$,
$c_E$, and $c_S$ are supposed smooth (at least first differentiability
is required), but may be nonconvex.
The constraints $c_S(x)=0$ look like standard equality constraints, but
are considered by SQPlab in a different manner. They are supposed to
express state constraints in an optimal
control setting. We mean by this that, once they are present,
the variables $x$ are split in state variables $y$ and
control variables $u$; hence $x = (y,u)$. There are as
many state variables as state equations, i.e., $m_S$. In addition, it is
assumed that the Jacobian of the state constraints with respect to the
state variables is uniformly nonsingular. The state
constraints are then used to reduce the size of the optimization problem
to $n-m_S$. SQPlab offers in fact more flexibility than in this short
description; more is given in the documentation.