  In this paper, a class of optimization problems with nonlinear inequality
constraints is discussed. Based on the ideas of sequential quadratic
programming algorithm and the method of strongly sub-feasible directions, a new
superlinearly convergent algorithm is proposed. The initial iteration point can
be chosen arbitrarily for the algorithm. At each iteration, the new algorithm
solves one quadratic programming subproblem which is always feasible, and one
or two systems of linear equations with a common coefficient matrix. Moreover,
the coefficient matrix is uniformly nonsingular. After finite iterations, the
iteration points can always enter into the feasible set of the problem, and the
search direction is obtained by solving one quadratic programming subproblem
and only one system of linear equations. The new algorithm possesses global and
superlinear convergence under some suitable assumptions without the strict
complementarity. Finally, some preliminary numerical experiments are reported
to show that the algorithm is promising.
