Preprint A227/2003
On the sequential quadratically constrained quadratic programming methods

Mikhail Solodov

**Keywords: **
Quadratically constrained quadratic programming | nonsmooth penalty function | Maratos effect.

An iteration of the
sequential quadratically constrained quadratic programming
method (SQCQP) consists of minimizing a quadratic approximation
of the objective function subject to quadratic approximation
of the constraints, followed by a linesearch in the
obtained direction. Methods of this class are receiving
attention due to the development of
efficient interior point techniques for solving
subproblems with this structure, via formulating them as
second-order cone programs.
Recently, Fukushima, Luo and Tseng \cite{flt:03} proposed
a SQCQP method for convex minimization with twice continuously
differentiable data. Their method possesses
global and locally quadratic convergence, and it is free
of the Maratos effect.
The feasibility of subproblems in their method is enforced by
switching between the linear and quadratic approximations of
the constraints. This strategy requires computing a strictly feasible
point, as well as choosing some further parameters.
We propose a SQCQP method where feasibility
of subproblems is ensured by introducing a slack variable
and, hence, is automatic. In addition, we do not assume convexity
of the objective function or twice differentiability of the
problem data. While our method has all the
desirable convergence properties, it is easier to
implement.
Among other things, it does not require computing
a strictly feasible point, which is a nontrivial task.
In addition, its global convergence requires weaker assumptions.