A numerical comparison of Hessian update techniques for an SQCQP method

This short note considers Hessian updates for a sequential quadratically constrained quadratic programming (SQCQP) method. The SQCQP method is an iterative method to solve an inequality constrained optimization problem. At each iteration, the SQCQP method solves a quadratically constrained quadratic programming subproblem whose objective function and constraints are approximations to the objective function and constraints of the original problem. We typically use Hessian update to construct this subproblem and Hessian update has an important role in establishing global convergence of the SQCQP method. In this short note, we introduce some different Hessian updates and apply them to the feasible SQCQP method by Solodov [12]. The presented update methods are the Hessian matrix, Levenberg Marquardt type modification, the modified BFGS method by Powell [10], the cautious BFGS proposed by Li and Fukushima [8] and the limited-memory BFGS proposed by Wei, Yu, Yuan and Lian [15]. We present the algorithms which integrates the feasible SQCQP method and these Hessian updates. Moreover, we examine the numerical behavior of these algorithms.