15h30 - 15h55
Numerical Observation of Maratos Effect with Multiple Precision Arithmetic
When we use the sequential quadratic programming (SQP) method to solve nonlinear programming problems (NLPs), the unit stepsize is required for superlinear convergence. However, we may suffer from the Maratos effect, which is a phenomenon in which the unit stepsize is rejected, even if an iteration point is sufficiently close to the optimal solution and the assumptions for fast convergence are satisfied. In this research, we observe this phenomenon numerically and seek some clues to overcome it efficiently. Especially, we utilize multiple precision arithmetic to observe it closely.
15h55 - 16h20
Adaptive matrix algebras in unconstrained optimization
In this communication we will introduce some recent techniques which involve structured matrix spaces in the reduction of time and space complexity of BFGS-type minimization algorithms [1,2]. Some general results for the global convergence of algorithms for unconstrained optimization based on a BFGS-type Hessian approximation scheme are introduced and it is shown how the constructibility of convergent algorithms suitable for large scale problems can be tackled using projections onto low complexity matrix algebras.
 C.Di Fiore, S.Fanelli, F.Lepore, P.Zellini, Matrix algebras in Quasi-Newton
methods for unconstrained minimization, Numerische Mathematik, 94, (2003).
 S.Cipolla, C.Di Fiore, F.Tudisco, P.Zellini, Adaptive matrix algebras in un-
constrained minimization, Linear Algebra and its Application, 471, (2015).
16h20 - 16h45
Efficient modifications on the parameters of Yabe and Takano’s conjugate gradient algorithm
Our main concern in this work is to make some modifications on the Yabe and Takano’s conjugate gradient (CG) algorithm [Comput. Optim. Appl. 28 (2004), pp. 203-225] in order for receiving some appealing results in theory and practice. In fact, we propose an efficient adaptive updating formula for the parameters of Yabe and Takano CG algorithm leading to the well promising theoretical and numerical results. Global convergence property of the new proposed CG algorithm in Dai-Liao family is established under standard assumptions on uniformly convex and general functions. Numerical performance of the new algorithm on some test problems from CUTEr collection shows the efficiency and effectiveness of the proposed method in practice.
16h45 - 17h10
A superquadratic variant of Newton's method
We present the first Q-superquadratically convergent version of Newton's method for solving operator equations in Banach spaces that requires only one operator value and one inverse of the Frechet derivative per iteration. The R-order of convergence is at least 2.4142. A semi-local analysis provides sufficient conditions for existence of a solution and convergence. The local analysis assumes that a solution exists and shows that the method converges from any starting point belonging to an explicitly defined neighbourhood of the solution called the ball of attraction.