Gould, Nicholas I. M. and Robinson, Daniel P. (2008) A second derivative SQP method: local convergence. Technical Report. SIAM Journal on Optimization. (Submitted)

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Abstract
In [19], we gave global convergence results for a secondderivative SQP method for minimizing the exact ℓ1merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the socalled Cauchy step, which was itself computed from the socalled predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm.
Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positivedefinite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positivedefinite matrix Bk—a simple diagonal approximation and a more sophisticated limitedmemory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1penalty function over a sequence of increasing values of the penalty parameter.
Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the socalled Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set.
Item Type:  Technical Report (Technical Report) 

Subjects:  O  Z > Operations research, mathematical programming A  C > Calculus of variations and optimal control H  N > Numerical analysis 
Research Groups:  Numerical Analysis Group 
ID Code:  869 
Deposited By:  Lotti Ekert 
Deposited On:  17 Dec 2009 08:08 
Last Modified:  29 May 2015 18:33 
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