We are very glad to note that other researchers and users of optimization have recognized the powerful and unique features of Pshenichnyi's algorithm for engineering applications. Beltracchi and Gabriele's paper has also clearly shown that there is considerable room for improvement of the numerical performance of the algorithm. They have pinpointed the areas needing further investigation. The purpose of this discussion is to augment the study with our experience with the algorithm (and its

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... cements) since 1980 when the algorithm was originally discovered and its unique features were recognized. Performance of any algorithm is greatly affected by the details of the numerical implementation of each of its steps. A theoretically "good" algorithm can be implemented badly such that it does not work at all, and an approximate algorithm can be implemented such that it solves most of the problems. Therefore, efficiency and robustness of algorithms is quite dependent on the details of implementation as well as the ease with which they are implemented on the computer. Nevertheless, it is important to choose theoretically robust algorithms, study numerical implementation details and evaluate their numerical performance. This is particularly true for large scale applications in an industrial environment where robustness as well as efficiency are extremely important. Pshenichnyi's algorithm has proof of convergence and potential of being efficient. Therefore it is extremely useful to study the algorithm and evaluate its numerical performance. Its unique feature is the way in which "potential constraint set" (a subset of original constraints used in the subproblem) is defined. Also, the ease with which various improvements can be incorporated makes it a good candidate for further analysis and investigation. For example, it is relatively easy to incorporate Hessian updating into the algorithm [1][2] [3] [4] [5] . However, the difficulty of using it with a potential constraint strategy was recognized early [3] and some re-start procedures were devised, as has been suggested in the paper. Each numerical algorithm has vagueness and uncertainty in its computational steps. Pshenichnyi's algorithm is no different in this regard. To study improvements in the algorithm, its steps must be analyzed and procedures developed to implement them robustly. This has been done in several recent studies [3] [4] [5] [6] [7] [8] [9] [10] [11] using small scale as well as large scale problems. A modified algorithm has been developed and evaluated. We