Research Progress Report on

-- I received the acceptance of our paper Abstract Convexity and Augmented Lagrangians (Burachik, R. S. and Rubinov, A.) just hours before my talk in the Workshop on Continuous Optimization in Ballarat, in Alex’s memory, in November 2006. This paper was not the last, there is still a submitted paper with Alex, for which we didn't get the reports yet.

Research with Alex was always full of excitement. Alex always had a very clear idea of what kind of results he wanted, and how these results should “look” from the mathematical point of view. Very often he would discard a new result, just because it was not addressing the problem he had in mind. He would write to me detailed emails where he would describe very clearly the problem and which kind of result he was expecting from our study together. It was important for him to devise a duality theory of augmented Lagrangians which would use the ideas and the concepts of abstract convexity. Alex and I did this (for a general class of augmented Lagrangians) in a paper which was accepted by SIAM-Optimization in November 2006.

Alex was amongst the most patient co-workers I have ever had. He was so kind that he could explain the same thing as many times as necessary, with the same patience and calmness as if he would be explaining it for the first time.

We started working on the project which gave rise to the above mentioned paper in June 2005. Because I was based in Adelaide, we worked by email until November 2005, by which time the manuscript was pretty ready. I travelled to Ballarat at the end of November 2005, and, in a few days of intensive work, we completed the manuscript. We submitted in December 2005 and Alex was the corresponding author. In the beginning of 2006, Alex became sick. Because Alex was so much interested in this work, he wanted to stay as the corresponding author, in spite of his declining health. He only agreed to transfer to me the task of being the corresponding author after I insisted many times.

We received the referee reports in August 2006. He was very happy to learn that the reports were overall positive. Unfortunately, he was too ill at that time to participate on the (rather minor) revision, which I am sure would have been much better if I had his guidance. A strong, deep feeling of sadness and loneliness accompanied me during the whole revision process. Fortunately the referees were satisfied with the revised version and the acceptance was sent to me in November 2006, just hours before my talk in the Workshop on Continuous Optimization in Ballarat, which was held in Alex’s memory, in November 2006. The content of this paper was the material of my talk.

Alex wanted the subject of abstract convexity to reach the traditional optimization community and therefore he insisted that we should aim to publish our results in well-known optimization journals. I see this paper as a preliminary step in addressing Alex’s wish. Now it is our opportunity to continue the path he had so generously set for us, by taking up the ideas, enthusiasm, curiosity, and love for mathematics he so skillfully fomented on those of us who had the immense gift to share his sweet, precious and unforgettable company.