Solving Euclidean Distance Matrixcompletion Problems Viasemidefinite

Given a partial symmetric matrix A with only certain elements speciied, the Eu-clidean distance matrix completion problem (EDMCP) is to nd the unspeciied elements of A that make A a Euclidean distance matrix (EDM). In this paper, we follow the successful approach in 20] and solve the EDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primal-dual interior-point algorithm that solves an equivalent (quadratic objective function) semideenite programming problem (SDP). Numerical results are included which illustrate the eeciency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed. Dedicated to Olvi Mangasarian The rst time that I came across Olvi's work was as a graduate student in the 70's when I studied from his book on Nonlinear Programming (now a SIAM classic) and also used the Mangasarian-Fromovitz constraint qualii-cation. This is the constraint qualiication (CQ) in nonlinear programming (NLP), since it guarantees the existence of Lagrange multipliers and is equivalent to stability of the NLP. This CQ has since been extended to various generalizations of NLP and plays a crucial role in perturbation theory. was teaching a course in the Business College. While walking through the halls one day I noticed the name Fromovitz on one of the doors. I could not pass this by and knocked and asked if this was the Fromovitz. The reply was \yes"; and, this is the story of the now famous CQ. Stan Fromovitz had just received his PhD from Stanford and was working at Shell Development 2 Company in the Applied Math Dept. Olvi needed a special Theorem of the Alternative for his work on a CQ. Stan went digging into the Math Library at Berkeley and came up with exactly what was needed: Motzkin's Theorem of the Alternative. The end result of this was the MF CQ. I have followed Olvi's work very closely throughout my career. His work is marked by many beautiful and important results in various areas. Some of the ones I am aware of are: condition numbers for nonlinear programs; generalized convexity; complementarity problems; matrix splittings; solution of large scale linear programs. It is a pleasure and an honour to be able to contribute to this special issue. Henry