Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming

A partial pre-distance matrix A is a matrix with zero diagonal and with certain elements fixed to given nonnegative values; the other elements are considered free. The Euclidean distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a Euclidean distance matrix, EDM. The nearest (or approximate) Euclidean distance matrix problem is to find a Euclidean distance matrix, EDM, that is nearest in the Frobenius norm to the matrix A, when the free variables are discounted. In this paper we introduce two algorithms: one for the exact completion problem and one for the approximate completion problem. Both use a reformulation of EDM into a semidefinite programming problem, SDP. The first algorithm is based on an implicit equation for the completion that for many instances provides an explicit solution. The other algorithm is based on primal–dual interior-point methods that exploit the structure and sparsity. Included are results on maps that arise that keep the EDM and SDP cones invariant. We briefly discuss numerical tests. © 2005 Elsevier Inc. All rights reserved. ∗ Corresponding author. Tel.: +1 519 888 4567x5589. E-mail addresses: [email protected] (S. Al-Homidan), [email protected] (H. Wolkowicz). URL: http://orion.math.uwaterloo.ca/∼hwolkowi/henry/reports/ABSTRACTS.html 1 Research supported by King Fahd University of Petroleum and Minerals (KFUPM), Saudi Arabia. 2 Research supported by The Natural Sciences and Engineering Research Council of Canada. 0024-3795/$ see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2005.03.021 110 S. Al-Homidan, H. Wolkowicz / Linear Algebra and its Applications 406 (2005) 109–141