The hyper-scaled NLP bound for maximum-entropy remote sampling

The maximum-entropy remote sampling problem (MERSP) is to select a subset of s random variables from a set of n random variables, so as to maximize the information concerning a set of target random variables that are not directly observable. We assume that the set of all of these random variables follows a joint Gaussian distribution, and that we have the covariance matrix available. Finally, we measure information using Shannon's differential entropy. The main approach for exact solution of moderate-sized instances of MERSP has been branch-and-bound (B&B), and so previous work concentrated on upper bounds. Prior to our work, there were two upper-bounding methods for MERSP: the so-called ``complementary NLP bound'' and the ``spectral bound'', both introduced 25 years ago. We are able now to establish domination results between these two upper bounds. Further, we propose a novel and effective ``hyper-scaled NLP bound'' (hNLP bound) based on a subtle convex relaxation. The ``complementary'' version of hNLP bound for MERSP generalizes the previous complementary NLP bound for MERSP. We provide theoretical guarantees, giving sufficient conditions under which the complementary hNLP bound strictly dominates the complementary NLP bound. In addition, the hNLP formulation allows us to derive upper bounds for rank-deficient covariance matrices when they satisfy a technical condition. This is in contrast to the previous NLP bound that worked with only positive definite covariance matrices (because it was wedded to a complementary formulation). Additionally, we describe procedures for calculating hyper-scaling parameters. Finally, for B&B, we provide a variable-fixing methodology and results guiding the best way to construct subproblems. Numerical experiments on benchmark instances demonstrate the effectiveness of our approaches in advancing the algorithmic state-of-the-art for MERSP.