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fampa m maximum entropy sampling algorithms application 2022

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Textbook in PDF format This monograph presents a comprehensive treatment of the maximum-entropy sampling problem (MESP), which is a fascinating topic at the intersection of mathematical optimization and data science. The text situates MESP in information theory, as the algorithmic problem of calculating a sub-vector of pre-specificed size from a multivariate Gaussian random vector, so as to maximize Shannon's differential entropy. The text collects and expands on state-of-the-art algorithms for MESP, and addresses its application in the field of environmental monitoring. While MESP is a central optimization problem in the theory of statistical designs (particularly in the area of spatial monitoring), this book largely focuses on the unique challenges of its algorithmic side. From the perspective of mathematical-optimization methodology, MESP is rather unique (a 0/1 nonlinear program having a nonseparable objective function), and the algorithmic techniques employed are highly non-standard. In particular, successful techniques come from several disparate areas within the field of mathematical optimization; for example: convex optimization and duality, semidefinite programming, Lagrangian relaxation, dynamic programming, approximation algorithms, 0/1 optimization (e.g., branch-and-bound), extended formulation, and many aspects of matrix theory. The book is mainly aimed at graduate students and researchers in mathematical optimization and data analytics. Overview Notation The problem and basic properties Differential entropy The MESP and the CMESP Hardness A solvable case The complementary problem Scaling Masks Submodularity Branch-and-bound The branch-and-bound algorithmic framework for MESP Global upper bound for early termination Good lower bounds Greedy Swapping Approximation algorithm The branch-and-bound algorithmic framework for CMESP Upper bounds Spectral bounds Unconstrained Constrained Integer linear optimization An ILP-based diagonal bound for CMESP An ILP-based partition bound for MESP linx bound Convexity of linx Duality for linx Fixing variables in linx Computing linx and Dlinx solutions Scaling for linx The complementary problem of linx-gamma Factorization bound The Lagrangian dual of Fact Duality for DFact Fixing variables in DDFact Computing DDFact and DFact solutions Properties of the factorization bound NLP bound Convexity of NLP Scaling for NLP Good parameters for NLPgamma Strategies to select parameters for NLPgamma Duality and the logarithmic-barrier problem for gNLP Fixing variables in gNLP The logarithmic-barrier algorithm for gNLP NLP-gamma in the branch-and-bound algorithm BQP bound Convexity of BQP Duality for BQP Fixing variables in BQP A good feasible solution of DBQP from BQP Scaling for BQP Mixing bounds The mixing framework Optimizing the mixing parameters Duality for mixing Fixing variables in mix A good feasible solution of Dmix from mix Mixing the BQPgamma bound with the complementary BQPgamma bound Duality for smBQP Fixing variables in smBQP A good feasible solution of DsmBQP from smBQP Comparison of bounds Environmental monitoring The setting MESP within statistics and optimal experimental design MESP and environmental statistics From raw data to covariance matrices An example Opportunities Developing algorithmic advances for MESP/CMESP Variable fixing and branch-and-bound: state of the art Optimizing gamma for NLPgamma Solvable cases of MESP and mask optimization OA for CMESP MESP/CMESP variations and cousins Applications Basic formulae and inequalities Preliminary miscellany Square matrices Symmetric matrices Positive definite and semidefinite matrices

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fampa m maximum entropy sampling algorithms application 2022