Numerical Methods For Underdetermined Box Constrained Integer Least Squares Problems
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Numerical Methods for Underdetermined Box-constrained Integer Least Squares Problems
"Integer least squares (ILS) is an important class of optimization problems, which can arise in many applications, such as communications, cryptography and cryptanalysis and global navigation satellite systems. This thesis is concerned with solving the underdetermined box constrained ILS (UBILS) problems. For the two existing algorithms, the direct tree search (DTS) algorithm and the partial regularization (PR) algorithm, we propose to incorporate some lower bounds to speed up the search process. Simulation results show that the proposed lower bounds can make the search process of the DTS algorithm perform more efficiently than the original one. Then we propose a modified DTS algorithm by partially using a best-first search strategy in the search process. Numerical tests results indicate that the new search algorithm is very effective in improving the efficiency of the DTS algorithm with or without incorporating the proposed lower bounds." --
An ADMM Method for Underdetermined Box-constrained Integer Least Squares Problems
"Integer least squares (ILS) is an important class of optimization problems that may arise from estimating the integer parameter vector in a linear model with additive Gaussian noise in applications such as communications, control and global navigation satellite systems. This thesis is concerned with solving the underdetermined box constrained ILS (UBILS) problems. We propose a modified alternating direction method of multipliers (ADMM) algorithm as a heuristic approach. Simulation results show that such heuristic is much superior to original ADMM on UBILS problems. Based on the existing direct tree search (DTS) algorithm, we show how to incorporate ADMM methods for computing initial points and lower bounds with the aim of improving the efficiency. Numerical results indicate that in most cases our new approach is better than DTS and selected commercial solvers in terms of efficiency and accuracy. Besides, the advantage of our new approach over DTS becomes more significant when the search region becomes larger"--