An outer-approximation algorithm for generalized maximum entropy sampling

This paper presents an outer-approximation algorithm to address a generalized maximum entropy sampling (GMES) problem that determines a set of measurement locations providing the largest entropy reduction. A new mixed- integer semidefinite program (MISDP) formulation is proposed to handle a GMES problem with a jointly Gaussian distribution over the search space. This formulation employs binary variables to indicate if the corresponding measurement location is selected, and exploits the linear equivalent form of a bilinear term involving binary variables to ensure convexity of the objective function and linearity of the constraint functions. An outer- approximation algorithm is developed for this formulation that obtains the optimal solution by solving a sequence of mixed-integer linear programs. Numerical experiments are presented to verify the solution optimality and the computational effectiveness of the proposed algorithm by comparing it with an existing branch-and-bound method that utilizes nonlinear programming relaxation. Sensor selection for best tracking of a moving target under a communication budget constraint is specifically considered to validate the superiority of the suggested algorithm in handling quadratic constraints.