An equation-by-equation method for solving the multidimensional moment constrained maximum entropy problem.

*(English)*Zbl 1409.65034A novel equation solver is introduced that can be used to solve systems of nonlinear equations arising from the moment constrained maximum entropy problem of multidimensional variables. The proposed method, which is called the equation-by-equation (EBE) method, is an iterative method that solves a one-dimensional problem at the first iterate, a two-dimensional problem at the second iterate, a three-dimensional problem at the third iterate, and eventually solves the full system of nonlinear equations corresponding to the maximum entropy problem at the last iterate. Technically, this method combines Newton’s method with ideas from homotopy continuation.

It is shown that the EBE method is locally convergent under appropriate conditions. Furthermore, sufficient conditions for its global convergence are provided. Through the convergence analysis, it is shown that, geometrically, the proposed method finds the solution of the nonlinear system of equations by tracking along the surface corresponding to one component of the system of nonlinear equations. The EBE method automatically selects a subset of the prescribed constraints from which the maximum entropy solution can be estimated within the desired tolerance. This is an important feature since maximum entropy problems do not necessarily have solutions for general sets of moment constraints.

The robustness of the method is demonstrated with various numerical examples. In addition, the new procedure is compared with Newton’s method and other numerical methods to show its efficiency.

It is shown that the EBE method is locally convergent under appropriate conditions. Furthermore, sufficient conditions for its global convergence are provided. Through the convergence analysis, it is shown that, geometrically, the proposed method finds the solution of the nonlinear system of equations by tracking along the surface corresponding to one component of the system of nonlinear equations. The EBE method automatically selects a subset of the prescribed constraints from which the maximum entropy solution can be estimated within the desired tolerance. This is an important feature since maximum entropy problems do not necessarily have solutions for general sets of moment constraints.

The robustness of the method is demonstrated with various numerical examples. In addition, the new procedure is compared with Newton’s method and other numerical methods to show its efficiency.

Reviewer: José Manuel Gutiérrez Jimenez (Logrono)

##### MSC:

65H10 | Numerical computation of solutions to systems of equations |

65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |

94A17 | Measures of information, entropy |

49M15 | Newton-type methods |

##### Software:

Bertini
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\textit{W. Hao} and \textit{J. Harlim}, Commun. Appl. Math. Comput. Sci. 13, No. 2, 189--214 (2018; Zbl 1409.65034)

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