The paper under review is located in the mathematical theory of global optimization. It bases on a careful use of the problem structure and first-order information, and it is very much motivated by and applicable on real-world problems. In fact, the structure of the unconstrained problems is lying in the additive from of its objective function where the concerted but also limited interplay between the \(L\) many functions added has to be analytically and numerically exploited. The applied background of this paper is given by the theories of networks, especially, sensor networks, and of statistics and data mining, in particular, logistic regression. As the reviewers think, there can in future be much more practical uses, the potential of this promising study is not at all exhausted.
In the paper, an incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented. This method requires a single gradient evaluation per iteration and uses a constant step size. In case where the gradient is bounded and Lipschitz continuous, the authors show that the method visits infinitely often regions in which the gradient is small. Under certain unimodality assumptions, global convergence is established. For the quadratic case, a global linear rate of convergence is shown. The method is applied to distributed optimization problems arising in wireless sensor networks, and numerical experiments compare the new method with other incremental gradient methods.
This paper with its four sections is well written, structured, mathematically documented, illustrated and interpreted. It ranges from a careful discussion about the form of the gradient type of algorithm with the way how the \(L\) functions are involved to the sensor network, to applications on wireless sensor networks with the two items of robust estimation and source localization. The various illustrations are well prepared, clear and helpful.
The reviewers mention that real-world interpretations, applications and generalizations of this paper could come from many areas of science, social sciences, economics, Operational Research and hightech. One way on how to motivate the structure of the objective function could come from linear or nonlinear sums of squares (regression and classification with various link functions) and from the minimization of separable functions where each of the \(L\) functions would just depend on an own variable. Future studies could also address constrained optimization by the use of, e.g., active index set strategies or Lagrangean methods.