Pattern-Multiplicative Average of Nonnegative Matrices: When a Constrained Minimization Problem Requires Versatile Optimization Tools

Abstract

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern as well. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative nature of averaging. The concept of PMA was motivated in a number of application fields, of which we consider the matrix population models and illustrate solving the PMA problem with several sets of model matrices calibrated in particular botanic case studies. The patterns of those matrices are typically nontrivial (they contain both zero and nonzero elements), the PMA problem thus having no exact solution for a fundamental reason (an overdetermined system of algebraic equations). Therefore, searching for the approximate solution reduces to a constrained minimization problem for the approximation error, the loss function in optimization terms. We consider two alternative types of the loss function and present a general algorithm of searching the optimal solution: basin-hopping global search, then local descents by the method of conjugate gradients or that of penalty functions. Theoretical disadvantages and practical limitations of both loss functions are discussed and illustrated with a number of practical examples.

Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of Versatile Optimization Tools

“…So, we come to the constrained nonlinear minimization problem (8 and 9), where the bounds of B are defined in Table 2. Both standard routines for global optimization [18] and more sophisticated ones [14] can be applied for solving the problem (8 and 9).…”

“…− λv 5 = 0, i.e., for the ten unknown entries of G and the 11th formal variable λ. (14) The 11th variable has its own bounds: as the dominant eigenvalue of the average matrix, the unknown λ has to locate within the following bounds,…”

“…. ., l, m, λ] T , while the polygons B p ⊂ R 11 + (p = 1, 2) are defined in (15) and the equality-type constraints (14). The two inner problems (16) represent the standard LP ones to be solved by standard software routines, such as the function linprog in MATLAB ® [19] (technical details in Appendix B).…”

“…The solution to the constraint minimization problem (8) has been obtained in [14] by the "versatile optimization tools" (Table 4) for the case of 12 annual PPMs. The quality of approximation ranges from 0.002374 to 0.002379 as a function of the optimization method (ibidem).…”

“…This approach has led to a concept that was proposed 5 years ago with regard to matrix population models [13], and the task of averaging was reduced to a constrained nonlinear minimization problem for the matrix norm [8,13] (see Section 2.3). The norm has however not appeared quite fit for global optimization but required versatile optimization tools [14] to solve the problem.…”