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

Vladimir Yu. Protasov, Tatyana I. Zaitseva and Dmitrii O. Logofet ()
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Vladimir Yu. Protasov: Faculty DISIM, University of L’Aquila, 67100 L’Aquila, Italy
Tatyana I. Zaitseva: Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia
Dmitrii O. Logofet: Laboratory of Mathematical Ecology, A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, 119017 Moscow, Russia

Mathematics, 2022, vol. 10, issue 23, 1-15

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.

Keywords: matrix population model; population projection matrices; life cycle graph; loss functions; basin-hopping global search; matrix derivatives (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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