Regularization of a Parameter Estimation Problem using Monotonicity and Convexity Constraints ^*

Abstract

In marine science, it is usually assumed that there is a functional relationship between the parental population size and subsequent offsprings. The function is referred to as the Stock Recruitment Function (SRF). Determining the SRF translates to the optimization problem of estimating a set of parameters using past and sparse observation, which are usually of modest accuracy. The problem is challenging because several candidate functions exist in the literature, and the choice of best function is non-trivial, due to data sparsity and uncertainty.

This paper formulates the problem as a constrained optimization task, and uses B-spline basis functions to represent the functional family to which the SRF belongs. Regularized solutions are obtained by requiring that the derived functions are both monotone and convex.

The approach presents two major contributions to the existing computational challenges:

• •

  It avoids the non-trivial problem of choosing the functional form a priori.

• •

  Regularization of the problem using constraints ensures that parameter estimates are realistic.

Numerical examples are presented to compare ℓ₁ and ℓ₂-norm solutions.

Résumé

The paper presents a method that uses a priori information to regularize an inverse problem in fisheries science. It demonstrates the efficacy of the methodology in dealing with sparse and uncertain data, and limiting assumptions on the solution space.