In their original versions, nature-inspired search algorithms such as evolutionary algorithms and those based on swarm intelligence, lack a mechanism to deal with the constraints of a numerical optimization problem. Nowadays, however, there exists a considerable amount of research devoted to design techniques for handling constraints within a nature-inspired algorithm. This paper presents an analysis of the most relevant types of constraint-handling techniques that have been adopted with nature-inspired algorithms. From them, the most popular approaches are analyzed in more detail. For each of them, some representative instantiations are further discussed. In the last part of the paper, some of the future trends in the area, which have been only scarcely explored, are briefly discussed and then the conclusions of this paper are presented.

http://www.cs.cinvestav.mx/~constraint/papers/swevo-mezura.pdf

Constraint-Handling in Nature-Inspired Numerical Optimization: Past, Present and Future

In this paper we review 30 years of developments and applications of the variable projection method for solving separable nonlinear least-squares problems. These are problems for which the model function is a linear combination of nonlinear functions. Taking advantage of this special structure, the method of variable projections eliminates the linear variables obtaining a somewhat more complicated function that involves only the nonlinear parameters. This procedure not only reduces the dimension of the parameter space but also results in a better-conditioned problem. The same optimization method applied to the original and reduced problems will always converge faster for the latter. We present first a historical account of the basic theoretical work and its various computer implementations, and then report on a variety of applications from electrical engineering, medical and biological imaging, chemistry, robotics, vision, and environmental sciences. An extensive bibliography is included. The method is particularly well suited for solving real and complex exponential model fitting problems, which are pervasive in their applications and are notoriously hard to solve.

https://iopscience.iop.org/article/10.1088/0266-5611/19/2/201/pdf

Separable nonlinear least squares: the variable projection method and its applications

Computer Methods in Applied Mechanics and Engineering

In the present paper we use a recently proposed generalization of the Hukuhara difference for compact convex sets, to introduce and study a generalization of the Hukuhara differentiability to the case of interval-valued functions. We also consider several possible definitions for the derivative of an interval-valued function, recently proposed in the literature, and we study connections between them and their properties. Using these concepts we study interval differential equations. Local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ODE systems. Several examples are also shown.

/pdf/generalized-hukuhara-differentiability-of-interval-valued-17l940t36c.pdf

Generalized Hukuhara differentiability of interval-valued functions and interval differential equations

http://www.econ.uniurb.it/RePEc/urb/wpaper/WP_12_09.pdf

Generalized differentiability of fuzzy-valued functions

In the present paper we introduce and study a generalization of the Hukuhara differ- ence and also generalizations of the Hukuhara differentiability to the case of interval valued functions. We consider several possible definitions for the derivative of an interval valued function and we study connections between them and their proper- ties. Using these concepts we study interval differential equations. Local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ODE systems and by differential algebraic equations. We also show some connection with differential inclusions. The thoretical results are turned into practical algorithms to solve interval differential equations.

Luciano Stefanini

Papers

Generalized Hukuhara differentiability of interval-valued functions and interval differential equations

Generalized differentiability of fuzzy-valued functions

Generalized Hukuhara Differentiability of Interval-valued Functions and Interval Differential Equations

A generalization of Hukuhara difference and division for interval and fuzzy arithmetic

Parametric representation of fuzzy numbers and application to fuzzy calculus