# Some Theoretical Foundations of Bare-Simulation Optimization of Some Directed Distances between Fuzzy Sets Respectively Basic Belief Assignments

## TL;DR

This paper introduces new methods to measure and optimize differences between fuzzy sets and belief assignments, using a novel simulation approach.

## Contribution

The paper introduces generalized φ–divergences and applies a novel dimension-free simulation method to optimize them.

## Key findings

- Generalized φ–divergences are defined for fuzzy sets and basic belief assignments.
- A dimension-free simulation method is applied to solve constrained minimization problems.
- The approach is extended to vectors and rescaled belief assignments.

## Abstract

It is well known that in information theory—as well as in the adjacent fields of statistics, machine learning and artificial intelligence—it is essential to quantify the dissimilarity between objects of uncertain/imprecise/inexact/vague information; correspondingly, constrained optimization is of great importance, too. In view of this, we define the dissimilarity-measure-natured generalized φ–divergences between fuzzy sets, ν–rung orthopair fuzzy sets, extended representation type ν–rung orthopair fuzzy sets as well as between those fuzzy set types and vectors. For those, we present how to tackle corresponding constrained minimization problems by appropriately applying our recently developed dimension-free bare (pure) simulation method. An analogous program is carried out by defining and optimizing generalized φ–divergences between (rescaled) basic belief assignments as well as between (rescaled) basic belief assignments and vectors.