🚀 go-pugleaf

Here are more informations about the 2 new entropy-based vector optimization
methods and the 2 new simulated entropy methods I have developed. It explains
in more details the contents of my thesis with a brief introduction to the
subject.....

  This thesis explores the use of Shannon's (informational) entropy measure
and Jaynes' maximum entropy criterion in the solution of multicriteria
optimization problems, for Pareto set generation, and for seeking the
global minimum of single-criteria optimization problems. They all view
optimization problems in terms of topological domain defined
deterministically by function hypersurfaces. Part of this thesis is a
continuation of research aimed at developing a non deterministic approach
to optimization problems through the use of the Maximum Entropy Principle
(MEP) and the Minimax Entropy Principle (MMEP). We treat the
multicriteria optimization problem as a statistical system which can be
interpreted as transformations of the system to a sequence of equilibrium
states which are characterized by certain entropy maxima depending upon a
feasible entropy parameter.

  This thesis also introduces, for the first time, simulated entropy for
obtaining the global minima of single-criteria minimization problems.
Several ways of using entropy in the optimization problem context are
investigated. Two simulated entropy algorithms are developed.
Computational results demonstrate that the algorithms proposed in this
work are efficient and reliable. The following parts of the present work
are original:

1-)  An entropy-based method for generating Pareto set is presented. This
new method yields additional insight into the nature of entropy as
an information-theoretic basis and clarifies some ambiguities in the
literature about its use in optimization.

2-)  A new stochastic technique for reaching the global minimum of
constrained single-criteria minimization problems is developed. The
system may be interpreted as a statistical thermodynamic one which
is characterized by lowering the temperature of the system to its
limit along the entropy process transitions. However, in each
transition, equilibrium is characterized by maximizing the entropy
at that certain temperature. This is known as Simulated Entropy.

3-)  Two simulated entropy techniques are developed which seek the global
minimum of some deterministic objective function. The two techniques
can be applied to constrained minimization problems only.

  For multi-objective optimization problems, most, if not all, of the
available Pareto set techniques have several difficulties. Firstly, they
are computer-time consuming since in order to generate a representative
or entire Pareto set the preassigned vector of weighting coefficients,
bounds, etc. must be varied over a large number of combinations.
Secondly, Pareto solutions are obtained randomly since the distribution
characteristics of these solutions are unknown. Thirdly, where there are
many criteria the amount of computation required may, itself, become a
difficulty which dramatically affects the total number of Pareto
solutions obtained and the selection of a better preferred solution.

  For single-objective optimization problems, most, if not all, of the
available deterministic techniques terminate in a local minimum which
depends on an initial configuration given by the user and do not provide
any information as to the amount by which this local minimum deviates
from a global minimum.

  The first chapter of this thesis reviews previous work in Mathematical
Optimizaion. The motivations and specifications of the present
mathematical work are also examined.

  In the second chapter, the concept of single-criteria optimization ,
multi-criteria optimization, and their mathematical formulations are
introduced. A brief survey of some of the most usable methods is given.

  The third chapter introduces the concept of entropy and the
relationships between informational entropy and the much better known
classical thermodynamical entropy. Entropy is used as a measure of
uncertainty. The development of Shannon's informational entropy is
described and its further development into entropy-based minimax methods
for solving multi-criteria minimization problems is presented. The
relationships among, entropy, simulated annealing and free energy in
optimization are investigated theoretically and are used for development
of a new subject for solving global minimization problems. In summary,
two families of entropy-based methods are described in detail. The first
set of methods is used for generating Pareto solutions of multi-criteria
optimization problem while the second is used for seeking the global
minimum of single criteria optimization problems.

  In the fourth chapter, the set of entropy-based methods developed in
the previous chapter is tested, discussed and compared.

A. Sultan