Doutoramento em Matemática
Optimization with flexible objectives and constraints
Nam Van Tran

Orientação:  Imme van den Berg

Both linear programming and non-linear optimization are studied from the point of view of non-standard analysis, in cases where the objective function and/or the constraints are not fully specified, indeed allow for some imprecision or flexibility in terms of some limited shifts.
The order of magnitude of such shifts will be modelled by neutrices, additive convex subgroups of the nonstandard real line and external numbers, sums of a neutrix and a non-standard real number. This approach captures essential features of imprecision, maintaining rather strong and effective rules of calculation.
Functions, sequences and equations which involve external numbers are called flexible. We consider optimization problems with flexible objective functions and/or constraints.
Necessary and sufficient conditions for the existence of optimal or approximate optimal solutions are given for both linear and non-linear optimization problems with flexible objective functions and constraints.
To deal with linear programming in this setting, flexible systems of linear equations are studied. Conditions for the solvability of a flexible system by usual methods such as Cramer’s rule and Gauss-Jordan elimination are established. Also, a parameter method is considered to solve flexible systems. Formulas of solutions depending on parameters are presented. The set of solutions of a flexible system is expressed in terms of external vectors and neutrices.
In order to investigate non-linear optimization with flexible objectives and constraints, we develop tools of analysis for both flexible sequences and functions.

Keywords: Optimization, uncertainty, external number, flexible system, flexible function, non-standard analysis