A parameter method for linear algebra and optimization with uncertainties

ABSTRACT

Systems of linear equations are studied with uncertainties in the coefficients, often representing imprecise data. Such imprecisions are seen as orders of magnitude. The orders of magnitude are not functionally modelled in terms of $O(.)$ or $o(.)$, but within nonstandard analysis as neutrices and external numbers, convex external sets of nonstandard real numbers.

In this setting, linear systems with external coefficients are solved by a parameter method. This method assigns a parameter to each imprecision and specifies its range. Then we obtain an ordinary linear system and solve by common methods. At the end we substitute each parameter by its range. We present general solution formulas, for an arbitrary number of equations and variables. We illustrate with a concrete system that the parameter method gives more precise results, under more general conditions, than dealing with propagation of errors in direct Gauss–Jordan elimination or the Cramer rule.

We apply the results to linear optimization with uncertainties. In a general setting, nearly optimal solutions are obtained and the shape of the domains of validity of the nearly optimal solutions.

KEYWORDS:

• Flexible systems
• linear optimization
• uncertainties
• external numbers

AMS CLASSIFICATIONS:

• 03H05
• 15A06
• 65G40
• 90C05
• 90C31

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Imme van den Berg http://orcid.org/0000-0001-8485-3091