# Glossary

Contains definitions of terms used in eGovPoliNet partly based on DCMI Metadata Terms.

- Linear program
- According to Luptáčik (2010) it is the simplest and most widely spread model of convex programming. A linear program or linear programming problem is an optimisation problem for which we attempt to maximise or minimise a linear function of the decision variables (so called objective function), where the value of the decision variables must satisfy a set of constraints, each of which must be a linear inequality or linear equality.

A linear program is a disarmingly simple object. According to Denardo (2011) its definition entails the terms, "linear expression" and "linear constraint". For instance 3x-2.5y+2z is a linear expression where its variables are x, y and z, and the dependence of this expression on x, y and z is linear. A linear constraint requires a linear inequality to take any of the proposed forms, or in other words, a linear constraints requires a linear expression to be less/greater than or equal to a number. A linear program either maximises or minimises a linear expression subject to finitely many linear constraints.

References:

Denardo, E. V. Linear Programming and Generalizations. A Problem-based Introduction with Spreadsheets. 1st Edition. New York: Springer Science+Business Media, 2011. ISBN 978-1-4419-6490-8.

Luptáčik, M. Mathematical Optimization and Economic Analysis. 1st Edition. New York: Springer Science+Business Media, 2010. ISBN 978-0-387-89552-9. - Linear Programming
- Linear programming describes the family of mathematical tools that are used to analyse linear programs (see definition of linear program in the glossary). The word „linear“ results from character of the objective function and the constraints and the word „programming“ results from applications in areas of planning or action scheduling.

Linear programming was first designed as planning and decision tool in setting where a central decision-maker fully in control of the various quantity variables in the system has to make consistent or optimal decision. The linear programming was developed by Kantorovich (1939) and Dantzig (1982) as a tool for optimal central decision making for primarily military purposes.

It is quite clear that the standard linear programming formulation is best suited to problems where a single decision maker optimises a central welfare function subject to technological and physical constraints. Unfortunately the standard formulation does not appear so well suited to modelling situations where many agents independently maximise their own welfare functions and jointly but inadvertently determine an outcome that can only be affected indirectly by the planner or policy maker.

References:

Dantzig, G. B. (1982). Reminiscences about the origins of linear programming, in Mathematical programming : the state of the art, Bonn, 1982 (New York, 1983), 78-86.

Kantorovich, L. V. (1939). "Mathematical Methods of Organizing and Planning Production" Management Science, Vol. 6, No. 4 (Jul., 1960), pp. 366–422.