Characterizing optimality in mathematical programming models

Abstract

This paper is a survey of basic results that characterize optimality in single- and multi-objective mathematical programming models. Many people believe, or want to believe, that the underlying behavioural structure of management, economic, and many other systems, generates basically ‘continuous’ processes. This belief motivates our definition and study of optimality, termed ‘structural’ optimality. Roughly speaking, we say that a feasible point of a mathematical programming model is structurally optimal if every improvement of the optimal value function, with respect to parameters, results in discontinuity of the corresponding feasible set of decision variables. This definition appears to be more suitable for many applications and it is also more general than the usual one: every optimum is a structural optimum but not necessarily vice versa. By characterizing structural optima, we obtain some new, and recover the familiar, optimality conditions in nonlinear programming. The paper is self-contained. Our approach is geometric and inductive: we develop intiution by studying finite-dimensional models before moving on to abstract situations.