Reformulations in Mathematical Programming: Definitions and Systematics

Abstract

Abstract. A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (opt-reformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type. Keywords

Feasibility-Based Bounds Tightening via Fixed Points

“…We also let O ′ = {+, −, a×} be the set of linear operators. Endowed with the operator set O, the MINLP formulation (1) (1) can also encode Semidefinite Programming (SDP) [14]. Black-box optimization problems, however, cannot be described by formulation (1).…”

“…We employ the latter approach: each non-leaf node z in the vertex set of D is replaced by an added variable w z and a corresponding constraint (4) is adjoined to to the formulation (usually, two variables w v , w u corresponding to identical defining constraints are replaced by one single added variable). The resulting reformulation, sometimes called Smith standard form [18], is exact [14]. A linear relaxation of (1) can automatically be obtained by the Smith standard form by replacing each nonlinear defining constraints by lower and upper linear approximations.…”

Transforming Mathematical Models Using Declarative Reformulation Rules

“…This system matches the capabilities of the framework envisioned in [11,12], which covers a large number of real-life problems and reformulation techniques. However, our system also allows to deal with algorithmic reformulation rules that are out of reach for frameworks based exclusively on algebraic techniques, and it makes explicit use of the concept of structure to allow exploiting reformulation rules based on the semantic (as opposed to purely syntactic) meaning of each block.…”

“…[4,9,16,18]), there are few formal definitions and theoretical characterizations of the concept. Some are limited to syntactic reformulations, i.e., those that can be obtained by application of algebraic rewriting rules to the elements of a given model [11]. These reformulations are capable of exploiting syntactical structure of the model, such as presence of particular algebraic terms in parts of its algebraic description [7].…”

“…Limited to MIP problems, general ideas based on variable redefinition were proposed by [13,14], without finding wide application due to its complexity, remaining unknown (or unused) by the "average" user. Only recently a wider attempt at formalizing the definition of formulation has been done which covers several techniques such as reformulation based on the preservation of the optimality information, changes of variables, narrowing, approximation and relaxation [11,12].…”