Links Between Linear Bilevel and Mixed 0–1 Programming Problems

Optimal configuration of gamma ray machine radiosurgery units: the sphere covering subproblem

“…Reformulations that are linked to a change in the problem definition (and not only to the problem formulation) are called reformulations in the sense of Hansen [1,14]. We prove that the reformulation given above is a reformulation in the sense of Hansen.…”

“…The problem, which involves covering an ellipsoid by spheres, is naturally cast as a MINLP; however, as is well known, solving MINLPs is much more difficult than solving MILPs. This is why we exhibit a sequence of reformulations [1,14] that either transform exactly or approximate the original MINLP into a cMINLP and eventually into different types of MILPs. By formulation we mean a formal mathematical description of an optimization problem consisting of parameters, decision variables, an objective function and constraints.…”

Transforming Mathematical Models Using Declarative Reformulation Rules

“…that of [17]. On the other hand, [2] manages the idea of mapping functions; while in theory it has the same power that our reformulation system has, we propose a reformulation system defined over a precise modeling language, that allows us to algorithmically and algebraically deduce reformulations. i-dare(t) offers a way of determining which structures can be reformulated and how they will be reformulated, obtaining at the end of the process valid formulations and data ready to be given to the solvers.…”

“…A view based on complexity theory was proposed in [2], but since it requires a polynomial time mapping between the problems it already cuts off a number of well-known reformulation techniques where the mapping is pseudo-polynomial [6] or even exponential in theory [3,5,8], but quite effective in practice. 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.…”

Reformulations in Mathematical Programming: Definitions and Systematics

“…Curiously, the term "reformulation" appears in conjunction with "mathematical programming" over 400,000 times on Google; and yet there are surprisingly few attempts to formally define what a reformulation in mathematical programming actually is [10,71]. Further motives in support of a unified study of reformulations in mathematical programming are that there is a remarkable lack of literature reviews on the topic of reformulations [46] and that modelling languages such as AMPL [24] or GAMS [17] offer very limited automatic reformulation capabilities.…”