Practical issues on the projection of polyhedral sets

Abstract

Projection of polyhedral sets is a fundamental operation in both geometry and symbolic computation. In most cases, however, it is not practically feasible to generate projections as the size of the output can be exponential in the size of the input. Even when the size of the output is manageable, we still face two serious problems: overwhelming redundancy and degeneracy. Here, we address these problems from a practical point of view. We discuss three algorithms based on algebraic and geometric techniques and we compare their performance in order to assess the feasibility of these approaches.

Energy-Aware Profit Maximizing Scheduling Algorithm for Heterogeneous Computing Systems

“…This convex set is in the space of x and is projected onto the energy and makespan subspace. This projection was computed with the convex hull method described in [20]. When there is no idle power it is feasible to increase the makespan indefinitely when a minimum energy solution is sought.…”

Iterative optimization in the polyhedral model

“…Integer linear programming relates to the task of checking the satisfiability of a set of Presburger formulas, an NP-complete problem, and is the core of many optimization algorithms in particular for polyhedral program optimization. Although numerous previous work addressed the problem of checking the emptiness of a polyhedron by eliminating quantified variables, such as Ancourt and Irigoin [6], Irigoin et al with the PIPS system [61] or Lassez [59], the Omega Test developed by Pugh [92] is a powerful technique based on Fourier-Motzkin elimination to check the emptiness of an integer set. This technique is implemented in the Omega library.…”

Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

“…Other projection methods, i.e., the Extreme Point Method (see Lassez (1990)) and Convex Hull Method (see ), are evaluated in Huynh et al (1992). Jones et al (2004) develop a new algorithm for obtaining the projection of polytopes, which is suited for problems in which the number of vertices far exceeds the number of facets.…”