Ray projection for optimizing polytopes with prohibitively many constraints in set-covering column generation

نویسنده

  • Daniel Cosmin Porumbel
چکیده

A recurrent task in mathematical programming requires optimizing polytopes with prohibitivelymany constraints, e.g., the primal polytope in cutting-plane methods or the dual polytope in Column Generation (CG). This paper is devoted to the ray projection technique for optimizing such polytopes: start from a feasible solution and advance on a given ray direction until intersecting a polytope facet. The resulting intersection point is determined by solving the intersection sub-problem: given ray r ∈ Z, find the maximum t∗ ≥ 0 such that t∗r is feasible. We focus on dual polytopes associated to CG: if the CG (separation) sub-problem can be solved by Dynamic Programming (DP), so can be the intersection sub-problem. The convergence towards the CG optimum is realized through a sequence of intersection points t∗r (feasible dual solutions) determined from such rays r. Our method only uses integer rays r, so as to render the intersection sub-problem tractable by r-indexed DP. We show that in such conditions, the intersection sub-problem can be even easier than the CG sub-problem, especially when no other integer data is available to index states in DP, i.e., if the CG sub-problem input only consists of fractional (or large-range) values. As such, the proposed method can tackle scaled instances (with large-range weights) of capacitated problems that seem prohibitively hard for classical CG. This is confirmed by numerical experiments on various capacitated Set-Covering problems: Capacitated Arc-Routing, Cutting-Stock and other three versions of Elastic Cutting-Stock (i.e., a problem that include Variable Size Bin Packing). We also prove the theoretical convergence of the proposed method.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Parallel computation framework for optimizing trailer routes in bulk transportation

We consider a rich tanker trailer routing problem with stochastic transit times for chemicals and liquid bulk orders. A typical route of the tanker trailer comprises of sourcing a cleaned and prepped trailer from a pre-wash location, pickup and delivery of chemical orders, cleaning the tanker trailer at a post-wash location after order delivery and prepping for the next order. Unlike traditiona...

متن کامل

Convergent Dual Bounds Using an Aggregation of Set-Covering Constraints for Capacitated Problems

Extended formulations are now widely used to solve hard combinatorial optimization problems. Such formulations have prohibitively-many variables and are generally solved via Column Generation (CG). CG algorithms are known to have frequent convergence issues, and, up to a sometimes large number of iterations, classical Lagrangian dual bounds may be weak. This paper is devoted to set-covering pro...

متن کامل

Experiences with Enumeration of Integer Projections of Parametric Polytopes

Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints. In an extended problem (the “integer projection” of a parametric polytope), some of the variables that appear in the ...

متن کامل

Cockpit Crew Pairing Problem in Airline Scheduling: Shortest Path with Resources Constraints Approach

Increasing competition in the air transport market has intensified active airlines’ efforts to keep their market share by attaching due importance to cost management aimed at reduced final prices. Crew costs are second only to fuel costs on the cost list of airline companies. So, this paper attempts to investigate the cockpit crew pairing problem. The set partitioning problem has been used for ...

متن کامل

Convergent Lower Bounds for Packing Problems via Restricted Dual Polytopes

Cutting-stock and bin-packing problems have been widely studied in the operations research literature for their large range of industrial applications. Many integer programming models have been proposed for them. The most famous is from Gilmore and Gomory [5] and relies on a column generation scheme. Column generation methods are known to have convergence issues, and (when minimizing) no useful...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Math. Program.

دوره 155  شماره 

صفحات  -

تاریخ انتشار 2016