Method For Integer Programming Problems Research Articles

A New Nonparametric Filled Function Method for Integer Programming Problems with Constraints

In this paper, we investigate and develop a new filled function method for solving integer programming problems with constraints. By adopting the appropriate equivalent transformation method, these problems are transformed into a class of box-constrained integer programming problems. Then, an effective nonparametric filled function is constructed, and a new global optimization algorithm is designed using the discrete steepest descent method. Numerical experiments illustrate that this algorithm has effectiveness, feasibility, and better global optimization ability.

The blockwise coordinate descent method for integer programs

Blockwise coordinate descent methods have a long tradition in continuous optimization and are also frequently used in discrete optimization under various names. New interest in blockwise coordinate descent methods arises for improving sequential solutions for problems which consist of several planning stages. In this paper we systematically formulate and analyze the blockwise coordinate descent method for integer programming problems. We discuss convergence of the method and properties of the resulting solutions. We extend the notion of Pareto optimality for blockwise coordinate descent to the case that the blocks do not form a partition and compare Pareto optimal solutions to blockwise optimal and to global optimal solutions. Among others we derive a condition which ensures that the solution obtained by blockwise coordinate descent is weakly Pareto optimal and we confirm convergence of the blockwise coordinate descent to a global optimum in matroid polytopes. The results are interpreted in the context of multi-stage linear integer programming problems and illustrated for integrated planning in public transportation.

Fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems

Branch and cut methods for integer programming problems solve a sequence of linear programming problems. Traditionally, these linear programming relaxations have been solved using the simplex method. The reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one. If the solution to a relaxation is fractional, additional constraints can be generated which cut off the solution to the relaxation, but donot cut off any feasible integer points. Gomory cutting planes and other classes of cutting planes are generated from the final tableau. In this paper, we consider using an interior point method to solve the linear programming relaxations. We show that it is still possible to generate Gomory cuts and other cuts without having to recreate a tableau, and we also show how variables can be fixed without using the optimal reduced costs. The procedures we develop do not require that the current relaxation be solved to optimality; this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation.

General Purpose Heuristics for Integer Programming—Part II

In spite of the many special purpose heuristics for specific classes of integer programming (IP) problems, there are few developments that focus on general purpose integer programming heuristics. This stems partly from the perception that general purpose methods are likely to be less effective than specialized procedures for specific problems, and partly from the perception that there is no unifying theoretical basis for creating general purpose heuristics. Still, there is a general acknowledgment that methods which are not limited to solving IP problems on a “class by class” basis, but which apply to a broader range of problems, have significant value. We provide a theoretical framework and associated explicit proposals for generating general purpose IP heuristics. Our development, makes use of cutting plane derivations that also give a natural basis for marrying heuristics with exact branch and cut methods for integer programming problems.

Technical Note—A Flexible Tree-Search Method for Integer Programming Problems

This paper presents an enumerative method for solving integer programming problems. In the enumerative methods so far available, the variables are ranked in a rigid order by the choices in the forward steps, and backtracking always takes place on the last variable in this order. The proposed method allows a greater flexibility in the backtracking step that makes it possible to devise suitable rules of choice to improve the efficiency of the searching process. A numerical example is devised to illustrate the enumerative method in the direct-search case.