The value function of a mixed integer program: II

We prove that the gap in optimal value, between a mixed-integer program in rationals and its corresponding linear programming relaxation, is bounded as the right-hand-side is varied. In addition, a variant of value iteration is shown to construct subadditive functions which resolve a pure-integer program when no dual degeneracy occurs. These subadditive functions provide solutions to subadditive dual programs for integer programs which are given here, and for which the values of primal and dual problems are equal.     

The b-hull of an integer program

Given a linear integer program: max{cx:Ax=b, x≥0 and integer},A rational, it is known that it can be solved in theory for all values of c, either by testing a finite number of solutions for optimality, or by adding a finite number of valid inequalities, each generated from a superadditive function, and solving the resulting linear program.The main result of this paper is to show that (analogously) the integer program can be solved for all values of b, either by testing a finite number of solutions for feasibility and optimality, or by adding a finite number of valid inequalities, each generated from a superadditive function, and solving the resulting linear program.     

The integrality number of an integer program

Mathematical Programming - We introduce the integrality number of an integer program (IP). Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an...     

整数分解式的指数函数及其均值的计算

给出了整数标准分解式的数论函数β(n，p)和它的均值∑n≤Nβ(n，p)的定义，推出了整数n标准分解式的指数函数计算公式和均值∑n≤Nβ(n,p)的精确计算公式．     

A characterisation of all feasible solutions to an integer program

It is shown how the dual of Fourier–Motzkin elimination can be applied to eliminating the constraints of an Integer Linear Program. The result will, in general, be to reduce the Integer Program to a single Diophantine equation together with a series of Linear homogeneous congruences. Extreme continuous solutions to the Diophantine equation give extreme solutions to the Linear Programming relaxation. Integral solutions to the Diophantine equation which also satisfy the congruences give all the solutions to the Integer Program.     

Optimizing a linear function over an integer efficient set

In this paper, a method for optimizing a linear function over the integer Pareto-optimal set without having to determine all integer efficient solutions is presented. The proposed algorithm is based on a simple selection technique that improves the linear objective value at each iteration. Two types of cuts are performed successively until the optimal value is obtained and the current truncated region contains no integer feasible solution.     

The fleet assignment problem: Solving a large-scale integer program

Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.This work was supported by NSF and AFORS grant DDM-9115768 and NSF grant SES-9122674.Corresponding author.     

On the smallest value of the maximal modulus of an algebraic integer

The house of an algebraic integer of degree is the largest modulus of its conjugates. For , we compute the smallest house of degree , say m. As a consequence we improve Matveev's theorem on the lower bound of m We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer whose house is equal to m is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer whose house is small.

     

The value function of an infinite-horizon single-item lot-sizing problem

We characterize the value function of a discounted infinite-horizon version of the single-item lot-sizing problem. As corollaries, we show that this value function inherits several properties of finite, mixed-integer program value functions; namely, it is subadditive, lower semicontinuous, and piecewise linear.     

Computing the integer partition function

In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of for for primes up to and small powers of and .

     

Minimum variance of a random integer value

The statement on a variance minimum of a random integer value with fixed mathematical expectation is proved.     

The “best” partition of an integer

An algorithm, based on a conjecture, to compute a permutation whose repeated application to a given set will yield a maximum number of different orderings of that set is presented. The algorithm gives the lengths of the cycles required. This problem turns out to be equivalent to the problem of determining a partitionB(n) ofn for which the least common multiple (l.c.m.) of the numbers ofB(n) is maximal.     

A hard integer program made easy by lexicography

A small but notoriously hard integer program formulated by Donald Knuth fifty years ago is solved by three versions of a lexicographic algorithm using Gomory cuts. The lexicographic cutting plane algorithms are faster than CPLEX on this problem by a factor of at least 10.