Duality and a Farkas lemma for integer programs (original) (raw)

Generating functions and duality for integer programs

Discrete Optimization, 2004

We consider the integer program P→ max{c x|Ax = y; x ∈ N n }. Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables λ ∈ R m have discrete analogues z ∈ C m , related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P * , the analogue of the dual LP in linear programming.

On the augmented Lagrangian dual for integer programming

Mathematical Programming, 2014

We consider the augmented Lagrangian dual for integer programming, and provide a primal characterization of the resulting bound. As a corollary, we obtain proof that the augmented Lagrangian is a strong dual for integer programming. We are able to show that the penalty parameter applied to the augmented Lagrangian term may be placed at a fixed, large value and still obtain strong duality for pure integer programs. 2000) Primary: 47H05 · Secondary: 49J53 1 Introduction Duality has long been a powerful tool in Integer Programming (IP). Several dual problems have been proposed for IP (see, e.g. Nemhauser and Wolsey [34] Chapter II.3), including the Lagrangian dual [17], the surrogate dual [18], the subadditive dual , and more recently, the inference dual . The Lagrangian dual has given rise to a wealth of algorithms, both as a heuristic and through the provision of bounds for use in exact methods such as branch-and-bound. Surrogate duality has been less well explored, but methods have been developed for solving surrogate duals, and for embedding them in methods for solving the primal IP, notably those of Karwan

Duality for mixed-integer linear programs

2007

Abstract The theory of duality for linear programs is well-developed and has been successful in advancing both the theory and practice of linear programming. In principle, much of this broad framework can be extended to mixed-integer linear programs, but this has proven difficult, in part because duality theory does not integrate well with current computational practice.

Kalai and Muller’s possibility theorem: a simplified integer programming version

Review of Economic Design, 2018

We provide a respecification of an integer programming characterization of Arrovian social welfare functions introduced by Sethuraman et al. (2003). By exploiting this respecification, we give a new and simpler proof of Theorem 2 in Kalai and Muller (1977). Journal of Economic Literature Classification Number: D71.

Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs

Operations Research, 2011

In this paper, we give a finite disjunctive programming procedure to obtain the convex hull of general mixed-integer linear programs (MILP) with bounded integer variables. We propose a finitely convergent convex hull tree algorithm which constructs a linear program that has the same optimal solution as the associated MILP. In addition, we combine the standard notion of sequential cutting planes with ideas underlying the convex hull tree algorithm to help guide the choice of disjunctions to use within a cutting plane method. This algorithm, which we refer to as the cutting plane tree algorithm, is shown to converge to an integral optimal solution in finitely many iterations. Finally, we illustrate the proposed algorithm on three well-known examples in the literature that require an infinite number of elementary or split disjunctions in a rudimentary cutting plane algorithm.

An Algorithmic Theory of Integer Programming

2019

We study the general integer programming problem where the number of variables nnn is a variable part of the input. We consider two natural parameters of the constraint matrix AAA: its numeric measure aaa and its sparsity measure ddd. We show that integer programming can be solved in time g(a,d)textrmpoly(n,L)g(a,d)\textrm{poly}(n,L)g(a,d)textrmpoly(n,L), where ggg is some computable function of the parameters aaa and ddd, and LLL is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by aaa and ddd, and is solvable in polynomial time for every fixed aaa and ddd. Our results also extend to nonlinear separable convex objective functions. Moreover, for linear objectives, we derive a strongly-polynomial algorithm, that is, with running time g(a,d)textrmpoly(n)g(a,d)\textrm{poly}(n)g(a,d)textrmpoly(n), independent of the rest of the input data. We obtain these results by developing an algorithmic framework based on the idea of iterative augmentation: starting from an initial feasible solution, w...

Integral Farkas Type Lemmas for Systems with Equalities and Inequalities

SSRN Electronic Journal, 2007

A central result in the theory of integer optimization states that a system of linear Diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, y T A integral such that y T b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, Cx ≤ d}. We show that a certificate of integral infeasibility is a linear system with rank (C) variables containing no integral point.