174 research outputs found
Mixing sets linked by bidirected paths
Recently there has been considerable research on simple mixed-integer sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot- sizing. This in turn has led to study of more general sets, called network-dual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the network-dual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a network-dual set in the original space of variables are known only for some special cases. Here we study two new cases, in which the continuous variables of the network-dual set are linked by a bi- directed path. In the first case, which is motivated by lot-sizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2^n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomial-time separation algorithm.mixing sets, extended formulations, mixed integer programming, lot-sizing with sales
Lot-sizing with stock upper bounds and fixed charges
Here we study the discrete lot-sizing problem with an initial stock variable and an associated variable upper bound constraint. This problem is of interest in its own right, and is also a natural relaxation of the constant capacity lot-sizing problem with upper bounds and fixed charges on the stock variables. We show that the convex hull of solutions of the discrete lot-sizing problem is obtained as the intersection of two simpler sets, one involving just 0-1 variables and the second a mixing set with a variable upper bound constraint. For these two sets we derive both inequality descriptions and polynomial-size extended formulations of their respective convex hulls. Finally we carry out some limited computational tests on single-item constant capacity lot-sizing problems with upper bounds and fixed charges on the stock variables in which we use the extended formulations derived above to strengthen the initial mixed integer programming formulations.mixed integer programming, discrete lot-sizing, stock fixed costs, mixing sets
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
On the convergence of the affine hull of the Chv\'atal-Gomory closures
Given an integral polyhedron P and a rational polyhedron Q living in the same
n-dimensional space and containing the same integer points as P, we investigate
how many iterations of the Chv\'atal-Gomory closure operator have to be
performed on Q to obtain a polyhedron contained in the affine hull of P. We
show that if P contains an integer point in its relative interior, then such a
number of iterations can be bounded by a function depending only on n. On the
other hand, we prove that if P is not full-dimensional and does not contain any
integer point in its relative interior, then no finite bound on the number of
iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra
chapter has been adde
The role of rationality in integer-programming relaxations
For a finite set XâZd that can be represented as X=Qâ©Zd for some polyhedron Q, we call Q a relaxation of X and define the relaxation complexity rc(X) of X as the least number of facets among all possible relaxations Q of X. The rational relaxation complexity rcQ(X) restricts the definition of rc(X) to rational polyhedra Q. In this article, we focus on X=Îd, the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in Rd. We show that rc(Îd)â€d for every dâ„5. That is, since rcQ(Îd)=d+1, irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Math Program 154(1):407â425, 2015). Moreover, we prove the asymptotic statement rc(Îd)âO(dlog(d)â/), which shows that the ratio rc(Îd)rcQ(Îd)/ goes to 0, as dââ
Towards Lower Bounds on the Depth of ReLU Neural Networks
We contribute to a better understanding of the class of functions that is
represented by a neural network with ReLU activations and a given architecture.
Using techniques from mixed-integer optimization, polyhedral theory, and
tropical geometry, we provide a mathematical counterbalance to the universal
approximation theorems which suggest that a single hidden layer is sufficient
for learning tasks. In particular, we investigate whether the class of exactly
representable functions strictly increases by adding more layers (with no
restrictions on size). This problem has potential impact on algorithmic and
statistical aspects because of the insight it provides into the class of
functions represented by neural hypothesis classes. However, to the best of our
knowledge, this question has not been investigated in the neural network
literature. We also present upper bounds on the sizes of neural networks
required to represent functions in these neural hypothesis classes.Comment: Camera-ready version for NeurIPS 2021 conferenc
Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective
We consider the integer points in a unimodular cone K ordered by a
lexicographic rule defined by a lattice basis. To each integer point x in K we
associate a family of inequalities (lex-cuts) that defines the convex hull of
the integer points in K that are not lexicographically smaller than x. The
family of lex-cuts contains the Chvatal-Gomory cuts, but does not contain and
is not contained in the family of split cuts. This provides a finite cutting
plane method to solve the integer program min{cx : x \in S \cap Z^n }, where S
\subset R^n is a compact set and c \in Z^n . We analyze the number of
iterations of our algorithm.Comment: 16 pages, 1 figur
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