278 research outputs found
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
A learning-based algorithm to quickly compute good primal solutions for Stochastic Integer Programs
We propose a novel approach using supervised learning to obtain near-optimal
primal solutions for two-stage stochastic integer programming (2SIP) problems
with constraints in the first and second stages. The goal of the algorithm is
to predict a "representative scenario" (RS) for the problem such that,
deterministically solving the 2SIP with the random realization equal to the RS,
gives a near-optimal solution to the original 2SIP. Predicting an RS, instead
of directly predicting a solution ensures first-stage feasibility of the
solution. If the problem is known to have complete recourse, second-stage
feasibility is also guaranteed. For computational testing, we learn to find an
RS for a two-stage stochastic facility location problem with integer variables
and linear constraints in both stages and consistently provide near-optimal
solutions. Our computing times are very competitive with those of
general-purpose integer programming solvers to achieve a similar solution
quality
Mirror-Descent Methods in Mixed-Integer Convex Optimization
In this paper, we address the problem of minimizing a convex function f over
a convex set, with the extra constraint that some variables must be integer.
This problem, even when f is a piecewise linear function, is NP-hard. We study
an algorithmic approach to this problem, postponing its hardness to the
realization of an oracle. If this oracle can be realized in polynomial time,
then the problem can be solved in polynomial time as well. For problems with
two integer variables, we show that the oracle can be implemented efficiently,
that is, in O(ln(B)) approximate minimizations of f over the continuous
variables, where B is a known bound on the absolute value of the integer
variables.Our algorithm can be adapted to find the second best point of a
purely integer convex optimization problem in two dimensions, and more
generally its k-th best point. This observation allows us to formulate a
finite-time algorithm for mixed-integer convex optimization
Orthonormal sequences in and time frequency localization
We study uncertainty principles for orthonormal bases and sequences in
. As in the classical Heisenberg inequality we focus on the product
of the dispersions of a function and its Fourier transform. In particular we
prove that there is no orthonormal basis for for which the time and
frequency means as well as the product of dispersions are uniformly bounded.
The problem is related to recent results of J. Benedetto, A. Powell, and Ph.
Jaming.
Our main tool is a time frequency localization inequality for orthonormal
sequences in . It has various other applications.Comment: 18 page
Exact Solution Methods for the -item Quadratic Knapsack Problem
The purpose of this paper is to solve the 0-1 -item quadratic knapsack
problem , a problem of maximizing a quadratic function subject to two
linear constraints. We propose an exact method based on semidefinite
optimization. The semidefinite relaxation used in our approach includes simple
rank one constraints, which can be handled efficiently by interior point
methods. Furthermore, we strengthen the relaxation by polyhedral constraints
and obtain approximate solutions to this semidefinite problem by applying a
bundle method. We review other exact solution methods and compare all these
approaches by experimenting with instances of various sizes and densities.Comment: 12 page
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
Hypercontractive measures, Talagrand's inequality, and influences
We survey several Talagrand type inequalities and their application to
influences with the tool of hypercontractivity for both discrete and
continuous, and product and non-product models. The approach covers similarly
by a simple interpolation the framework of geometric influences recently
developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb
decompositions are also considered in this context
GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs
We present a prototype of a software tool for exploration of multiple
combinatorial optimisation problems in large real-world and synthetic complex
networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial
Explorer), provides a unified framework for scalable computation and
presentation of high-quality suboptimal solutions and bounds for a number of
widely studied combinatorial optimisation problems. Efficient representation
and applicability to large-scale graphs and complex networks are particularly
considered in its design. The problems currently supported include maximum
clique, graph colouring, maximum independent set, minimum vertex clique
covering, minimum dominating set, as well as the longest simple cycle problem.
Suboptimal solutions and intervals for optimal objective values are estimated
using scalable heuristics. The tool is designed with extensibility in mind,
with the view of further problems and both new fast and high-performance
heuristics to be added in the future. GraphCombEx has already been successfully
used as a support tool in a number of recent research studies using
combinatorial optimisation to analyse complex networks, indicating its promise
as a research software tool
Nonlinear spectral calculus and super-expanders
Nonlinear spectral gaps with respect to uniformly convex normed spaces are
shown to satisfy a spectral calculus inequality that establishes their decay
along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to
behave sub-multiplicatively under zigzag products. These results yield a
combinatorial construction of super-expanders, i.e., a sequence of 3-regular
graphs that does not admit a coarse embedding into any uniformly convex normed
space.Comment: Typos fixed based on referee comments. Some of the results of this
paper were announced in arXiv:0910.2041. The corresponding parts of
arXiv:0910.2041 are subsumed by the current pape
- …