7,268 research outputs found
An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles
We address combinatorial optimization problems with uncertain coefficients
varying over ellipsoidal uncertainty sets. The robust counterpart of such a
problem can be rewritten as a second-oder cone program (SOCP) with integrality
constraints. We propose a branch-and-bound algorithm where dual bounds are
computed by means of an active set algorithm. The latter is applied to the
Lagrangian dual of the continuous relaxation, where the feasible set of the
combinatorial problem is supposed to be given by a separation oracle. The
method benefits from the closed form solution of the active set subproblems and
from a smart update of pseudo-inverse matrices. We present numerical
experiments on randomly generated instances and on instances from different
combinatorial problems, including the shortest path and the traveling salesman
problem, showing that our new algorithm consistently outperforms the
state-of-the art mixed-integer SOCP solver of Gurobi
An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming
Powerful interior-point methods (IPM) based commercial solvers, such as
Gurobi and Mosek, have been hugely successful in solving large-scale linear
programming (LP) problems. The high efficiency of these solvers depends
critically on the sparsity of the problem data and advanced matrix
factorization techniques. For a large scale LP problem with data matrix
that is dense (possibly structured) or whose corresponding normal matrix
has a dense Cholesky factor (even with re-ordering), these solvers may require
excessive computational cost and/or extremely heavy memory usage in each
interior-point iteration. Unfortunately, the natural remedy, i.e., the use of
iterative methods based IPM solvers, although can avoid the explicit
computation of the coefficient matrix and its factorization, is not practically
viable due to the inherent extreme ill-conditioning of the large scale normal
equation arising in each interior-point iteration. To provide a better
alternative choice for solving large scale LPs with dense data or requiring
expensive factorization of its normal equation, we propose a semismooth Newton
based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different
from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can
efficiently be used to solve simpler yet better conditioned semismooth Newton
linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic
superlinear convergence but is also proven to enjoy a finite termination
property. Numerical comparisons with Gurobi have demonstrated encouraging
potential of {\sc Snipal} for handling large-scale LP problems where the
constraint matrix has a dense representation or has a dense
factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920
characters", the abstract appearing here is slightly shorter than that in the
PDF fil
A General Large Neighborhood Search Framework for Solving Integer Programs
This paper studies how to design abstractions of large-scale combinatorial optimization problems that can leverage existing state-of-the-art solvers in general purpose ways, and that are amenable to data-driven design. The goal is to arrive at new approaches that can reliably outperform existing solvers in wall-clock time. We focus on solving integer programs, and ground our approach in the large neighborhood search (LNS) paradigm, which iteratively chooses a subset of variables to optimize while leaving the remainder fixed. The appeal of LNS is that it can easily use any existing solver as a subroutine, and thus can inherit the benefits of carefully engineered heuristic approaches and their software implementations. We also show that one can learn a good neighborhood selector from training data. Through an extensive empirical validation, we demonstrate that our LNS framework can significantly outperform, in wall-clock time, compared to state-of-the-art commercial solvers such as Gurobi
Nonnegative/binary matrix factorization with a D-Wave quantum annealer
D-Wave quantum annealers represent a novel computational architecture and
have attracted significant interest, but have been used for few real-world
computations. Machine learning has been identified as an area where quantum
annealing may be useful. Here, we show that the D-Wave 2X can be effectively
used as part of an unsupervised machine learning method. This method can be
used to analyze large datasets. The D-Wave only limits the number of features
that can be extracted from the dataset. We apply this method to learn the
features from a set of facial images
An Integer Linear Programming Solution to the Telescope Network Scheduling Problem
Telescope networks are gaining traction due to their promise of higher
resource utilization than single telescopes and as enablers of novel
astronomical observation modes. However, as telescope network sizes increase,
the possibility of scheduling them completely or even semi-manually disappears.
In an earlier paper, a step towards software telescope scheduling was made with
the specification of the Reservation formalism, through the use of which
astronomers can express their complex observation needs and preferences. In
this paper we build on that work. We present a solution to the discretized
version of the problem of scheduling a telescope network. We derive a solvable
integer linear programming (ILP) model based on the Reservation formalism. We
show computational results verifying its correctness, and confirm that our
Gurobi-based implementation can address problems of realistic size. Finally, we
extend the ILP model to also handle the novel observation requests that can be
specified using the more advanced Compound Reservation formalism.Comment: Accepted for publication in the refereed conference proceedings of
the International Conference on Operations Research and Enterprise Systems
(ICORES 2015
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