3,024 research outputs found
Multi-objective integer programming: An improved recursive algorithm
This paper introduces an improved recursive algorithm to generate the set of
all nondominated objective vectors for the Multi-Objective Integer Programming
(MOIP) problem. We significantly improve the earlier recursive algorithm of
\"Ozlen and Azizo\u{g}lu by using the set of already solved subproblems and
their solutions to avoid solving a large number of IPs. A numerical example is
presented to explain the workings of the algorithm, and we conduct a series of
computational experiments to show the savings that can be obtained. As our
experiments show, the improvement becomes more significant as the problems grow
larger in terms of the number of objectives.Comment: 11 pages, 6 tables; v2: added more details and a computational stud
An exact method for a discrete multiobjective linear fractional optimization
Integer linear fractional programming problem with multiple objective MOILFP is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated
A Scalable Algorithm For Sparse Portfolio Selection
The sparse portfolio selection problem is one of the most famous and
frequently-studied problems in the optimization and financial economics
literatures. In a universe of risky assets, the goal is to construct a
portfolio with maximal expected return and minimum variance, subject to an
upper bound on the number of positions, linear inequalities and minimum
investment constraints. Existing certifiably optimal approaches to this problem
do not converge within a practical amount of time at real world problem sizes
with more than 400 securities. In this paper, we propose a more scalable
approach. By imposing a ridge regularization term, we reformulate the problem
as a convex binary optimization problem, which is solvable via an efficient
outer-approximation procedure. We propose various techniques for improving the
performance of the procedure, including a heuristic which supplies high-quality
warm-starts, a preprocessing technique for decreasing the gap at the root node,
and an analytic technique for strengthening our cuts. We also study the
problem's Boolean relaxation, establish that it is second-order-cone
representable, and supply a sufficient condition for its tightness. In
numerical experiments, we establish that the outer-approximation procedure
gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin
On Minimal Valid Inequalities for Mixed Integer Conic Programs
We study disjunctive conic sets involving a general regular (closed, convex,
full dimensional, and pointed) cone K such as the nonnegative orthant, the
Lorentz cone or the positive semidefinite cone. In a unified framework, we
introduce K-minimal inequalities and show that under mild assumptions, these
inequalities together with the trivial cone-implied inequalities are sufficient
to describe the convex hull. We study the properties of K-minimal inequalities
by establishing algebraic necessary conditions for an inequality to be
K-minimal. This characterization leads to a broader algebraically defined class
of K- sublinear inequalities. We establish a close connection between
K-sublinear inequalities and the support functions of sets with a particular
structure. This connection results in practical ways of showing that a given
inequality is K-sublinear and K-minimal.
Our framework generalizes some of the results from the mixed integer linear
case. It is well known that the minimal inequalities for mixed integer linear
programs are generated by sublinear (positively homogeneous, subadditive and
convex) functions that are also piecewise linear. This result is easily
recovered by our analysis. Whenever possible we highlight the connections to
the existing literature. However, our study unveils that such a cut generating
function view treating the data associated with each individual variable
independently is not possible in the case of general cones other than
nonnegative orthant, even when the cone involved is the Lorentz cone
A Neural Benders Decomposition for the Hub Location Routing Problem
In this study, we propose an imitation learning framework designed to enhance
the Benders decomposition method. Our primary focus is addressing degeneracy in
subproblems with multiple dual optima, among which Magnanti-Wong technique
identifies the non-dominant solution. We develop two policies. In the first
policy, we replicate the Magnanti-Wong method and learn from each iteration. In
the second policy, our objective is to determine a trajectory that expedites
the attainment of the final subproblem dual solution. We train and assess these
two policies through extensive computational experiments on a network design
problem with flow subproblem, confirming that the presence of such learned
policies significantly enhances the efficiency of the decomposition process
Algorithms for Highly Symmetric Linear and Integer Programs
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title
slightly change
A decomposition approach to the integrated vehicle-crew-rostering problem
The problem addressed in this paper is the integrated vehicle-crew-rostering problem (VCRP) aiming to define the schedules for the buses and the rosters for the drivers of a public transit company. The VCRP is described by a bi-objective mixed binary linear programming model with one objective function aggregating vehicle and crew scheduling costs and the other the rostering features. The VCRP is solved by a heuristic approach based on Benders decomposition where the master problem is partitioned into daily integrated vehicle-crew scheduling problems and the sub-problem is a rostering problem. Computational experience with data from a bus company in Lisbon shows the ability of the decomposition approach for producing a variety of potentially efficient solutions for the VCRP within low computing times
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