4,251 research outputs found
Branch-and-bound for biobjective mixed-integer linear programming
We present a generic branch-and-bound method for finding all the Pareto
solutions of a biobjective mixed integer program. Our main contribution is new
algorithms for obtaining dual bounds at a node, for checking node fathoming,
presolve and duality gap measurement. Our various procedures are implemented
and empirically validated on instances from literature and a new set of hard
instances. We also perform comparisons against the triangle splitting method of
Boland et al. [\emph{INFORMS Journal on Computing}, \textbf{27} (4), 2015],
which is a objective space search algorithm as opposed to our variable space
search algorithm. On each of the literature instances, our branch-and-bound is
able to compute the entire Pareto set in significantly lesser time. Most of the
instances of the harder problem set were not solved by either algorithm in a
reasonable time limit, but our algorithm performs better on average on the
instances that were solved.Comment: 35 pages, 12 figures. Original preprint at Optimization Online,
October 201
A Computational study of branching rules for multi-commodity fixed-charged network flow problems
Branch and bound based algorithms are used by many commercial mixed integer programming solvers for solving complex optimization problems. In a branch and bound based method, a feasible region is divided into smaller sub-problems. This is called branching and various branching strategies have been developed to improve the performance of branch and bound based algorithms. However, their performance has primarily been studied on general mixed integer programs. Thus, in the first phase of this thesis, we study the performance of these branching strategies on a specific, structured mixed integer program, the capacitated multi-commodity fixed charge network flow (MCFCNF) problem. We also develop new branching strategies using the pool of available feasible solutions for solving the mixed integer program for MCFCNF. We present the computational results for testing various branching rules with four different variants of the network design problem studied with SCIP and GLPK mathematical solvers
Parameterizing Branch-and-Bound Search Trees to Learn Branching Policies
Branch and Bound (B&B) is the exact tree search method typically used to
solve Mixed-Integer Linear Programming problems (MILPs). Learning branching
policies for MILP has become an active research area, with most works proposing
to imitate the strong branching rule and specialize it to distinct classes of
problems. We aim instead at learning a policy that generalizes across
heterogeneous MILPs: our main hypothesis is that parameterizing the state of
the B&B search tree can aid this type of generalization. We propose a novel
imitation learning framework, and introduce new input features and
architectures to represent branching. Experiments on MILP benchmark instances
clearly show the advantages of incorporating an explicit parameterization of
the state of the search tree to modulate the branching decisions, in terms of
both higher accuracy and smaller B&B trees. The resulting policies
significantly outperform the current state-of-the-art method for "learning to
branch" by effectively allowing generalization to generic unseen instances.Comment: AAAI 2021 camera-ready version with supplementary materials, improved
readability of figures in main article. Code, data and trained models are
available at https://github.com/ds4dm/branch-search-tree
Optimization as an analysis tool for human complex decision making
We present a problem class of mixed-integer nonlinear programs (MINLPs) with nonconvex continuous relaxations which stem from economic test scenarios that are used in the analysis of human complex problem solving. In a round-based scenario participants hold an executive function. A posteriori a performance indicator is calculated and correlated to personal measures such as intelligence, working memory, or emotion regulation. Altogether, we investigate 2088 optimization problems that differ in size and initial conditions, based on real-world experimental data from 12 rounds of 174 participants. The goals are twofold. First, from the optimal solutions we gain additional insight into a complex system, which facilitates the analysis of a participant’s performance in the test. Second, we propose a methodology to automatize this process by providing a new criterion based on the solution of a series of optimization problems. By providing a mathematical optimization model and this methodology, we disprove the assumption that the “fruit fly of complex problem solving,” the Tailorshop scenario that has been used for dozens of published studies, is not mathematically accessible—although it turns out to be extremely challenging even for advanced state-of-the-art global optimization algorithms and we were not able to solve all instances to global optimality in reasonable time in this study. The publicly available computational tool Tobago [TOBAGO web site https://sourceforge.net/projects/tobago] can be used to automatically generate problem instances of various complexity, contains interfaces to AMPL and GAMS, and is hence ideally suited as a testbed for different kinds of algorithms and solvers. Computational practice is reported with respect to the influence of integer variables, problem dimension, and local versus global optimization with different optimization codes
Learning to Optimize Computational Resources: Frugal Training with Generalization Guarantees
Algorithms typically come with tunable parameters that have a considerable
impact on the computational resources they consume. Too often, practitioners
must hand-tune the parameters, a tedious and error-prone task. A recent line of
research provides algorithms that return nearly-optimal parameters from within
a finite set. These algorithms can be used when the parameter space is infinite
by providing as input a random sample of parameters. This data-independent
discretization, however, might miss pockets of nearly-optimal parameters: prior
research has presented scenarios where the only viable parameters lie within an
arbitrarily small region. We provide an algorithm that learns a finite set of
promising parameters from within an infinite set. Our algorithm can help
compile a configuration portfolio, or it can be used to select the input to a
configuration algorithm for finite parameter spaces. Our approach applies to
any configuration problem that satisfies a simple yet ubiquitous structure: the
algorithm's performance is a piecewise constant function of its parameters.
Prior research has exhibited this structure in domains from integer programming
to clustering
Industrial and Tramp Ship Routing Problems: Closing the Gap for Real-Scale Instances
Recent studies in maritime logistics have introduced a general ship routing
problem and a benchmark suite based on real shipping segments, considering
pickups and deliveries, cargo selection, ship-dependent starting locations,
travel times and costs, time windows, and incompatibility constraints, among
other features. Together, these characteristics pose considerable challenges
for exact and heuristic methods, and some cases with as few as 18 cargoes
remain unsolved. To face this challenge, we propose an exact branch-and-price
(B&P) algorithm and a hybrid metaheuristic. Our exact method generates
elementary routes, but exploits decremental state-space relaxation to speed up
column generation, heuristic strong branching, as well as advanced
preprocessing and route enumeration techniques. Our metaheuristic is a
sophisticated extension of the unified hybrid genetic search. It exploits a
set-partitioning phase and uses problem-tailored variation operators to
efficiently handle all the problem characteristics. As shown in our
experimental analyses, the B&P optimally solves 239/240 existing instances
within one hour. Scalability experiments on even larger problems demonstrate
that it can optimally solve problems with around 60 ships and 200 cargoes
(i.e., 400 pickup and delivery services) and find optimality gaps below 1.04%
on the largest cases with up to 260 cargoes. The hybrid metaheuristic
outperforms all previous heuristics and produces near-optimal solutions within
minutes. These results are noteworthy, since these instances are comparable in
size with the largest problems routinely solved by shipping companies
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