836 research outputs found
Dynamic Programming on Nominal Graphs
Many optimization problems can be naturally represented as (hyper) graphs,
where vertices correspond to variables and edges to tasks, whose cost depends
on the values of the adjacent variables. Capitalizing on the structure of the
graph, suitable dynamic programming strategies can select certain orders of
evaluation of the variables which guarantee to reach both an optimal solution
and a minimal size of the tables computed in the optimization process. In this
paper we introduce a simple algebraic specification with parallel composition
and restriction whose terms up to structural axioms are the graphs mentioned
above. In addition, free (unrestricted) vertices are labelled with variables,
and the specification includes operations of name permutation with finite
support. We show a correspondence between the well-known tree decompositions of
graphs and our terms. If an axiom of scope extension is dropped, several
(hierarchical) terms actually correspond to the same graph. A suitable
graphical structure can be found, corresponding to every hierarchical term.
Evaluating such a graphical structure in some target algebra yields a dynamic
programming strategy. If the target algebra satisfies the scope extension
axiom, then the result does not depend on the particular structure, but only on
the original graph. We apply our approach to the parking optimization problem
developed in the ASCENS e-mobility case study, in collaboration with
Volkswagen. Dynamic programming evaluations are particularly interesting for
autonomic systems, where actual behavior often consists of propagating local
knowledge to obtain global knowledge and getting it back for local decisions.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Constraint-based protocols for distributed problem solving
AbstractDistributed Problem Solving (DPS) approaches decompose problems into subproblems to be solved by interacting, cooperative software agents. Thus, DPS is suitable for solving problems characterized by many interdependencies among subproblems in the context of parallel and distributed architectures. Concurrent Constraint Programming (CCP) provides a powerful execution framework for DPS where constraints define local problem solving and the exchange of information among agents declaratively. To optimize DPS, the protocol for constraint communication must be tuned to the specific kind of DPS problem and the characteristics of the underlying system architecture. In this paper, we provide a formal framework for modeling different problems and we show how the framework applies to simple yet generalizable examples
Learning with Clustering Structure
We study supervised learning problems using clustering constraints to impose
structure on either features or samples, seeking to help both prediction and
interpretation. The problem of clustering features arises naturally in text
classification for instance, to reduce dimensionality by grouping words
together and identify synonyms. The sample clustering problem on the other
hand, applies to multiclass problems where we are allowed to make multiple
predictions and the performance of the best answer is recorded. We derive a
unified optimization formulation highlighting the common structure of these
problems and produce algorithms whose core iteration complexity amounts to a
k-means clustering step, which can be approximated efficiently. We extend these
results to combine sparsity and clustering constraints, and develop a new
projection algorithm on the set of clustered sparse vectors. We prove
convergence of our algorithms on random instances, based on a union of
subspaces interpretation of the clustering structure. Finally, we test the
robustness of our methods on artificial data sets as well as real data
extracted from movie reviews.Comment: Completely rewritten. New convergence proofs in the clustered and
sparse clustered case. New projection algorithm on sparse clustered vector
Detection of permutation symmetries in numerical constraint satisfaction problems
Technical reportThis technical report summarizes the work done by Ieva Dauzickaite during her traineeship at IRI from 2016-10-03 to 2017-03-31. The main objectives of the stay at IRI were to study detection of variable symmetries in numerical constraint satisfaction problems and to implement a method that detects such symmetries.Peer ReviewedPostprint (published version
BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization
Despite the success of neural-based combinatorial optimization methods for
end-to-end heuristic learning, out-of-distribution generalization remains a
challenge. In this paper, we present a novel formulation of Combinatorial
Optimization Problems (COPs) as Markov Decision Processes (MDPs) that
effectively leverages common symmetries of COPs to improve out-of-distribution
robustness. Starting from a direct MDP formulation of a constructive method, we
introduce a generic way to reduce the state space, based on Bisimulation
Quotienting (BQ) in MDPs. Then, for COPs with a recursive nature, we specialize
the bisimulation and show how the reduced state exploits the symmetries of
these problems and facilitates MDP solving. Our approach is principled and we
prove that an optimal policy for the proposed BQ-MDP actually solves the
associated COPs. We illustrate our approach on five classical problems: the
Euclidean and Asymmetric Traveling Salesman, Capacitated Vehicle Routing,
Orienteering and Knapsack Problems. Furthermore, for each problem, we introduce
a simple attention-based policy network for the BQ-MDPs, which we train by
imitation of (near) optimal solutions of small instances from a single
distribution. We obtain new state-of-the-art results for the five COPs on both
synthetic and realistic benchmarks. Notably, in contrast to most existing
neural approaches, our learned policies show excellent generalization
performance to much larger instances than seen during training, without any
additional search procedure
Decomposition, Reformulation, and Diving in University Course Timetabling
In many real-life optimisation problems, there are multiple interacting
components in a solution. For example, different components might specify
assignments to different kinds of resource. Often, each component is associated
with different sets of soft constraints, and so with different measures of soft
constraint violation. The goal is then to minimise a linear combination of such
measures. This paper studies an approach to such problems, which can be thought
of as multiphase exploitation of multiple objective-/value-restricted
submodels. In this approach, only one computationally difficult component of a
problem and the associated subset of objectives is considered at first. This
produces partial solutions, which define interesting neighbourhoods in the
search space of the complete problem. Often, it is possible to pick the initial
component so that variable aggregation can be performed at the first stage, and
the neighbourhoods to be explored next are guaranteed to contain feasible
solutions. Using integer programming, it is then easy to implement heuristics
producing solutions with bounds on their quality.
Our study is performed on a university course timetabling problem used in the
2007 International Timetabling Competition, also known as the Udine Course
Timetabling Problem. In the proposed heuristic, an objective-restricted
neighbourhood generator produces assignments of periods to events, with
decreasing numbers of violations of two period-related soft constraints. Those
are relaxed into assignments of events to days, which define neighbourhoods
that are easier to search with respect to all four soft constraints. Integer
programming formulations for all subproblems are given and evaluated using ILOG
CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table
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
- …