321,363 research outputs found
Enumerative Branching with Less Repetition
We can compactly represent large sets of solutions for problems with discrete decision variables by using decision diagrams. With them, we can efficiently identify optimal solutions for different objective functions. In fact, a decision diagram naturally arises from the branch-and-bound tree that we could use to enumerate these solutions if we merge nodes from which the same solutions are obtained on the remaining variables. However, we would like to avoid the repetitive work of finding the same solutions from branching on different nodes at the same level of that tree. Instead, we would like to explore just one of these equivalent nodes and then infer that the same solutions would have been found if we explored other nodes. In this work, we show how to identify such equivalences—and thus directly construct a reduced decision diagram—in integer programs where the left-hand sides of all constraints consist of additively separable functions. First, we extend an existing result regarding problems with a single linear constraint and integer coefficients. Second, we show necessary conditions with which we can isolate a single explored node as the only candidate to be equivalent to each unexplored node in problems with multiple constraints. Third, we present a sufficient condition that confirms if such a pair of nodes is indeed equivalent, and we demonstrate how to induce that condition through preprocessing. Finally, we report computational results on integer linear programming problems from the MIPLIB benchmark. Our approach often constructs smaller decision diagrams faster and with less branching
Теория выпуклых продолжений в задачах комбинаторной оптимизации
Для задач евклидовой комбинаторной оптимизации выделены классы вершинно расположенных и полиэдрально-сферических множеств, для которых обобщены результаты теории выпуклых продолжений. На
основе теорем о существовании дифференцируемых выпуклых продолжений для вершинно расположенных
множеств сформулирована эквивалентная задача дискретной оптимизации выпуклой функции при выпуклых функциональных ограничениях. Описаны свойства релаксационных задач как задач выпуклого программирования.Для задач евклідової комбінаторної оптимізації виділені класи вершинно розташованих і поліедрально-
сферичних множин, для яких узагальнено результати теорії опуклих продовжень. З використанням теорем про існування диференційованих опуклих продовжень для вершинно розташованих множин сформульовано еквівалентну задачу дискретної оптимізації опуклої функції при опуклих функціональних обмеженнях. Описано властивості релаксаційних задач опуклого програмування, що виникають.The results of the theory of convex extensions for vertex located and polyhedral-spherical sets are summarized.
In view of the theorems of existence of convex differentiable extensions, the problem is equivalent to a discrete
optimization problem of convex functions under convex functional constraints. The convex nonlinear relaxation
problem is considered
Ergodic Control and Polyhedral approaches to PageRank Optimization
We study a general class of PageRank optimization problems which consist in
finding an optimal outlink strategy for a web site subject to design
constraints. We consider both a continuous problem, in which one can choose the
intensity of a link, and a discrete one, in which in each page, there are
obligatory links, facultative links and forbidden links. We show that the
continuous problem, as well as its discrete variant when there are no
constraints coupling different pages, can both be modeled by constrained Markov
decision processes with ergodic reward, in which the webmaster determines the
transition probabilities of websurfers. Although the number of actions turns
out to be exponential, we show that an associated polytope of transition
measures has a concise representation, from which we deduce that the continuous
problem is solvable in polynomial time, and that the same is true for the
discrete problem when there are no coupling constraints. We also provide
efficient algorithms, adapted to very large networks. Then, we investigate the
qualitative features of optimal outlink strategies, and identify in particular
assumptions under which there exists a "master" page to which all controlled
pages should point. We report numerical results on fragments of the real web
graph.Comment: 39 page
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
Constraint Programming (CP) has proved an effective paradigm to model and
solve difficult combinatorial satisfaction and optimisation problems from
disparate domains. Many such problems arising from the commercial world are
permeated by data uncertainty. Existing CP approaches that accommodate
uncertainty are less suited to uncertainty arising due to incomplete and
erroneous data, because they do not build reliable models and solutions
guaranteed to address the user's genuine problem as she perceives it. Other
fields such as reliable computation offer combinations of models and associated
methods to handle these types of uncertain data, but lack an expressive
framework characterising the resolution methodology independently of the model.
We present a unifying framework that extends the CP formalism in both model
and solutions, to tackle ill-defined combinatorial problems with incomplete or
erroneous data. The certainty closure framework brings together modelling and
solving methodologies from different fields into the CP paradigm to provide
reliable and efficient approches for uncertain constraint problems. We
demonstrate the applicability of the framework on a case study in network
diagnosis. We define resolution forms that give generic templates, and their
associated operational semantics, to derive practical solution methods for
reliable solutions.Comment: Revised versio
Discrete time optimal control with frequency constraints for non-smooth systems
We present a Pontryagin maximum principle for discrete time optimal control
problems with (a) pointwise constraints on the control actions and the states,
(b) frequency constraints on the control and the state trajectories, and (c)
nonsmooth dynamical systems. Pointwise constraints on the states and the
control actions represent desired and/or physical limitations on the states and
the control values; such constraints are important and are widely present in
the optimal control literature. Constraints of the type (b), while less
standard in the literature, effectively serve the purpose of describing
important spectral properties of inertial actuators and systems. The
conjunction of constraints of the type (a) and (b) is a relatively new
phenomenon in optimal control but are important for the synthesis control
trajectories with a high degree of fidelity. The maximum principle established
here provides first order necessary conditions for optimality that serve as a
starting point for the synthesis of control trajectories corresponding to a
large class of constrained motion planning problems that have high accuracy in
a computationally tractable fashion. Moreover, the ability to handle a
reasonably large class of nonsmooth dynamical systems that arise in practice
ensures broad applicability our theory, and we include several illustrations of
our results on standard problems
Controller synthesis with very simplified linear constraints in PN model
This paper addresses the problem of forbidden states for safe Petri net
modeling discrete event systems. We present an efficient method to construct a
controller. A set of linear constraints allow forbidding the reachability of
specific states. The number of these so-called forbidden states and
consequently the number of constraints are large and lead to a large number of
control places. A systematic method for constructing very simplified controller
is offered. By using a method based on Petri nets partial invariants, maximal
permissive controllers are determined.Comment: Dependable Control of discrete Systems, Bari : Italie (2009
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