Topic
Discrete optimization
About: Discrete optimization is a research topic. Over the lifetime, 4598 publications have been published within this topic receiving 158297 citations. The topic is also known as: discrete optimisation.
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TL;DR: In this article, the authors consider and extend MPEC formulations for the optimization of a class of hybrid dynamic models, where the differential states remain continuous over time, and particular care is required in the formulation in order to preserve smoothness properties of the dynamic system.
66 citations
TL;DR: In this article, an interior penalty function is used to convert the original constrained problem into an unconstrained parametric problem, and then the search for the optimal solution to the parametric problems is based on a discrete direction gradient.
Abstract: A new method for solving discrete structural optimization problems is presented. An interior penalty function is used to convert the original constrained problem into an unconstrained parametric problem. Then the search for the optimal solution to the parametric problem is based on a discrete direction gradient. Solving an appropriate sequence of these unconstrained parametric problems is equivalent to solving the original constrained optimization problem. This method is illustrated first on a small reinforced concrete problem, and then to the design of steel building frames which are made up of standard sections. Results for a one-story four-bay unsymmetrical frame and an eight-story three-bay symmetrical frame are described.
66 citations
TL;DR: A dynamic domain decomposition‐based parallel strategy for combined finite/discrete element analysis of multi‐fracturing solids and discrete systems and two graph representation models for discrete objects in contact are proposed which lay the foundation of the current development.
Abstract: This paper outlines a dynamic domain decomposition‐based parallel strategy for combined finite/discrete element analysis of multi‐fracturing solids and discrete systems. Attention is focused on the parallelised interaction detection between discrete objects. Two graph representation models for discrete objects in contact are proposed which lay the foundation of the current development. In addition, a load imbalance detection and re‐balancing scheme is also suggested to enhance the parallel performance. Finally, numerical examples are provided to illustrate the parallel performance achieved with the current implementation.
65 citations
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TL;DR: In this paper, a strong relation between the discrete problem and its continuous relaxation, obtained through the extension by expectation of the submodular function, was established, and the best known approximation ratio for the problem was improved to a polynomial size domain.
Abstract: Submodular maximization generalizes many fundamental problems in discrete optimization, including Max-Cut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks.
In this paper we consider the problem of maximizing any submodular function subject to $d$ knapsack constraints, where $d$ is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through {\em extension by expectation} of the submodular function. Formally, we show that, for any non-negative submodular function, an $\alpha$-approximation algorithm for the continuous relaxation implies a randomized $(\alpha - \eps)$-approximation algorithm for the discrete problem. We use this relation to improve the best known approximation ratio for the problem to $1/4- \eps$, for any $\eps > 0$, and to obtain a nearly optimal $(1-e^{-1}-\eps)-$approximation ratio for the monotone case, for any $\eps>0$. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.
65 citations
TL;DR: This paper presents a classification of formulations for distributed system optimization based on formulation structure, and identifies nested and alternating formulations, which play a crucial role in the theoretical and computational properties of distributed optimization methods.
Abstract: This paper presents a classification of for- mulations for distributed system optimization based on formulation structure. Two main classes are identi- fied: nested formulations and alternating formulations. Nested formulations are bilevel programming problems where optimization subproblems are nested in the func- tions of a coordinating master problem. Alternating formulations iterate between solving a master problem and disciplinary subproblems in a sequential scheme. Methods included in the former class are collaborative optimization and BLISS2000. The latter class includes concurrent subspace optimization, analytical target cas- cading, and augmented Lagrangian coordination. Al- though the distinction between nested and alternating formulations has not been made in earlier comparisons, it plays a crucial role in the theoretical and computa- tional properties of distributed optimization methods. The most prominent general characteristics for each class are discussed in more detail, providing valuable insights for the theoretical analysis and further devel- opment of distributed optimization methods.
65 citations