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.

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Abstract: Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness.

A Survey of Swarm Algorithms Applied to Discrete Optimization Problems

Abstract: Most swarm intelligence algorithms were devised for continuous optimization problems. However, they have been adapted for discrete optimization as well with applications in different domains. This survey aims at providing an updated review of research of swarm intelligence algorithms for discrete optimization problems, comprising combinatorial or binary. The biological inspiration that motivated the creation of each swarm algorithm is introduced, and later, the discretization and encoding methods are used to adapt each algorithm for discrete problems. Methods are compared for different classes of problems and a critical analysis is provided, pointing to future trends.

Topology optimization using a dual method with discrete variables

Abstract: This paper deals with topology optimization of continuous structures in static linear elasticity. The problem consists in distributing a given amount of material in a specified domain modelled by a fixed finite element mesh in order to minimize the compliance. As the design variables can only take two values indicating the presence or absence of material (1 and 0), this problem is intrinsicallydiscrete. Here, it is solved by a mathematical programming method working in the dual space and specially designed to handle discrete variables. This method is very wellsuited to topology optimization, because it is particularly efficient for problems with a large number of variables and a small number of constraints. To ensure the existence of a solution, the perimeter of the solid parts is bounded. A computer program including analysis and optimization has been developed. As it is specialized for regular meshes, the computational time is drastically reduced. Some classical 2-D and new 3-D problems are solved, with up to 30,000 design variables. Extensions to multiple load cases and to gravity loads are also examined.

Method and device for placement of transmitters in wireless networks

Abstract: From layout and attenuation data of an area and an initial guess, the placement and power levels of base stations and the resulting attenuation and base station ranges are calculated based on any chosen propagation model. A cost function is calculated which indicates the merit of the initial guess placement. The cost function is a function of the transmitter locations and power levels and can be calculated as a linear combination of the uncovered and interference areas. Other cost functions can also be considered. The cost function is optimized by one of several optimization methods to give the optimal base station placement. The optimization can be continuous or discrete. Continuous optimization methods include the modified steepest descent method and the downhill simplex method, while discrete optimization methods include the Hopfield neural network method. The optimization can be performed several times for different initial guess placements to achieve a global, rather than simply local, optimization. The entire optimization process is packaged in the form of an interactive software tool that permits the designer to steer and adjust the solution according to any criteria that the designer may choose.

Distributed neural dynamics algorithms for optimization of large steel structures

Abstract: Optimization of large structures consisting of thousands of members subjected to the highly non­ linear constraints of the actual commonly used design codes, such as the American Institute of Steel Construction (AISC), Allowable Stress Design (ASD), or Load and Resistance Factor Design (LRFD) specifications (AISC 1989, 1994), requires high-performance computing resources. We have previously developed parallel optimi­ zation algorithms on shared memory multiprocessors where a few powerful processors are connected to a single shared memory. In contrast, in a distributed memory machine, a relatively large number of microprocessors are connected to their own locally distributed memories without globally shared memory. In this article, we present distributed nonlinear neural dynamics algorithms for discrete optimization of large steel structures. The algo­ rithms are implemented on a recently introduced distributed memory machine, the CRAY T3D, and applied to the minimum weight design of three large space steel structures ranging in size from 1,310 to 8,904 members. The stability, convergence, and efficiency of the algorithms are demonstrated through examples. For an 8,904­ member structure, a high parallel processing efficiency of 94% is achieved using a 32-processor configuration.