Showing papers on "Discrete optimization published in 2009"

Practical Methods of Optimization

Abstract: Preface Table of Notation Part 1: Unconstrained Optimization Introduction Structure of Methods Newton-like Methods Conjugate Direction Methods Restricted Step Methods Sums of Squares and Nonlinear Equations Part 2: Constrained Optimization Introduction Linear Programming The Theory of Constrained Optimization Quadratic Programming General Linearly Constrained Optimization Nonlinear Programming Other Optimization Problems Non-Smooth Optimization References Subject Index.

A review of recent advances in global optimization

Abstract: This paper presents an overview of the research progress in deterministic global optimization during the last decade (1998---2008). It covers the areas of twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization with grey box/nonfactorable models, and bilevel nonlinear optimization.

ClearPath: highly parallel collision avoidance for multi-agent simulation

Abstract: We present a new local collision avoidance algorithm between multiple agents for real-time simulations. Our approach extends the notion of velocity obstacles from robotics and formulates the conditions for collision free navigation as a quadratic optimization problem. We use a discrete optimization method to efficiently compute the motion of each agent. This resulting algorithm can be parallelized by exploiting data-parallelism and thread-level parallelism. The overall approach, ClearPath, is general and can robustly handle dense scenarios with tens or hundreds of thousands of heterogeneous agents in a few milli-seconds. As compared to prior collision avoidance algorithms, we observe more than an order of magnitude performance improvement.

Reliability-Based Optimization Using Evolutionary Algorithms

Abstract: Uncertainties in design variables and problem parameters are often inevitable and must be considered in an optimization task if reliable optimal solutions are sought. Besides a number of sampling techniques, there exist several mathematical approximations of a solution's reliability. These techniques are coupled in various ways with optimization in the classical reliability-based optimization field. This paper demonstrates how classical reliability-based concepts can be borrowed and modified and, with integrated single and multiobjective evolutionary algorithms, used to enhance their scope in handling uncertainties involved among decision variables and problem parameters. Three different optimization tasks are discussed in which classical reliability-based optimization procedures usually have difficulties, namely (1) reliability-based optimization problems having multiple local optima, (2) finding and revealing reliable solutions for different reliability indices simultaneously by means of a bi-criterion optimization approach, and (3) multiobjective optimization with uncertainty and specified system or component reliability values. Each of these optimization tasks is illustrated by solving a number of test problems and a well-studied automobile design problem. Results are also compared with a classical reliability-based methodology.

Hybrid discrete dynamically dimensioned search (HD-DDS) algorithm for water distribution system design optimization.

Abstract: [1] The dynamically dimensioned search (DDS) continuous global optimization algorithm by Tolson and Shoemaker (2007) is modified to solve discrete, single-objective, constrained water distribution system (WDS) design problems. The new global optimization algorithm for WDS optimization is called hybrid discrete dynamically dimensioned search (HD-DDS) and combines two local search heuristics with a discrete DDS search strategy adapted from the continuous DDS algorithm. The main advantage of the HD-DDS algorithm compared with other heuristic global optimization algorithms, such as genetic and ant colony algorithms, is that its searching capability (i.e., the ability to find near globally optimal solutions) is as good, if not better, while being significantly more computationally efficient. The algorithm's computational efficiency is due to a number of factors, including the fact that it is not a population-based algorithm and only requires computationally expensive hydraulic simulations to be conducted for a fraction of the solutions evaluated. This paper introduces and evaluates the algorithm by comparing its performance with that of three other algorithms (specific versions of the genetic algorithm, ant colony optimization, and particle swarm optimization) on four WDS case studies (21- to 454-dimensional optimization problems) on which these algorithms have been found to perform well. The results obtained indicate that the HD-DDS algorithm outperforms the state-of-the-art existing algorithms in terms of searching ability and computational efficiency. In addition, the algorithm is easier to use, as it does not require any parameter tuning and automatically adjusts its search to find good solutions given the available computational budget.

Optical flow estimation on coarse-to-fine region-trees using discrete optimization

Abstract: In this paper, we propose a new region-based method for accurate motion estimation using discrete optimization. In particular, the input image is represented as a tree of over-segmented regions and the optical flow is estimated by optimizing an energy function defined on such a region-tree using dynamic programming. To accommodate the sampling-inefficiency problem intrinsic to discrete optimization compared to the continuous optimization based methods, both spatial and solution domain coarse-to-fine (C2F) strategies are used. That is, multiple region-trees are built using different over-segmentation granularities. Starting from a global displacement label discretization, optical flow estimation on the coarser level region-tree is used for defining region-wise finer displacement samplings for finer level region-trees. Furthermore, cross-checking based occlusion detection and correction and continuous optimization are also used to improve accuracy. Extensive experiments using the Middlebury benchmark datasets have shown that our proposed method can produce top-ranking results.

A classification of methods for distributed system optimization based on formulation structure

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.

Design Optimization With Discrete and Continuous Variables of Aleatory and Epistemic Uncertainties

Abstract: Reliability based design optimization has received increasing attention for satisfying high requirements on reliability and safety in structure design. However, in practical engineering design, there are both continuous and discrete design variables. Moreover, both aleatory uncertainty and epistemic uncertainty may associate with design variables. This paper proposes the formulation of random/fuzzy continuous/discrete variables design optimization (RFCDV-DO) and two different approaches for uncertainty analysis (probability/possibility analysis). A method named random/fuzzy sequential optimization and reliability assessment is proposed based on the idea of sequential optimization and reliability assessment to improve efficiency in solving RFCDV-DO problems. An engineering design problem is utilized to demonstrate the approaches and the efficiency of the proposed method.

An Active-set Algorithm for Discrete Network Design Problems

Abstract: In this paper, we formulate a discrete network design problem as a mathematical program with complementarity constraints and propose an active set algorithm to solve the problem. Each complementarity constraint requires the product of a pair of nonnegative variables to be zero. Instead of dealing with this type of constraints directly, the proposed algorithm assigns one of the nonnegative variables in each pair a value of zero. Doing so reduces the design problem to a regular nonlinear program. Using the multipliers associated with the constraints forcing nonnegative variables to be zero, the algorithm then constructs and solves binary knapsack problems to make changes to the zero-value assignments in order to improve the system delay. Numerical experiments with data from networks in the literature indicate that the algorithm is effective and has the potential for solving larger network design problems.

Global optimization of truss topology with discrete bar areas--Part II: Implementation and numerical results

Abstract: A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume The design variables describe the cross sectional areas of the bars This class of problems is well-studied for continuous bar areas We consider here the difficult situation that the truss must be built from pre-produced bars with given areas This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained We focus on the implementation details but also establish finite convergence of the branch-and-bound method The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function This is one reason for making the resulting branch-and-bound method very efficient The paper closes with several large-scale numerical examples These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization

Multi-objective optimization of engineering systems using game theory and particle swarm optimization

Abstract: This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.

Discrete Concepts in PDE Constrained Optimization

Abstract: In the present chapter we give an introduction to discrete concepts for optimization problems with PDE constraints. As models for the state we consider elliptic and parabolic PDEs which are well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss and compare the approaches First discretize, then optimize and First optimize, then discretize, and introduce a variational discrete concept which avoids explicit discretization of the controls. We investigate problems with general constraints on the control, and also consider pointwise bounds on the state, and on the gradient of the state. We present error analysis for the variational discrete concept and accomplish our analytical findings with numerical examples which confirm our analytical results.

Optimization Theory and Algorithms

Abstract: In this book, we have considered features which are defined by some optimality properties, such as maximum sparseness. In this chapter, we briefly explain how those optimal features can be numerically computed. The solutions are based either on general-purpose optimization methods, such as gradient methods, or specific tailor-made methods such as fixed-point algorithms.

GlobSol user guide

Abstract: We explain the installation and use of the GlobSol package for mathematically rigorous bounds on all solutions to constrained and unconstrained global optimization problems, as well as non-linear systems of equations. This document should be of use both to people with optimization problems to solve and to people incorporating GlobSol's components into other systems or providing interfaces to GlobSol.

Optimization of aircraft structural components by using nature-inspired algorithms and multi-fidelity approximations

Abstract: In this work, a flat pressure bulkhead reinforced by an array of beams is designed using a suite of heuristic optimization methods (Ant Colony Optimization, Genetic Algorithms, Particle Swarm Optimization and LifeCycle Optimization), and the Nelder-Mead simplex direct search method. The compromise between numerical performance and computational cost is addressed, calling for inexpensive, yet accurate analysis procedures. At this point, variable fidelity is proposed as a tradeoff solution. The difference between the low-fidelity and high-fidelity models at several points is used to fit a surrogate that corrects the low-fidelity model at other points. This allows faster linear analyses during the optimization; whilst a reduced set of expensive non-linear analyses are run "off-line," enhancing the linear results according to the physics of the structure. Numerical results report the success of the proposed methodology when applied to aircraft structural components. The main conclusions of the work are (i) the variable fidelity approach enabled the use of intensive computing heuristic optimization techniques; and (ii) this framework succeeded in exploring the design space, providing good initial designs for classical optimization techniques. The final design is obtained when validating the candidate solutions issued from both heuristic and classical optimization. Then, the best design can be chosen by direct comparison of the high-fidelity responses.

A new efficient algorithm for optimal assignment of smart weapons to targets

Abstract: In the modern battlefields smart weapons inherently rely on the sensors. The benefit of assigning a given weapon to a target often depends on the pre-assigned sensor. In this paper we present an efficient algorithm to optimally assign sensors and weapons to targets. This algorithm is derived from the well-known auction algorithm, and it is named as Swt-opt. We prove that Swt-opt converges to an optimal solution.

Wireless sensor deployment for 3D coverage with constraints

Abstract: We consider the problem of deploying wireless sensors in a three dimensional space to achieve a desired degree of coverage, while minimizing the number of sensors placed. Typical sensor deployment scenarios impose constraints on possible locations of the sensors, and on the desired coverage, but currently there is no unified way to handle these constraints in optimizing the number of sensors placed. We present a novel approach called discretization which allows us to cast the sensor deployment problem as a discrete optimization problem, and hence apply well-understood and flexible discrete optimization techniques for sensor deployment. Our results show that this approach yields solutions that nearly minimize the number of sensors used, while providing a high degree of coverage. Further, unlike typical approaches to sensor deployment, where 3D coverage is significantly more complex than 2D coverage, discretization is equally easy to apply for 2D as well as 3D coverage.

Simulation-Based Discrete Optimization of Stochastic Discrete Event Systems Subject to Non Closed-Form Constraints

Abstract: This technical note addresses the discrete optimization of stochastic discrete event systems for which both the performance function and the constraint function are not known but can be evaluated by simulation and the solution space is either finite or unbounded. Our method is based on random search in a neighborhood structure called the most promising area proposed in and a moving observation area. The simulation budget is allocated dynamically to promising solutions. Simulation-based constraints are taken into account in an augmented performance function via an increasing penalty factor. We prove that under some assumptions, the algorithm converges with probability 1 to a set of true local optimal solutions. These assumptions are restrictive and difficult to verify but we hope that the encouraging numerical results would motivate future research exploiting ideas of this technical note.

Discrete Optimization in Public Transportation

Abstract: The mathematical treatment of planning problems in public transit has made significant advances in the last decade. Among others, the classical problems of vehicle and crew scheduling can nowadays be solved on a routine basis using combinatorial optimization methods. This is not yet the case for problems that pertain to the design of public transit networks, and for the problems of operations control that address the implementation of a schedule in the presence of disturbances. The article gives a sketch of the state and important developments in these areas, and it addresses important challenges. The vision is that mathematical tools of computer aided scheduling (CAS) will soon play a similar role in the design and operation of public transport systems as CAD systems in manufacturing.

Discrete tracking of parametrized curves

Abstract: A novel scheme for deformable tracking of curvilinear structures in image sequences is presented. The approach is based on B-spline snakes defined by a set of control points whose optimal configuration is determined through efficient discrete optimization. Each control point is associated with a discrete random variable in a MAP-MRF formulation where a set of labels captures the deformation space. In such a context, generic terms are encoded within this MRF in the form of pairwise potentials. The use of pairwise potentials along with the B-spline representation offers nearly perfect approximation of the continuous domain. Efficient linear programming is considered to recover the approximate optimal solution. The method is successfully applied to the tracking of guide-wires in fluoroscopic X-ray sequences of several hundred frames which requires extremely robust techniques.

Mixed–discrete structural optimization using a rank-niche evolution strategy

Abstract: In this study, the evolution strategy, which is one of the evolutionary algorithms, is modified to solve mixed–discrete optimization problems. Three approaches are proposed for handling discrete variables. The first approach is to treat discrete variables as continuous variables and replace the latter with discrete variables that are closest to the continuous variables. The second approach is to compress the difference between discrete variables so that discrete variables far away from the current value will have a higher probability of being selected. The third approach is to represent the discrete variables as integers. As a result, the difference between neighbouring discrete variables becomes equal. This also increases the probability of selection of discrete variables far away from the current value through the mutation operation. Five examples are tested representing single objective, multi-objective, unconstrained, constrained, pure discrete and mixed–discrete variable problems. From the results ob...