Hands-on tutorial for parallel computing with R

Due to the increasing availability of powerful hardware resources, parallel computing is becoming an important issue, as a noticeable speedup may be achieved. The statistical programming language R allows for parallel computing on computer clusters as well as multicore systems through several packages. This tutorial gives a short, practical overview of four, in view of the authors, important packages for parallel computing in R, namely multicore, snow, snowfall and nws. First, the general principle of parallelizing simple tasks is briefly illustrated based on a statistical cross-validation example. Afterwards, the usage of each of the introduced packages is being demonstrated on the example. Furthermore, we address some specific features of the packages and provide guidance for selecting an adequate package for the computing environment at hand.     

Benchmarking nonlinear optimization software in technical computing environments

Our strategic objective is to develop a broadly categorized, expandable collection of test problems, to support the benchmarking of nonlinear optimization software packages in integrated technical computing environments (ITCEs). ITCEs—such as Maple, Mathematica, and MATLAB—support concise, modular and scalable model development: their built-in documentation and visualization features can be put to good use also in test model selection and analysis. ITCEs support the flexible inclusion of both new models and general-purpose solver engines for future studies. Within this broad context, in this article we review a collection of global optimization problems coded in Mathematica, and present illustrative and summarized numerical results obtained using the MathOptimizer Professional software package.     

OPTAC: a portable software package for analyzing and comparing optimization methods by visualization

A software package for analyzing and comparing optimization methods is presented. This package displays, using different colors, the regions of convergence to the minima of a given function for various optimization methods. It displays also the rate of their convergence as well as the regions of divergence of these methods. Moreover, this package gives quantitative information regarding the total convergence area in a specific domain for various minima.     

Risk-sensitive portfolio optimization on infinite time horizon

We consider a continuous time portfolio optimization problems on an infinite time horizon for a factor model, recently treated by Bielecki and Pliska ["Risk-sensitive dynamic asset management", Appl. Math. Optim. , 39 (1990) 337-360], where the mean returns of individual securities or asset categories are explicitly affected by economic factors. The factors are assumed to be Gaussian processes. We see new features in constructing optimal strategies for risk-sensitive criteria of the portfolio optimization on an infinite time horizon, which are obtained from the solutions of matrix Riccati equations.     

On implementing a primal-dual interior-point method for conic quadratic optimization

 Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for our implementation. Computational results are also presented to document that the implementation can solve very large problems robustly and efficiently. Received: November 18, 2000 / Accepted: January 18, 2001 Published online: September 27, 2002 Key Words. conic optimization – interior-point methods – large-scale implementation     

Globally optimal solutions of max–min systems

A variety of problems in computer science, operations research, control theory, etc., can be modeled as non-linear and non-differentiable max–min systems. This paper introduces the global optimization into such systems. The criteria for the existence and uniqueness of the globally optimal solutions are established using the high matrix, optimal max-only projection set and k ^s -control vector of max–min functions. It is also shown that the global optimization can be accomplished through the partial max-only projection representation with algebraic and combinatorial features. The methods are constructive and lead to an algorithm of finding all globally optimal solutions.     

Necessary Optimality Conditions and a New Approach to Multiobjective Bilevel Optimization Problems

Multiobjective optimization problems typically have conflicting objectives, and a gain in one objective very often is an expense in another. Using the concept of Pareto optimality, we investigate a multiobjective bilevel optimization problem (say, P). Our approach consists of proving that P is locally equivalent to a single level optimization problem, where the nonsmooth Mangasarian–Fromovitz constraint qualification may hold at any feasible solution. With the help of a special scalarization function introduced in optimization by Hiriart–Urruty, we convert our single level optimization problem into another problem and give necessary optimality conditions for the initial multiobjective bilevel optimization problem P.     

Neurodynamical Optimization

Dynamical (or ode) system and neural network approaches for optimization have been co-existed for two decades. The main feature of the two approaches is that a continuous path starting from the initial point can be generated and eventually the path will converge to the solution. This feature is quite different from conventional optimization methods where a sequence of points, or a discrete path, is generated. Even dynamical system and neural network approaches share many common features and structures, yet a complete comparison for the two approaches has not been available. In this paper, based on a detailed study on the two approaches, a new approach, termed neurodynamical approach, is introduced. The new neurodynamical approach combines the attractive features in both dynamical (or ode) system and neural network approaches. In addition, the new approach suggests a systematic procedure and framework on how to construct a neurodynamical system for both unconstrained and constrained problems. In analyzing the stability issues of the underlying dynamical (or ode) system, the neurodynamical approach adopts a new strategy, which avoids the Lyapunov function. Under the framework of this neurodynamical approach, strong theoretical results as well as promising numerical results are obtained.     

A Lagrangian relaxation approach to the mixed-product assembly line sequencing problem: A case study of a door-lock company in Taiwan

In mixed-product assembly line sequencing, the production resources required for the assembly lines should be scheduled to minimize the overall cost and meet customer demand. In this paper, we study an assembly line sequencing problem for the door-lock industry in Taiwan and develop an integer programming formulation with realistic constraints. The complex solution space makes the resulting program difficult to solve using commercial optimization packages. Therefore, a heuristic based on the Lagrangian relaxation principle is developed to solve this problem efficiently. We evaluate the efficiency of the developed Lagrangian relaxation heuristic by comparing its solutions with those obtained using a commercial optimization package: the computational results show that the developed heuristic solves the real-world problem faster than the optimization package by almost 15 times in CPU time at a comparable solution quality.     

Theoretical and Empirical Properties of the Genetic Algorithm as a Numerical Optimizer

Abstract

We investigate the basic form of the genetic algorithm as an optimization technique. Its failure to maximize a simple function of a string of 50 binary variables prompts a closer study of Holland's “schema theorem” and we find the implications of this result to be much weaker than are often claimed. Further theoretical results and exact calculations for simple problems provide an understanding of how the genetic algorithm works and why it failed in our original application. We show that the algorithm can be fine tuned to succeed in that problem but only by introducing features that could cause serious difficulties in harder problems.     

The One-Shot Method in SU2

For PDE-constrained optimization tasks the one-shot approach is well-suited. The solution of the state equation is augmented with an adjoint solver to provide approximate reduced derivatives that can be used in an optimization iteration to change the design. In this paper different one-shot strategies for SU2 are implemented and tested for aerodynamic shape optimization in the open-source multi-physics package SU2 to extend the choice of optimization algorithms. These include the application of the one-shot approach when considering additional constraints apart from the state equation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementation of the finite-element method in the numerical package front for calculation of supported thin-walled structures

The application of the FRONT package for calculation of supported thin-walled structures is considered. The main focus is on the specific features of the package which permit using the finiteelement method to analyze large problems for complex structures. Issues of automatic computation are considered. The high computational efficiency of the package is illustrated by some examples.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 44–47, 1986.     

Interacting with local and remote data repositories using the stashR package

The stashR package (a Set of Tools for Administering Shared Repositories) for R implements a basic versioned key-value style database where character string keys are associated with data values. Using the S4 classes ‘localDB’ and ‘remoteDB’, and associated methods, versioned key-value databases can be either created locally on the user’s computer or accessed remotely via the Internet. The stashR package can enhance reproducible research by providing a ‘localDB’ database format for the caching of computations which can subsequently be stored on the Internet. To reproduce a particular computation, a reader can access the ‘remoteDB’ database and obtain the associated R objects. This work was supported in part by the Johns Hopkins Training Program in the Epidemiology and Biostatistics of Aging (NIA T32 AG00247) and the Faculty Innovation Fund, Johns Hopkins Bloomberg School of Public Health.     

UTV Expansion Pack: Special-purpose rank-revealing algorithms

This collection of Matlab 7.0 software supplements and complements the package UTV Tools from 1999, and includes implementations of special-purpose rank-revealing algorithms developed since the publication of the original package. We provide algorithms for computing and modifying symmetric rank-revealing VSV decompositions, we expand the algorithms for the ULLV decomposition of a matrix pair to handle interference-type problems with a rank-deficient covariance matrix, and we provide a robust and reliable Lanczos algorithm which – despite its simplicity – is able to capture all the dominant singular values of a sparse or structured matrix. These new algorithms have applications in signal processing, optimization and LSI information retrieval. AMS subject classification 65F25     

Parallel deterministic and stochastic global minimization of functions with very many minima

The optimization of three problems with high dimensionality and many local minima are investigated under five different optimization algorithms: DIRECT, simulated annealing, Spall’s SPSA algorithm, the KNITRO package, and QNSTOP, a new algorithm developed at Indiana University.     

Greedy primal-dual algorithm for dynamic resource allocation in complex networks

In Stolyar (Queueing Systems 50 (2005) 401–457) a dynamic control strategy, called greedy primal-dual (GPD) algorithm, was introduced for the problem of maximizing queueing network utility subject to stability of the queues, and was proved to be (asymptotically) optimal. (The network utility is a concave function of the average rates at which the network generates several “commodities.”) Underlying the control problem of Stolyar (Queueing Systems 50 (2005) 401–457) is a convex optimization problem subject to a set of linear constraints. In this paper we introduce a generalized GPD algorithm, which applies to the network control problem with additional convex (possibly non-linear) constraints on the average commodity rates. The underlying optimization problem in this case is a convex problem subject to convex constraints. We prove asymptotic optimality of the generalized GPD algorithm. We illustrate key features and applications of the algorithm on simple examples. AMS Subject Classifications: 90B15 · 90C25 · 60K25 · 68M12     

Weak Subdifferential of Set-Valued Mappings

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.     

A computer software package for the simulation and optimization of an industrial ammonia converter based on a rigorous heterogeneous model

A computer package for the modelling, simulation and optimization of an ammonia reactor is developed for use on personal computers. The package is easy to use with user friendly interface utilizing the graphical capabilities of the personal computer. The model used in the package is a heterogeneous model, which has been checked against an industrial reactor having three adiabatic beds with interstage cooling, and gave excellent agreement with the industrial data. The two-point boundary value differential equation for the catalyst particles is solved using the orthogonal collocation method and three internal collocation points are necessary to obtain accurate results. The model is used to investigate the optimal behavior of the reactor.     

The Discrete Wavelet Transform in S

Abstract

The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one- or two-dimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The package's wavelet transform “engine” is written in C for speed and the object-oriented functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the wavethresh software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and wavethresh includes functions to perform thresholding according to methods in the literature.     

OPTIMALITY CONDITIONS AND APPROXIMATE OPTIMALITY CONDITIONS IN LOCALLY LIPSCHITZ VECTOR OPTIMIZATION

Abstract

In this paper, we study constrained locally Lipschitz vector optimization problems in which the objective and constraint spaces are Hilbert spaces, the decision space is a Banach space, the dominating cone and the constraint cone may be with empty interior. Necessary optimality conditions for this type of optimization problems are derived. A sufficient condition for the existence of approximate efficient solutions to a general vector optimization problem is presented. Necessary conditions for approximate efficient solutions to a constrained locally Lipschitz optimization problem is obtained.