Block-simultaneous direction method of multipliers: a proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints

We introduce a generalization of the linearized Alternating Direction Method of Multipliers to optimize a real-valued function f of multiple arguments with potentially multiple constraints \(g_\circ\) on each of them. The function f may be nonconvex as long as it is convex in every argument, while the constraints \(g_\circ\) need to be convex but not smooth. If f is smooth, the proposed Block-Simultaneous Direction Method of Multipliers (bSDMM) can be interpreted as a proximal analog to inexact coordinate descent methods under constraints. Unlike alternative approaches for joint solvers of multiple-constraint problems, we do not require linear operators \({{\mathsf {L}}}\) of a constraint function \(g({{\mathsf {L}}}\ \cdot )\) to be invertible or linked between each other. bSDMM is well-suited for a range of optimization problems, in particular for data analysis, where f is the likelihood function of a model and \({{\mathsf {L}}}\) could be a transformation matrix describing e.g. finite differences or basis transforms. We apply bSDMM to the Non-negative Matrix Factorization task of a hyperspectral unmixing problem and demonstrate convergence and effectiveness of multiple constraints on both matrix factors. The algorithms are implemented in python and released as an open-source package.     

Dynamic scaling in the mesh adaptive direct search algorithm for blackbox optimization

Blackbox optimization deals with situations in which the objective function and constraints are typically computed by launching a time-consuming computer simulation. The subject of this work is the mesh adaptive direct search (mads) class of algorithms for blackbox optimization. We propose a way to dynamically scale the mesh, which is the discrete spatial structure on which mads relies, so that it automatically adapts to the characteristics of the problem to solve. Another objective of the paper is to revisit the mads method in order to ease its presentation and to reflect recent developments. This new presentation includes a nonsmooth convergence analysis. Finally, numerical tests are conducted to illustrate the efficiency of the dynamic scaling, both on academic test problems and on a supersonic business jet design problem.     

Linear complexity and correlation of a class of binary cyclotomic sequences

Let $p_1,p_2,\ldots ,p_n$ be distinct odd primes and let $e_1,e_2,\ldots ,e_n$ be positive integers. Based on cyclotomic classes proposed by Ding and Helleseth (Finite Fields Appl 4:140–166, 1998), a binary cyclotomic sequence of period $p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}$ is defined and denoted by $\underline{s_\Upsilon }$ . The linear complexity of $\underline{s_\Upsilon }$ is determined and is proved to be greater than or equal to $(p_1^{e_{1}}p_{2}^{e_{2}}\ldots p_{n}^{e_n}-1)/2$ . The autocorrelation function of $\underline{s_\Upsilon }$ is explicitly computed. Let $\ell \in \{1,2,\ldots ,n\}$ . We also explicitly compute the crosscorrelation function of $\underline{s_\Upsilon }$ and the Legendre sequence $\underline{L_{p_\ell }}$ with respect to $p_\ell $ . It is shown that $\underline{s_\Upsilon }$ and $\underline{L_{p_\ell }}$ have two-level or three-level crosscorrelation, and all their two-level crosscorrelation functions are determined.     

Log-symmetric regression models under the presence of non-informative left- or right-censored observations

In this paper, an extension to allow the presence of non-informative left- or right-censored observations in log-symmetric regression models is addressed. Under such models, the log-lifetime distribution belongs to the symmetric class and its location and scale parameters are described by semi-parametric functions of explanatory variables, whose nonparametric components are approximated using natural cubic splines or P-splines. An iterative process of parameter estimation by the maximum penalized likelihood method is presented. The large sample properties of the maximum penalized likelihood estimators are studied analytically and by simulation experiments. Diagnostic methods such as deviance-type residuals and local influence measures are derived. The package ssym, which includes an implementation in the computational environment R of the methodology addressed in this paper, is also discussed. The proposed methodology is illustrated by the analysis of a real data set.     

A robust cellular manufacturing system design for dynamic part population using a genetic algorithm

As changing conditions prevail in the manufacturing environment, the design of cellular manufacturing systems, which involves the formation of part families and machine cells, is difficult. This is due to the fact that the machines need to be relocated as per the requirements if adaptive designs are used. This paper presents a new approach (robust design) for forming part families and machine cells, which can handle all the changes in demands and product mixes without any relocations. This method suggests fixed machine cells for the dynamic nature of the production environment by considering a multi-period forecast of product mix and demand. A genetic algorithm based solution procedure is adopted to solve the problem. The results thus obtained were compared with the adaptive design proposed by Wicks and Reasor (1999 Wicks, EM and Reasor, RJ. 1999. Designing cellular manufacturing systems with dynamic part populations. IIE Trans., 31: 11–20. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). It is found that the robust design performs better than the adaptive design for the problems attempted.     

Derivative-free global ship design optimization using global/local hybridization of the DIRECT algorithm

The application of global/local hybrid DIRECT algorithms to the simulation-based hull form optimization of a military vessel is presented, aimed at the reduction of the resistance in calm water. The specific features of the black-box-type objective function make the problem suitable for the application of DIRECT-type algorithms. The objective function is given by numerical iterative procedures, which could lead to inaccurate derivative calculations. In addition, the presence of local minima cannot be excluded a priori. The algorithms proposed (namely DIRMIN and DIRMIN-2) are hybridizations of the classic DIRECT algorithm, with deterministic derivative-free local searches. The algorithms’ performances are first assessed on a set of test problems, and then applied to the ship optimization application. The numerical results show that the local hybridization of the DIRECT algorithm has beneficial effects on the overall computational cost and on the efficiency of the simulation-based optimization procedure.     

Exact Analysis of a One-Dimensional Weakly Repulsive Bose-Fermi Mixture

We study a one-dimensional system of Bose–Fermi mixture with repulsive $\delta $ -function interactions using the nested Bethe ansatz method. This system is integrable when the masses of bonsons and fermions are equal and the interactions between Bose–Bose and Bose–Fermi particles are equal. By use of the power series expansion method, the Surtherland integral equation, which describes the ground state properties, is solved analytically in the weak coupling regime. Physical quantities such as the ground state energy, the sound velocity, and the chemical potential are explicitly expressed in terms of a dimensionless interaction parameter $\gamma =c/D$ and boson fraction $\alpha =N_{b}/N$ , where $c$ is the interaction strength, $D$ is the number density, $N_{b}$ is the number of bosons, and N is the total number of particles.     

A membership function approach for aggregate production planning problems in fuzzy environments

This paper proposes a novel approach for the aggregate production planning (APP) problem with fuzzy parameters. Different from the results of previous studies, in this paper the membership function of the fuzzy minimal total cost is constructed based on Zadeh's extension principle and fuzzy solutions are provided. A pair of mathematical programs parameterised by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy total cost at α. By enumerating different values of α, the membership function of the fuzzy total cost is constructed. To illustrate the validity of the proposed approach, the example studied by Lai and Hwang (1992 Lai, YJ and Hwang, CL. 1992. Fuzzy mathematical programming: methods and applications, Berlin: Springer.  [Google Scholar]) using Chanas's approach is investigated. Since the objective value is expressed by a membership function rather than by a crisp value, the proposed approach can represent APP systems more accurately, thus obtained solutions which contain more information can offer more chance to achieve the feasible disaggregate plan, and it is beneficial to the decision-maker in practical applications. The proposed approach can also be applied to APP problems with other characteristics.     

On transitive parallelisms of PG(3,4)

A parallelism in $PG(n,q)$ P G ( n , q ) is transitive if it has an automorphism group which is transitive on the spreads. A parallelism is regular if all its spreads are regular. In $PG(3,4)$ P G ( 3 , 4 ) no examples of transitive and no regular parallelisms are known. Transitive parallelisms in $PG(3,4)$ P G ( 3 , 4 ) must have automorphisms of order 7. That is why we construct all 482 parallelisms with automorphisms of order 7 and establish that there are neither transitive, nor regular ones among them. We conclude that there are no transitive parallelisms in $PG(3,4)$ P G ( 3 , 4 ) . The investigation is computer-aided. We use GAP (Groups, Algorithms, Programming—a System for Computational Discrete Algebra) to find a subgroup of order 7 and its normalizer in the automorphism group of $PG(3,4)$ P G ( 3 , 4 ) . For all the other constructions and tests we use our own software written in C++.     

Bracelet monoids and numerical semigroups

Given positive integers \(n_1,\ldots ,n_p\), we say that a submonoid M of \(({\mathbb N},+)\) is a \((n_1,\ldots ,n_p)\)-bracelet if \(a+b+\left\{ n_1,\ldots ,n_p\right\} \subseteq M\) for every \(a,b\in M\backslash \left\{ 0\right\} \). In this note, we explicitly describe the smallest \(\left( n_1,\ldots ,n_p\right) \)-bracelet that contains a finite subset X of \({\mathbb N}\). We also present a recursive method that enables us to construct the whole set \(\mathcal B(n_1,\ldots ,n_p)=\left\{ M|M \quad \text {is a} \quad (n_1,\ldots ,n_p)\text {-bracelet}\right\} \). Finally, we study \((n_1,\ldots ,n_p)\)-bracelets that cannot be expressed as the intersection of \((n_1,\ldots , n_p)\)-bracelets properly containing it.     

Statistical Engineering and Variation Reduction

ABSTRACT

Statistical engineering as proposed by Hoerl and Snee (2010a Hoerl , R. W. , Snee , R. D. ( 2010a ). Closing the gap . Quality Progress , 43 ( 5 ): 52 – 53 . [Google Scholar]) aims to develop a discipline devoted to better understanding how to use statistical tools to support project goals. Existing examples abound but more work is needed. We discuss the use of statistical engineering to improve problem solving—that is, reducing variation in processes—and note that this requires a series of empirical investigations where we should use information gained to help plan subsequent investigations. The systematic use of prior/existing information, especially baseline information, in problem solving is illustrated using a crossbar dimension case study. The baseline results are used to help plan and analyze all subsequent investigations both when looking for a dominant cause of the variation and when assessing a possible solution. The effective use of prior statistical information and the consequences of its use in the variation reduction context are not commonly taught and thus opportunities for more efficient problem solving are lost.     

Trisection for genus 2 curves in odd characteristic

We provide trisection (division by 3) algorithms for Jacobians of genus 2 curves over finite fields \(\mathbb {F}_q\) of odd characteristic which rely on the factorization of a polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the Galois structure of the 3-torsion subgroup. We also generalize these ideas for \(\ell \in \{5,7\}\).     

Balancing and sequencing of parallel mixed-model assembly lines

In this paper, a simulated annealing approach is developed for the parallel mixed-model assembly line balancing and model sequencing (PMMAL/BS) problem which is an extension of the parallel assembly line balancing (PALB) problem introduced by Gökçen et al. (2006 Gökçen, H and A?pak, K. 2006. A goal programming approach to simple U-line balancing problem. European Journal of Operational Research, 171(2): 577–585. [Crossref], [Web of Science ®] , [Google Scholar]). In PALB, the aim is to balance more than one assembly line together. Balancing of the lines simultaneously with a common resource is very important in terms of resource minimisation. The proposed approach maximises the line efficiency and distributes the workloads smoothly across stations. The proposed approach is illustrated with two numerical examples and its performance is tested on a set of test problems. The computational results show that the proposed approach is very effective for PMMAL/BS.     

Flowshop-scheduling problems with makespan criterion: a review

This paper is a complete survey of flowshop-scheduling problems and contributions from early works of Johnson of 1954 Johnson, SM. 1954. Optimal two- and three-stage production schedules with set-up times included. Naval Res. Logist. Quart., 1: 61–68.   to recent approaches of metaheuristics of 2004. It mainly considers a flowshop problem with a makespan criterion and it surveys some exact methods (for small size problems), constructive heuristics and developed improving metaheuristic and evolutionary approaches as well as some well-known properties and rules for this problem. Each part has a brief literature review of the contributions and a glimpse of that approach before discussing the implementation for a flowshop problem. Moreover, in the first section, a complete literature review of flowshop-related scheduling problems with different assumptions as well as contributions in solving these other aspects is considered. This paper can be seen as a reference to past contributions (particularly in n/m/p/c _max or equivalently F/prmu/c _max) for future research needs of improving and developing better approaches to flowshop-related scheduling problems.     

Relaxing high-dimensional constraints in the direct solution space method for early phase development

Early phase distributed system design can be accomplished using solution spaces that provide an interval of permissible values for each functional parameter. The feasibility property guarantees fulfillment of all design requirements for all possible realizations. Flexibility denotes the size measure of the intervals, with higher flexibility benefiting the design process. Two methods are available for solution space identification. The direct method solves a computationally cheap optimization problem. The indirect method employs a sampling approach that requires a relaxation of the feasibility property through re-formulation as a chance constraint. Even for high probabilities of fulfillment, \(P>0.99\), this results in substantial increases in flexibility, which offsets the risk of infeasibility. This work implements the chance constraint formulation into the direct method for linear constraints by showing that its problem statement can be understood as a linear robust optimization problem. Approximations of chance constraints from the literature are transferred into the context of solution spaces. From this, we derive a theoretical value for the safety parameter \(\varOmega\). A further modification is presented for use cases, where some intervals are already predetermined. A problem from vehicle safety is used to compare the modified direct and indirect methods and discuss suitable choices of \(\varOmega\). We find that the modified direct method is able to identify solution spaces with similar flexibility, while maintaining its cost advantage.     

A note on an industrial strategy for stock management in supply chains: modelling and performance evaluation

An analytic formulation of consignment stock (CS) policy has been proposed in Braglia and Zavanella (2003 Braglia M Zavanella L 2003 An industrial strategy for stock management in Supply Chains: modelling and performance evaluation International Journal of Production Research 2003 41 3793 3808  , International Journal of Production Research, 41, 3793) where an implicit analytical solution is given. In this note it is shown that this solution has properties that enable it to be developed into a completely explicit form, allowing for a joint optimization of all decision variables governing the delivery management.     

Large Area Transition Edge Sensor X-ray Microcalorimeters for Diffuse X-ray Background Studies

We are developing transition edge sensor (TES) mirocalorimeters with large area (0.72 mm \(^2\) ) absorbers to study the keV diffuse X-ray background. The goal is to develop a 2 cm \(^2\) array of 256 pixels for a sounding rocket payload. We present a pixel design which includes a Mo/Au bilayer TES coupled to a large (850 x 850 x 0.2 \(\upmu \) m \(^3\) ) gold absorber. Our simulations indicate that such a design can achieve energy resolution as good as 1.6 eV FWHM in our target bandpass of 0.05–1 keV. Additionally, thermal modelling shows that for typical gold layers, the position-dependent variation of the pulse shape over the large area of the absorber is not expected to significantly degrade this energy resolution. An array of devices will be fabricated in late 2013 to test this design.     

STEP-based feature modeller for computer-aided process planning

Feature-based modelling has been considered as an indispensable tool for integrating CAD/CAPP systems. One of its advantages over conventional geometric modelling is the ability to associate not only geometric and topological information, but also form features, tolerances, material properties and other information which may be used during the process planning. This paper presents a STEP-based feature modeller (STEP-FM) for prismatic parts. High-level three-dimensional solid features are used as the basic entities for part design. The modeller relies on three main steps; (1) selection of the part base shape and overall size, (2) selection of the features to be added to or subtracted from the part being designed, and (3) providing information needed to define feature size, position, orientation and other attributes such as surface finish, tolerances, etc. The designed part is then exported as a STEP XML data format (according to ISO 2000 International Standards Organization ISO10303 — Part 224, Mechanical Product Definition for Process Planning Using Machining Features 2000 ISO Geneva ). This file can be used to integrate into CAPP/CAM systems without using a complex feature recognition process. An object-oriented approach is used in the definition and implementation of the product model. This approach offers advantages of incremental system development and reusability. An example is given to demonstrate the application of the developed modeller.