Latter Class Of Problems Research Articles

A survey on Benders Decomposition methods applied to Assembly Line Balancing Problems

The assembly line balancing problem (ALBP) assigns tasks to (work)stations to manufacture products. It divides the station loads as evenly as possible since any bottleneck defines the production rate of a flow shop system. The sum of task processing times represents the station load in the classical Simple Assembly Line Balancing Problem (SALBP). Thus, the most loaded station imposes a lower bound on the line's cycle time. However, the simple sum of processing times is only valid under several assumptions. For instance, more realistic ALBPs may contain stochastic data, multiple workers per station, or depend on the production sequence. Hence, the station load computation involves further scheduling decisions for this latter class of problems. In order to tackle these ALBPs with practical extensions, several literature contributions have recently employed Benders decomposition (BD) algorithms. The BD structure allows a division of the formulation into two or more levels. More specifically, this framework permits the decoupling of task assignment decisions and cycle time assessments. In the field of ALBPs, various stochastic and sequence-dependent problems have successfully applied this approach. This paper provides a literature review of these methods, the different formulations, implementation details, and improvement ideas.

A unified FFT-based approach to maximum assignment problems related to transitive finite group actions

This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The “simplest” nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach (SODA, 2010) to the exact solution of the asymmetric maximum quadratic assignment problem.

On Radically Expanding the Landscape of Potential Applications for Automated-Proof Methods

In this paper, we examine the potential of optimization-based computer-assisted proof methods to be applied much more widely than commonly recognized by engineers and computer scientists. More specifically, we contend that there are vast opportunities to derive valuable mathematical results and properties that may be narrow in scope, such as in highly specialized engineering control applications, that are presently overlooked, because they have characteristics atypical of those that are conventionally studied in the areas of pure and applied mathematics. As a concrete example, we demonstrate use of sum-of-squares (SOS) optimization for certifying polynomial nonnegativity as a part of a proposed dimension-pinning strategy to prove that the inverse of the relative gain array (RGA) of a d-dimensional positive-definite matrix is doubly stochastic for \(d\le 4\). However, it is not specifically this result and solution method that are of principal interest in this paper but rather how they illustrate the relevance of optimization-based proof techniques to engineering system design more broadly. We believe that our paper is the first to explicitly emphasize the fundamental distinction between methods that can be applied to prove results/properties over a fixed number of dimensions versus those that hold generally. The latter class of problems is the conventional domain of mathematicians, but the former is what we propose to be a fertile and largely unrecognized class of problems that are amenable to automated-proof technologies, e.g., as we demonstrate using our novel dimension-pinning approach.

Computing B-Stationary Points of Nonsmooth DC Programs

Motivated by a class of applied problems arising from physical layer based security in a digital communication system, in particular, by a secrecy sum-rate maximization problem, this paper studies a nonsmooth, difference-of-convex (dc) minimization problem. The contributions of this paper are (i) clarify several kinds of stationary solutions and their relations; (ii) develop and establish the convergence of a novel algorithm for computing a d-stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions; (iii) extend the algorithm in several directions including a randomized choice of the subproblems that could help the practical convergence of the algorithm, a distributed penalty approach for problems whose objective functions are sums of dc functions, and problems with a specially structured (nonconvex) dc constraint. For the latter class of problems, a pointwise Slater constraint qualification is introduced that facilitates the verification and computation of a B(ouligand)-stationary point.

A metadynamics study of the fcc–bcc phase transition in Xenon at high pressure and temperature

To theoretically find a stable solid phase is not a trivial task even at 0K. The difficulties multiply at high temperature (T) because even more elaborate crystal structure prediction methods fail in the vicinity of the melting transition. Moreover, if the submelting phase is dynamically unstable at low T some methods cannot be applied at all. The method of metadynamics allows finding local minima of Gibbs free energy without additional simplifications. However, so far this method has been mainly used for study of pressure-induced solid–solid phase transitions and not in searching for T-induced ones. Here we study the applicability of the technique to the latter class of problems as well as to the approximate determination of the transition temperature. We apply the metadynamics method to study the solid–solid phase transition in Xe described by the Buckingham potential at high temperature and observe the fcc–bcc phase transition in a pressure–temperature range consistent with earlier results.

Enhanced parallel Differential Evolution algorithm for problems in computational systems biology

Many key problems in computational systems biology and bioinformatics can be formulated and solved using a global optimization framework. The complexity of the underlying mathematical models require the use of efficient solvers in order to obtain satisfactory results in reasonable computation times. Metaheuristics are gaining recognition in this context, with Differential Evolution (DE) as one of the most popular methods. However, for most realistic applications, like those considering parameter estimation in dynamic models, DE still requires excessive computation times.Here we consider this latter class of problems and present several enhancements to DE based on the introduction of additional algorithmic steps and the exploitation of parallelism. In particular, we propose an asynchronous parallel implementation of DE which has been extended with improved heuristics to exploit the specific structure of parameter estimation problems in computational systems biology. The proposed method is evaluated with different types of benchmarks problems: (i) black-box global optimization problems and (ii) calibration of non-linear dynamic models of biological systems, obtaining excellent results both in terms of quality of the solution and regarding speedup and scalability.

A Polyhedral Frobenius Theorem with Applications to Integer Optimization

We prove a representation theorem of projections of sets of integer points by an integer matrix $W$. Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of nonlinear integer optimization problems in variable dimension. Concretely, we aim to optimize $f(Wx)$ over a set $\mathcal{F} = P\cap \mathbb{Z}^n$, where $f$ is a nonlinear function, $P\subset \mathbb{R}^n$ is a polyhedron, and $W\in \mathbb{Z}^{d\times n}$. As a consequence of our representation theorem, we obtain a general efficient transformation from the latter class of problems to integer linear programming. Our bounds depend polynomially on various important parameters of the input data leading, among others, to first polynomial time algorithms for several classes of nonlinear optimization problems.

Adhesion and delamination boundary conditions for elastic plates with arbitrary contact shape

The adhesion of two heterogeneous, thin-walled structures is shown to be controlled by a boundary condition that balances mechanical energies with the work of adhesion at the edge of the contact zone. This boundary condition is well-known in fracture mechanics but is here rederived with plate theory and represented in a form that is easy to use. This formulation is applicable either to problems in which the contact area is a priori unknown, or in problems in which the bonded area is predefined and it is the onset of debonding that is of interest. The simplified boundary condition is shown to be very useful and simple to use in both cases, but particularly in the latter class of problems. In the case of one-dimensional or axisymmetric problems where one of the bodies is rigid, this representation is equivalent to the Moment-Discontinuity-Method (MDM) introduced by Pamp and Adams and Majidi and Adams.

A heuristic approach to solving a class of bilinear matrix inequality problems

Many canonical and modern control problems can be recast into the problem of a group of matrix inequalities. Some of them are in the form of linear matrix inequalities (LMIs), which can be solved very efficiently by the powerful LMI toolbox in Matlab, but some others are in the form of bilinear matrix inequalities. The characteristic of this latter class of problems is that when the so called “communicating variables” are fixed, the overall problem will be reduced to the problem in LMIs. Thus, how to find the communicating variables is the key to solve the whole problem. In this paper, an optimal estimate for the communicating variables is presented. We will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems. Numerical examples are provided to show the effectiveness of the proposed method.

Reconstructions and predictions of nonlinear dynamical systems: a hierarchical Bayesian approach

An attempt is made to reconstruct model nonlinear dynamical systems from scalar time series data via a hierarchical Bayesian framework. Reconstruction is performed by fitting given training data with a parameterized family of functions without overfitting. The reconstructed model dynamical systems are compared with respect to (approximated) model marginal likelihood, which is a natural Bayesian information criterion. The best model is selected with respect to this criterion and is utilized to make predictions. The results are applied to two problems: (i) chaotic time series prediction and (ii) building air-conditioning load prediction. The former is a very good class of problems for checking the abilities of prediction algorithms for at least two reasons. First, since no linear dynamical systems can admit chaotic behavior, an algorithm must capture the nonlinearities behind the time series. Second, chaotic dynamical systems are sensitive to initial conditions. More precisely, the error grows exponentially with respect to time so that crispness of capturing nonlinearities is also important. Experimental results appear to indicate that the proposed scheme can capture difficult nonlinearities behind the chaotic time series data. The latter class of problems (air conditioning load prediction) is motivated by a great amount of demand for reducing CO/sub 2/ emissions associated with electric power generation. The authors won a prediction competition using the proposed algorithm; therefore, it appears to be reasonably sound.

Geometric Complexity Theory I: An Approach to thePvs.NPand Related Problems

We suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size. Specifically, we introduce the notion of a partially stable point in a reductive-group representation, which generalizes the notion of stability in geometric invariant theory due to Mumford [Geometric Invariant Theory, Springer-Verlag, Berlin, 1965]. Then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points. We also suggest an approach to tackle the latter class of problems via construction of explicit obstructions.

The Pen and the Barn

A standard differential calculus problem involves showing that the rectangle with fixed perimeter and maximal area is a square. Textbooks often present this in terms of a farmer who wishes to build the most efficient rectangular pen with a fixed length of fencing. A variant uses one wall of a barn for all or part of one side of the pen (Figure 1). Let's take a new look at this latter class of problems, because there is an attractive way to show how the lengths of the fence and the barn wall affect the shape of the optimal pen.

FUNDAMENTAL ISSUES IN FINITE ELEMENT ANALYSES OF LOCALIZATION OF DEFORMATION

Classical continuum models, i.e. continuum models that do not incorporate an internal length scale, suffer from excessive mesh dependence when strain‐softening models are used in numerical analyses and cannot reproduce the size effect commonly observed in quasi‐brittle failure. In this contribution three different approaches will be scrutinized which may be used to remedy these two intimately related deficiencies of the classical theory, namely (i) the addition of higher‐order deformation gradients, (ii) the use of micropolar continuum models, and (iii) the addition of rate dependence. By means of a number of numerical simulations it will be investigated under which conditions these enriched continuum theories permit localization of deformation without losing ellipticity for static problems and hyperbolicity for dynamic problems. For the latter class of problems the crucial role of dispersion in wave propagation in strain‐softening media will also be highlighted.

Two-Metric Projection Methods for Constrained Optimization

This paper is concerned with the problem min $\{ f(x)\mid x \in X\} $ where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. We propose the class of methods $x_{k + 1} = P(x_k - \alpha _k g_k )$, where P denotes projection on X with respect to a Hilbert space norm $\| \cdot \|$ , $g_k $ denotes the Frechet derivative of f at $x_k $ with respect to another Hilbert space norm $\| \cdot \|_k $ on H, and $\alpha _k $ is a positive scalar stepsize. We thus remove an important restriction in the original proposal of Goldstein [1] and Levitin and Poljak [2], where the norms $\| \cdot \|$ and $\| \cdot \|_k $ must be the same. It is therefore possible to match the norm $\| \cdot \|$ with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose $\| \cdot \|_k $on the basis of approximations to the Hessian of f so as to attain a typically superlinear rate of convergence. The resulting methods are particularly attractive for large-scale problems with specially structured constraint sets such as optimal control and nonlinear multi-commodity network flow problems. The latter class of problems is discussed in some detail.

Seismic response of structures by the response spectrum method

The problems of the acceleration profile at the lower elevations of cantilever structures and the response of relatively rigid structures are explored. It is shown that the use of the conventional methods for the above problems provide very approximate results. An alternate combination of the modal responses is proposed that not only resolves the above problems but also provides better estimates of response for the complete range of structure frequencies. The procedure treats the relative and rigid body responses separately and then appropriately combines the two results. For the rigid range of frequencies (fundamental frequencies greater than about 2 Hz), the proposed procedure does not encounter any numerical difficulties because of the additive nature of the component responses; however, the application of the proposed procedure for very flexible structures causes accuracy problems since the rigid body effects tend to be subtractive from the flexural response of about equal magnitude. For this latter class of problems, the conventional approach of modal combination provides adequate results and avoids the above mentioned numerical difficulties.

Bernstein-Bézier Methods for the Computer-Aided Design of Free-Form Curves and Surfaces

The m th degree Bernstein polynomial approximation to a function ƒ defined over [0, 1] is Σ m μ =0 ƒ( μ / m ) φ μ ( s ), where the weights φ μ ( s ) are binomial density functions. The Bernstein approximations inherit many of the global characteristics of ƒ, like monotonicity and convexity, and they always are at least as “smooth” as ƒ, where “smooth” refers to the number of undulations, the total variation, and the differentiability class of ƒ. Historically, their relatively slow convergence in the L ∞ -norm has tended to discourage their use in practical applications. However, in a large class of problems the smoothness of an approximating function is of greater importance than closeness of fit. This is especially true in connection with problems of computer-aided geometric design of curves and surfaces where aesthetic criteria and the intrinsic properties of shape are major considerations. For this latter class of problems, P. Bézier of Renault has successfully exploited the properties of parametric Bernstein polynomials. The purpose of this paper is to analyze the Bézier techniques and to explore various extensions and generalizations. In a sequel, the authors consider the extension of the results contained herein to free-form curve and surface design using polynomial splines . These B-spline methods have several advantages over the techniques described in the present paper.

On Some Properties of Programming Problems in Parametric form Pertaining to Fractional Programming

This paper presents results which apply to convex programming problem in parametric form. The results secured are also related to the problem of fractional programming in a way which indicates computational possibilities for the latter class of problems. The results are extended to general non-linear programming problems with special reference to continuous criterion functions. As a particular case, the linear fractional programming problem is considered and, in conclusion, the results secured here are pointed up by reference to existing algorithms for the latter class of problems.

BENDING OF A SQUARE PLATE WITH TWO ADJACENT EDGES FREE AND THE OTHERS CLAMPED OR SIMPLY SUPPORTED

T LITERATURE abounds with applications of classical thin elastic plate theory to rectangular plates having a multitude of combinations of boundary conditions and transverse loading. By far the greatest number of these papers deal with plates having only simply-supported and/or clamped boundary conditions, which requires restrictions upon the deflection and its first or second derivatives along the edges. Considerably fewer solutions have been obtained when a free edge is involved because of the relatively difficult combination of third derivatives of deflection which is encountered in the boundary conditions. When two adjacent free edges are involved, so that a free corner is created, the problem becomes exceedingly difficult. No closed form of solution is known to exist for any of this latter class of problems. Some solutions have been obtained for cantilevered rectangular plates—i.e., three edges free and the other clamped. Finite difference solutions for various aspect ratios and loadings were presented by Holl, Barton, MacNeal, Nash, and Livesly and Birchall. Nash also solved the problem of the uniformly loaded contilevered plate having a span-to-chord ratio of 1/2 by using a form of collocation known as point matching. This method depends upon choosing deflection functions which satisfy the partial differential equation of the continuum exactly, while matching the boundary conditions at only a finite number of discrete points. Nash solved the problem twice, once using an algebraic polynomial and, again, using a hyperbolic-trigonometric series. A recent paper by Leissa and Niedenfuhr presents the solution for the uniformly loaded cantilevered square plate using two other approaches: point matching, using an algebraic-trigonometric polynomial, and a Rayleigh-Ritz minimal-energy formulation. Doubtlessly, much of the foregoing work with cantilevered plates has been the result of the need for accurate information for the design of aircraft wings. However, approximate deflections and stresses may be obtained for these problems by using elementary beam theory. In marked contrast, the rectangular plate having two adjacent edges free and the others clamped or simply supported is in no way representable by beam theory. These problems are no more difficult than the cantilevered plate, but very little has been done with them up to the present. The problem of the uniformly loaded square plate with two adjacent edges free and the others clamped was solved by Huang and Conway. This involved a skillful superposition of five problems and the partial solution of an infinite set of simultaneous equations. Yeh generalized this same problem by including the reaction due to an elastic foundation. The Rayleigh-Ritz method was employed. In the present work the writers exhibit solutions for the four problems of a square plate with two adjacent edges free and the others clamped or simply supported subjected to either a uniform transverse loading or a concentrated force at the free corner. The method of point matching is used to solve the problems and the results are compared with other known solutions for two of the cases.