Optimality and duality with generalized convexity

Hanson and Mond have given sets of necessary and sufficient conditions for optimality and duality in constrained optimization by introducing classes of generalized convex functions, called type I and type II functions. Recently, Bector defined univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, optimality and duality results for several mathematical programs are obtained combining the concepts of type I and univex functions. Examples of functions satisfying these conditions are given.     

B-vex functions

A class of functions, called b-vex functions, is introduced by relaxing the definition of convexity of a function. Both the differentiable and nondifferentiable cases are presented. Members of this class satisfy most of the basic properties of convex functions. This class forms a subset of the sets of both semistrictly quasiconvex as well as quasiconvex functions, but are not necessarily included in the class of preinvex functions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319. The authors are also grateful to an anonymous referee for the constructive criticism of the first version of the paper, to Dr. Collen Knickerbocker of St. Lawrence University, and to Mrs. Meena K. Bector for their help in sharpening Examples 2.1 and 2.2, respectively.     

Nondifferentiable multiobjective programming under generalized d-univexity

In this paper, we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak quasi invex, weak pseudoinvex and strong quasi invex functions in Aghezzaf and Hachimi [Numer. Funct. Anal. Optim. 22 (2001) 775], d-invex functions in Antczak [Europ. J. Oper. Res. 137 (2002) 28] and univex functions in Bector et al. [Univex functions and univex nonlinear programming, Proc. Admin. Sci. Assoc. Canada, 1992, p. 115]. By utilizing the new concepts, we derive a Karush–Kuhn–Tucker sufficient optimality condition and establish Mond–Weir type and general Mond–Weir type duality results for the nondifferentiable multiobjective programming problem.     

An algorithm for the removal of noise and jitter in signals and its application to picosecond electrical measurement

The accuracy of data recorded by many measurement systems is limited both by uncertainty in the measured value as well as by uncertainty in the trigger input to the system which controls when a measurement is taken. The former effect, which appears as noise on the underlying signal, is due, in part, to the sampling process and can often be reduced to an acceptable level by averaging many measurements. Noise on the trigger input gives rise to uncertainty in time between the trigger and the measurement points. The effect, known as jitter, causes a distortion of the signal which cannot be removed by averaging.We describe and analyse the effects of noise and jitter on a waveform, and an algorithm for removing, or reducing, these effects is presented. The work is motivated by an application in picosecond electrical and optoelectronic metrology where a laser pulse is measured by a system consisting of a photodiode and sampling oscilloscope. Here, since the length of the pulse is so short, perhaps only tens of picoseconds in duration, the effect of jitter is as pronounced as that of measurement noise. Results obtained by applying the algorithm to simulated data obtained from this application are presented.     

OR as Technology: Some Issues and Implications

There has been a growing acceptance of the view that OR can be more appropriately understood as a technology rather than as a science. This view has, however, not been fully developed. This paper seeks to extend this area of investigation by addressing three issues: first, to consider why it is inappropriate to see OR as a science; second, if OR is to be seen as a technology, to ask what model of a technology, the classical or the modern, should be adopted; finally, to explore what implications for understanding OR are held by the most appropriate model of OR as technology. This paper shows that valuable insights might be gained for understanding and guiding the development of OR by seeing it as a technology.     

Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability

Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical results obtained using the two approaches are compared with estimates appearing in the literature.     

Is reliability a new science? A paper from the panel session held at the 10th International Conference on Mathematical Methods in Reliability

The 10th International Conference on Mathematical Methods in Reliability, MMR 2017, held in Grenoble, France during July 3‐7, entailed a panel discussion entitled “Is reliability a new science?” with Mark Brown, Regina Liu, William Meeker, Sheldon Ross, and Nozer Singpurwalla as panelists. Bill Meeker also doubled as a chair and as a moderator of the panel. The panel discussion was spawned by the recent appearance of a book by Professor Paolo Rocchi, Docent Emeritus of IBM, titled “Reliability is a new science: Gnedenko was right” published by Springer in 2017. The panel discussion was well attended and enthusiastically received and could serve as a forerunner to other such panel discussions at future MMR conferences. This paper presents some elements from the lively debate generated by this discussion.     

线性时滞微分方程解的渐近性态

本文用一个简单的方法证明了一类一阶线性时滞微分方程解的有界性帮必有非振动解，分析了振动解的性质。这个方法也被用来讨论一阶时滞方程组和中立型微分方程，所得结果均较简明。     

Simple Concepts

To talk about simple concepts presupposes that the notion of concept has been aptly explicated. I argue that a most adequate explication should abandon the set-theoretical paradigm and use a procedural approach. Such a procedural approach is offered by Tichý´s Transparent Intensional Logic (TIL). Some main notions and principles of TIL are briefly presented, and as a result, concepts are explicated as a kind of abstract procedure. Then it can be shown that simplicity, as applied to concepts, is well definable as a property relative to conceptual systems, each of which is determined by a finite set of simple (‘primitive’) concepts. Refinement as a method of replacing simple concepts by compound concepts is defined.     

Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator

Summary We study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method. Our technique is facilitated by a recent result of Lo and Singh (1986) which establishes a strong uniform approximation of the Kaplan-Meier estimator by an average of independent random variables. (Note that the approximation is carried out on the original probability space, which should be distinguished from the Hungarian embedding approach.) Pointwise strong consistency and a law of iterated logarithm are derived, as well as the mean squared error expression and asymptotic normality, which is obtain using a more traditional method, as compared with the Hajek projection employed by Tanner and Wong (1983).     

Geometric meanings of the parameters on rational conic segments

Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.     

Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate

The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained.     

Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems

The problem of estimating the time-dependent statistical characteristics of a random dynamical system is studied under two different settings. In the first, the system dynamics is governed by a differential equation parameterized by a random parameter, while in the second, this is governed by a differential equation with an underlying parameter sequence characterized by a continuous time Markov chain. We propose, for the first time in the literature, stochastic approximation algorithms for estimating various time-dependent process characteristics of the system. In particular, we provide efficient estimators for quantities such as the mean, variance and distribution of the process at any given time as well as the joint distribution and the autocorrelation coefficient at different times.     

Task Dynamics in a College Biology Course for Prospective Elementary Teachers

This paper explores the dynamic profile of a task as interpreted by a group of six prospective elementary teachers enrolled in a college biology course. Because of various constraints, such as lack of planning time, provision of materials and equipment, and lack of previous knowledge, the assigned task shifted from the planned or intended task (as defined by the instructor before implementation and presented to the students during the field trip) to a transitional or technical task (influenced by the list of materials available and on‐site conditions) and, finally, to an enacted task (tasks actually performed by the different students).     

A curve tracing algorithm for computing the pseudospectrum

The boundary curve of the pseudospectrum of a matrix is defined as a contour line of its resolvent norm. A rather simple and efficient continuation method is presented, which determines the implicitly given curve by a prediction-correction scheme, where the correction step is accomplished by one single Newton step. Besides its efficiency the algorithm turns out to be very accurate as long as the boundary of the pseudospectrum is a smooth curve. Problems may arise at bifurcation points where the resolvent norm is not differentiable.This work was partially supported by Deutsche Forschungsgemeinschaft.     

Residue formulation of the Chern character on smooth manifolds

The Chern character of a complex vector bundle is most conveniently defined as the exponential of a curvature of a connection. It is well known that its cohomology class does not depend on the particular connection chosen. It has been shown by Quillen that a connection may be perturbed by an endomorphism of the vector bundle, such as a symbol of some elliptic differential operator. This point of view, as we intend to show, allows one to relate Chern character to a noncommutative sibling formulated by Connes and Moscovici.     

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

The optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.     

Singular systems of linear forms and non-escape of mass in the space of lattices

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.     

Optimal Release Times in Single-Stage Manufacturing Systems with Blocking: Optimal Control Perspective

This paper concerns a due-date matching problem in a single-stage manufacturing system. Given a finite sequence of jobs and their service order, and given the delivery due date of each job, the problem is to choose the jobs release (arrival) times so as to match as closely as possible their completion times to their respective due dates. The system is modelled as a deterministic single-server FIFO queue with an output buffer for storing jobs whose service is completed prior to their due dates. The output buffer has a finite capacity; when it is full, the server is being blocked. Associated with each job there is a convex cost function penalizing its earliness as well as tardiness. The due-date matching problem is cast as an optimal control problem, whose objective is to minimize the sum of the above cost functions by the choice of the jobs arrival (release) times. Time-box upper-bound and lower-bound constraints are imposed on the jobs output (delivery) times. The optimal-control setting brings to bear on the development of fast and efficient algorithms having intuitive geometric appeal and potential for online implementation.Communicated by W. B. GongResearch supported in part by the National Science Foundation under Grant ECS-9979693 and by the Georgia Tech Manufacturing Research Center under Grant B01-D06.     

Student centred school timetabling

This paper reports on the formulation of a secondary school timetabling problem as a non-linear goal program, where students freely choose their courses of study from a complete list of subjects rather than the usual restricted sets of subjects. The problem as formulated is far too large to solve by traditional optimisation methods, so it is broken down into several stages for solution by heuristics to give good timetabling schedules which are at least as good as those built by manual methods. Timetable construction using a desktop computer is reduced from weeks to hours, giving schools the opportunity to construct timetables closer to the time when student choices and teaching staff are more settled.