On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification

This paper focuses on bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. We relax the constrained domain of the lower-level problem. Then, an approximate solution of the original bilevel program can be obtained by solving this perturbed bilevel program. As the lower-level problem of the perturbed bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a mathematical program with complementarity constraints which can be solved by standard algorithms. The lower convergence properties of the constraint set mapping and the solution set mapping of the lower-level problem of the perturbed bilevel program are expanded. We show that the solutions of a sequence of the perturbed bilevel programs are convergent to that of the original bilevel program under some appropriate conditions. And this convergence result is applied to simple trilevel programs.     

Is bilevel programming a special case of a mathematical program with complementarity constraints?

Bilevel programming problems are often reformulated using the Karush–Kuhn–Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is “No” even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater’s constraint qualification. We also show by examples that this correspondence can fail if the Slater’s constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.     

Teaching teachers mathematics through programming

There are two main arguments underlying the claims for the value of interactive computer programming used by students to model mathematical ideas. One is concerned with mathematical content, i.e. with mathematics as an object of study. The other is concerned with mathematical activity, i.e. doing mathematics, or ‘Mathematicking’ [1]. Both content and activity include processes and these provide the main links with programming. Examples of processes in the content of mathematics are addition, transformation and integration, and these can be described by instructions in a computer program. Examples of process in the activity are problem‐solving, proof generation and pattern finding which can be described by analogy to program building and debugging. We assess the arguments for programming, in relation to the training of teachers, and describe a pilot‐study in which student teachers with mathematical difficulties were taught the programming language LOGO. Observation of the students, learning the language and using it to manipulate computer models of mathematical ideas, which they had not understood previously, highlights both advantages and disadvantages in this approach. The problem of the representation of mathematical ideas within programming projects is discussed.     

Modelling a hormone-inspired controller for individual- and multi-modular robotic systems

For all living organisms, the ability to regulate internal homeostasis is a crucial feature. This ability to control variables around a set point is found frequently in the physiological networks of single cells and of higher organisms. Also, nutrient allocation and task selection in social insect colonies can be interpreted as homeostatic processes of a super-organism. And finally, behaviour can also represent such a control scheme. We show how a simple model of hormone regulation, inspired by simple biological organisms, can be used as a novel method to control the behaviour of autonomous robots. We demonstrate the formulation of such an artificial homeostatic hormone system (AHHS) by a set of linked difference equations and explain how the homeostatic control of behaviour is achieved by homeostatic control of the internal ‘hormonal’ state of the robot. The first task that we used to check the quality of our AHHS controllers was a very simple one, which is often a core functionality in controller programmes that are used in autonomous robots: obstacle avoidance. We demonstrate two implementations of such an AHHS controller that performs this task in differing levels of quality. Both controllers use the concept of homeostatic control of internal variables (hormones) and they extend this concept to also include the outside world of the robots into the controlling feedback loops: As they try to regulate internal hormone levels, they are forced to keep a homeostatic control of sensor values in a way that the desired goal ‘obstacle avoidance’ is achieved. Thus, the created behaviour is also a manifestation of the acts of homeostatic control. The controllers were evaluated using a stock-and-flow model that allowed sensitivity analysis and stability tests. Afterwards, we have also tested both controllers in a multi-agent simulation tool, which allowed us to predict the robots' behaviours in various habitats and group sizes. Finally, we demonstrate how this novel AHHS controller is suitable to control a multi-cellular robotic organism in an evolutionary robotics approach, which is used for self-programming in a gait-learning task. These examples shown in this article represent the first step in our research towards autonomous aggregation and coordination of robots to higher-level modular robotic organisms that consist of several joined autonomous robotic units. Finally, we plan to achieve such aggregation patterns and to control complex-shaped robotic organisms using AHHS controllers, as they are described here.     

Four Forms Make a Universe

Since Immanuel Kant’s Inaugural Dissertation of 1770 we assume that the concepts of space and time are not abstracted from sensations of external things. But outer experience is considered possible at all only through an inner representation of space and time within the cognitive system. In this work we describe a representation which is both inner and outer. We add to the Kantian imagination that “forms of nature, matter, space and time are intelligible, perceivable and comprehensible”, the idea that these four are indeed intelligent, perceiving, grasping and clear. They are active systems with their own intelligence. In this paper on the mind-matter interface we create the mathematical prerequisites for an appropriate system representation. We show that there is an oriented logic core within the space–time algebra. This logic core is a commutative subspace from which not only binary logic, but syntax with arbitrary real and complex truth classifiers can be derived. Space–time algebra too is obtained from this inner grammar by two rearrangements of four basic forms of connectives. When we conceive the existence of a few features like polarity between two appearances, identification and rearrangement of the latter as basic and primordial to human cognition and construction, the intelligence of space–time is prior to cognition, as it contains within its representation the basic self-reference necessary for the intelligible de-convolution of space–time. Thus the process of nature extends into the inner space.     

Robust Bayesian analysis in partially ordered plausibility calculi

In Robust Bayesian analysis one attempts to avoid the ‘Dogma of Precision’ in Bayesian analysis by entertaining a set of probability distributions instead of exactly one. The algebraic approach to plausibility calculi is inspired by Cox's and Jaynes' analyses of plausibility assessment as a logic of uncertainty. In the algebraic approach one is not so much interested in different ways to prove that precise Bayesian probability is inevitable but rather in how different sets of assumptions are reflected in the resulting plausibility calculus. It has repeatedly been pointed out that a partially ordered plausibility domain is more appropriate than a totally ordered one, but it has not yet been completely resolved exactly what such domains can look like. One such domain is the natural robust Bayesian representation, an indexed family of probabilities.We show that every plausibility calculus embeddable in a partially ordered ring is equivalent to a subring of a product of ordered fields, i.e., the robust Bayesian representation is universal under our assumptions, if extended rather than standard probability is used.We also show that this representation has at least the same expressiveness as coherent sets of desirable gambles with real valued payoffs, for a finite universe.     

Designing Effective Simulation-based Decision Support Systems: An Empirical Assessment of Three Types of Decision Support Systems

This paper addresses the design of effective simulation-based decision support systems (DSS). An experiment was conducted using three different DSS tools developed around three types of simulation model—traditional, conventional visual interactive simulation (VIS), and ‘paired-systems’ VIS. Subjects were asked to perform a decision making task and their performance was evaluated. Subjects who used the DSS based on a ‘paired systems’ VIS model were found to be both the most effective and the most efficient at the problem-solving task. Subjects provided with the DSS based upon a conventional VIS model were found to be more effective at the task than the group provided with the traditional simulation-based DSS.     

Model validation in operations research

Numerous articles have appeared in the literature expressing different degrees of concern with the methodology of OR in general and with the validation of OR models in particular. Suggestions have been formulated to remove some of the shortcomings of the methodology as currently practised and to introduce modifications in the approach because of the changing nature of the problems tackled. Advances in modeling capabilities and solution techniques have also had considerable impact on the way validation is perceived. Large scale computer-based mathematical models and especially simulation models have brought new dimensions to the notion of validation. Terms like ‘confidence’, ‘credibility and reliability’, ‘model assessment and evaluation’, ‘usefulness and usability of the model’ have become rather common. This paper is an attempt, through an interpretation of the literature, to put model validation and related issues in a framework that may be of use both to model-builders and to decision-makers.     

The Feeder-bus Network-design Problem

The potential for improving the cost-effectiveness of public transport operations by designing better integrated feeder-bus/rail rapid transit systems has been widely recognized. This paper defines the feeder-bus network-design problem (FBNDP) as that of designing a feeder-bus network to access an existing rail system. The FBNDP is considered under two different demand patterns, many-to-one (M-to-1) and many-to-many (M-to-M). We present a mathematical programming model for the M-to-1 FBNDP, and show that it can be generalized to the M-to-M FBNDP. The FBNDP is a large and difficult vehicle-routeing-type problem with an additional decision variable—operating frequency. A heuristic model is presented, which generalizes the ‘savings approach’ to incorporate operating frequency. The computational analysis shows that the proposed heuristic provides reasonable feeder-bus networks and consistent responses to ‘what if’ questions. A comparison indicates that the proposed heuristic provides solutions that are superior to manually designed networks. The advantages of this heuristic are particularly significant under variable demand.     

A diagrammatic view of the equals sign: arithmetical equivalence as a means,not an end

It is recommended in the mathematics education literature that pupils be presented with equality statements that can be assessed for numerical balance by attending to notational structure rather than computation. I describe an alternative, diagrammatic approach in which pupils do not assess statements but instead use them to make substitutions of notation. I report on two trials of a computer-based task conducted with pairs of pupils and highlight two findings. First, the pupils found it useful to articulate the distinct substitutive effects of commutative (‘swap’, ‘switch’) and partitional (‘split’, ‘separate’) statements when working on the task. Secondly, the pupils did not notice that some of the statements presented were in fact false, which suggests their substituting activities were independent of numerical equivalence conceptions. This demonstrates that making substitutions offers task designers a mathematical utility for equality statements that is distinct from, but complementary to, assessing numerical balance.     

O.R. and the Systems Movement: Mappings and Conflicts

The ‘crisis’ debate in O.R. expresses concern at the divergence between textbook O.R. and what practitioners actually do. The debate is examined by comparing O.R., systems analysis and systems engineering. They are all wedded to logic in situations in which logic may not be paramount. The science in O.R. applies only to aggregate results, but the practitioner must deal with a specific situation. The tradition of systems thinking which emerged from organismic biology is described. It leads to a way out of the O.R. ‘crisis’, by providing a formal structuring of a paradigm of learning rather than optimization. O.R. can aspire to match natural science, and pass the problems by; or it can close the textbook/practitioner gap by changing its concept of ‘being scientific’.     

“Can you help me count these pennies?”: Surfacing preschoolers’ understandings of counting

ABSTRACT

Capturing the breadth and variety of children’s understanding is critical if studies of children’s mathematical thinking are to inform policy and practice in early childhood education. This article presents an investigation of young children’s counting. Detailed coding and analyses of assessment interviews with 476 preschoolers revealed understandings that would be overlooked by solely assessing the accuracy of their responses. In particular, many children demonstrated understandings of counting principles on a challenging task that were not captured by other, simpler tasks. We conclude that common approaches to capturing young children’s mathematical understanding are likely underestimating their capabilities. This study contributes to researchers’ understanding of what making sense of counting looks and sounds like for preschool age children (3–5 years), the development and relations among counting principles (one-to-one, cardinal, and patterns of the number sequence), and the affordances of challenging, open-ended tasks. We close by considering the implications of recognizing and building from what children know and can do for researchers, practitioners, and policymakers.     

Numerical analysis of the generalized von Kármán equations

The ‘generalized von Kármán equations’ constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions ‘of von Kármán type’ only on a portion of its lateral face, the remaining portion being free. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J.-L. Lions and on Brouwer's fixed point theorem. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The Role of Heuristic Proof in Mathematics Teaching

With the great emphasis now being placed on the importance of ‘rigour’ in new mathematics programmes, many educators have been led to disparage intuition as the vitally important tool that it is in developing mathematical insights. Increasingly one sees evidence, even in technical schools, of pupils actually being discouraged from arriving at mathematical perceptions through unorthodox (and uncontrollable!) channels of analogy involving considerable divergent thinking or through consideration of physical models with which they are familiar.

As a mathematician, this pre‐occupation with ‘purism’ greatly disturbs the author. Mathematicians do not create through the formal apparatus ‐‐ they only apply formalism after ‘guessing’ results intuitionally. We have become overconcerned with the way the package is wrapped and less concerned with what is in it. Especially, the role of heuristic argument is widely misunderstood and misused in schools and colleges. The author hopes that this article will help to rectify this sorry state of affairs.     

Mathematics capital in the educational field: Bourdieu and beyond

Mathematics education needs a better appreciation of the dominant power structures in the educational field: Bourdieu's theory of capital provides a good starting point. We argue from Bourdieu's perspective that school mathematics provides capital that is finely tuned to generationally reproduce the social structures that serve to keep the powerful in power, while ensuring that less powerful groups are led to accept their own failure in mathematics. Bourdieu's perspective thereby highlights theoretical inadequacies in much mathematics education research, insofar as it presumes a consensus about a ‘what works agenda’ for improving achievement for all. Drawing on one case where we manufactured awkward facts, we illustrate a Bourdieusian interpretation of mathematics capital as reproductive, and the crucial role of its cultural arbitrary. We then criticise the Bourdieusian concept of ‘mathematical capital’ as the value of mathematical competence in practice and propose to extend his tools to include the contradictory ‘use’ and ‘exchange’ values of mathematics instead: we will show how this conceptualisation goes ‘beyond Bourdieu’ and helps explain how teaching-learning might (ideally) produce ‘cultural use value’ in mathematical competence, while still recognising the contradictions teachers and learners face. Finally, we suggest how critical education research generally can benefit from this theoretical framework: (1) in exposing the interest of the dominant classes; but also (2) in researching critical pedagogic alternatives that challenge orthodoxy in educational policy and practice both in mathematics education and more generally.     

Higher and lower-level knowledge discovery from Pareto-optimal sets

Innovization (innovation through optimization) is a relatively new concept in the field of multi-objective engineering design optimization. It involves the use of Pareto-optimal solutions of a problem to unveil hidden mathematical relationships between variables, objectives and constraint functions. The obtained relationships can be thought of as essential properties that make a feasible solution Pareto-optimal. This paper proposes two major extensions to innovization, namely higher-level innovization and lower-level innovization. While the former deals with the discovery of common features among solutions from different Pareto-optimal fronts, the latter concerns features commonly occurring among solutions that belong to a specified (or preferred) part of the Pareto-optimal front. The knowledge of such lower-level information is extremely beneficial to a decision maker, since it focuses on a preferred set of designs. On the other hand, higher-level innovization reveals interesting knowledge about the general problem structure. Neither of these crucial aspects concerning multi-objective designs has been addressed before, to the authors’ knowledge. We develop methodologies for handling both levels of innovization by extending the authors’ earlier automated innovization algorithm and apply them to two well-known engineering design problems. Results demonstrate that the proposed methodologies are generic and are ready to be applied to other engineering design problems.