Role Of Convexity Research Articles

Occupation Measure Relaxations in Variational Problems: The Role of Convexity

Occupation Measure Relaxations in Variational Problems: The Role of Convexity

The role of convexity in defining regular polyhedra

It is almost twenty years since Branko Grünbaum lamented that the ‘original sin’ in the theory of polyhedra is that from Euclid onwards “the writers failed to define what are the ‘polyhedra’ among which they are finding the ‘regular’ ones” ([1, p. 43]). Various definitions of ‘regular’ can be found in the literature with a condition of convexity often included (e.g. [2, p. 301], [3, p. 77], [4, p. 47], [5, p. 435], [6, p. 16]). The condition of convexity is usually cited to exclude regular self-intersecting polyhedra, i.e. the Kepler-Poinsot polyhedra, such as the ‘great dodecahedron’ consisting of twelve intersecting pentagonal faces shown in Figure 1 with one face shaded. Richeson also notes ([4, pp. 47-48]) that, for a particular definition of ‘regular’, convexity is needed to exclude the ‘punched-in’ icosahedron shown in Figure 2.

Nadler’s Theorem on Incomplete Modular Space

This manuscript is focused on the role of convexity of the modular, and some fixed point results for contractive correspondence and single-valued mappings are presented. Further, we prove Nadler’s Theorem and some fixed point results on orthogonal modular spaces. We present an application to a particular form of integral inclusion to support our proposed version of Nadler’s theorem.

Profit efficiency: Generalization, business accounting and the role of convexity

We present new theoretical results, showing that Nerlovian profit efficiency is a special case of the recently introduced general profit efficiency, deriving a new decomposition of profit efficiency, and outlining a simple way of estimating the generalized profit efficiency measure.

The Role of Convexity in Saddle-Point Dynamics: Lyapunov Function and Robustness

This paper studies the projected saddle-point dynamics associated to a convex–concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity–concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity–concavity of the saddle function. When strong convexity–concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability (ISS) of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.

The role of convexity in the adaptive control of rapidly time-varying systems

In classical adaptive control the parameters are assumed to be fixed or slowly time-varying. In order to facilitate parameter estimation/tuning it is desirable to have the set of admissible parameters lie in a convex set; if this set is not convex, a common trick is to replace it with its convex hull, but the adaptive control problem is challenging if stabilizability of the set of admissible parameters is lost. However, such a convexity assumption is an artifact of the approach to the problem, rather than an inherent constraint, since most logic-based and supervisory approaches to the problem make no such requirement. On the other hand, here we show that losing stabilizability on the convex hull of the set of admissible parameters plays an important role in the adaptive control of rapidly time-varying systems.

Convexity Bias and Perspective Cues in the Reverse-Perspective Illusion.

The present experiment was designed to examine the roles of painted linear perspective cues, and the convexity bias that are known to influence human observers’ perception of three-dimensional (3D) objects and scenes. Reverse-perspective stimuli were used to elicit a depth-inversion illusion, in which far points on the stimulus appear to be closer than near points and vice versa, with a 2 (Type of stimulus) × 2 (Fixation mark position) design. To study perspective, two types of stimuli were used: a version with painted linear perspective cues and a version with blank (unpainted) surfaces. To examine the role of convexity, two locations were used for the fixation mark: either in a locally convex or a locally concave part of each stimulus (painted and unpainted versions). Results indicated that the reverse-perspective illusion was stronger when the stimulus contained strong perspective cues and when observers fixated a locally concave region within the scene.

Comment on “Hypothesis testing by convex optimization”

With the growing size of problems at hand, convexity has become preponderant in modern statistics. Indeed, convex relaxations of NP-hard problems have been successfully employed in a variety of statistical problems such as classification [2, 16], linear regression [7, 5], matrix estimation [8, 12], graphical models [15, 9] or sparse principal component analysis (PCA) [10, 4]. The paper “Hypothesis testing by convex optimization” by Alexander Goldenshluger, Anatoli Juditsky and Arkadi Nemirovski, hereafter denoted by GJN, brings a new perspective on the role of convexity in a fundamental statistical problem: composite hypothesis testing. The role of this problem is illustrated in the light of several interesting applications in Section 4 of GJN. One of the key insights in GJN is that there exists a pair of distributions, one in each of the composite hypotheses and on which the statistician should focus her efforts. Indeed, Theorem 2.1(ii) guarantees that any test that is optimal for this simple hypothesis problem is also near optimal for the composite hypothesis problem. Moreover, this pair can be found by solving a convex optimization problem. While convexity does not necessarily imply tractability, the convex problem considered here may become simple to the point that closed solutions exist even though no succinct description of the hypothesis sets may be known. This point is illustrated below. Unlike the papers cited above, where the original problem to be solved is non-convex, GJN assumes given convex hypotheses (or finite unions of convex hypotheses). Hereafter, we investigate the performance of the proposed test when convexity is artificial and arises as a relaxation of a non-convex problem. Let us consider two examples that fall under the umbrella of combinatorial testing problems [1]. Such problems are defined as follows. Assume that one observes a Gaussian random vector X ∼ N (μ, In) for some μ ∈ R. Let p ∈ {0, 1}n be a sparsity pattern [17]. Given a class P ⊂ {0, 1}n of sparsity patterns,

The role of convexity in perception of symmetry and in visual short-term memory

Visual perception of shape is affected by coding of local convexities and concavities. For instance, a recent study reported that deviations from symmetry carried by convexities were easier to detect than deviations carried by concavities. We removed some confounds and extended this work from a detection of reflection of a contour (i.e., bilateral symmetry), to a detection of repetition of a contour (i.e., translational symmetry). We tested whether any convexity advantage is specific to bilateral symmetry in a two-interval (Experiment 1) and a single-interval (Experiment 2) detection task. In both, we found a convexity advantage only for repetition. When we removed the need to choose which region of the contour to monitor (Experiment 3) the effect disappeared. In a second series of studies, we again used shapes with multiple convex or concave features. Participants performed a change detection task in which only one of the features could change. We did not find any evidence that convexities are special in visual short-term memory, when the to-be-remembered features only changed shape (Experiment 4), when they changed shape and changed from concave to convex and vice versa (Experiment 5), or when these conditions were mixed (Experiment 6). We did find a small advantage for coding convexity as well as concavity over an isolated (and thus ambiguous) contour. The latter is consistent with the known effect of closure on processing of shape. We conclude that convexity plays a role in many perceptual tasks but that it does not have a basic encoding advantage over concavity.

The role of convexity and part structure in modal and amodal completion

The role of convexity and part structure in modal and amodal completion

On the role of convexity in isoperimetry, spectral gap and concentration

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-Émery.

On the role of convexity in functional and isoperimetric inequalities

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarliński. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature–dimension condition of Bakry–Émery.

Logarithmic Negativity: A Full Entanglement Monotone That is not Convex

It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is not a convex function this result is surprising, as it is generally considered that convexity describes the local physical process of losing information. The role of convexity and, in particular, its relation (or lack thereof) to physical processes is discussed and importance of continuity in this context is stressed.

Duality Theory of Non-convex Technologies

Duality Theory of production imposes a number of simplifying assumptions regarding the production technology, including various maintained convexity assumptions. Emphasizing the technological information content of alternative models, this paper challenges some widely held views on the role of convexity. The role of convexity in Duality Theory is asymmetric. While convexity is of importance in recovering technology information from economic models, cost functions are concave and profit functions are convex irrespective of convexity of the underlying technology. For recovering technology information from economic models and data, we discuss two alternative approaches: recovering inexact outer-bound approximations; and enriching standard economic models by additional quantity/financial constraints. The main conclusion is that non-convexities should not stop one from applying the Duality Theory.

The role of convexity in existence theorems for invariant and hyperinvariant subspaces in hilbert space

We use extreme point methods to obtain some existence theorems for invariant and hyperinvariant subspaces for certain classes of operators In Hilbert space. These convexity methods have, heretofore, not been used in this context, however it would appear that they provide useful alternatives to existing methods. Using the Krein-Milman theorem we prove several variants of the classical lemma of I. Schur. In one of the main results we prove the following: LetA ∈B (H), thenA andA* have a common non-trivial closed hyperinvariant subspace if and only if there exists a non-scalar operatorW eB(H) such that bothW andW* belong to the double commutant ofA.

The role of convexity in perceptual completion: beyond good continuation

Since the seminal work of the Gestalt psychologists, there has been great interest in understanding what factors determine the perceptual organization of images. While the Gestaltists demonstrated the significance of grouping cues such as similarity, proximity and good continuation, it has not been well understood whether their catalog of grouping cues is complete — in part due to the paucity of effective methodologies for examining the significance of various grouping cues. We describe a novel, objective method to study perceptual grouping of planar regions separated by an occluder. We demonstrate that the stronger the grouping between two such regions, the harder it will be to resolve their relative stereoscopic depth. We use this new method to call into question many existing theories of perceptual completion (Ullman, S. (1976). Biological Cybernetics, 25, 1–6; Shashua, A., & Ullman, S. (1988). 2nd International Conference on Computer Vision (pp. 321–327); Parent, P., & Zucker, S. (1989). IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 823–839; Kellman, P. J., & Shipley, T. F. (1991). Cognitive psychology, Liveright, New York; Heitger, R., & von der Heydt, R. (1993). A computational model of neural contour processing, figure-ground segregation and illusory contours. In Internal Conference Computer Vision (pp. 32–40); Mumford, D. (1994). Algebraic geometry and its applications, Springer, New York; Williams, L. R., & Jacobs, D .W. (1997). Neural Computation, 9, 837–858) that are based on Gestalt grouping cues by demonstrating that convexity plays a strong role in perceptual completion. In some cases convexity dominates the effects of the well known Gestalt cue of good continuation. While convexity has been known to play a role in figure/ground segmentation ( Rubin, 1927; Kanizsa & Gerbino, 1976), this is the first demonstration of its importance in perceptual completion.

The Equal Marginal Value Principle: A Graphical Analysis with Environmental Applications

The typical undergraduate intermediate microeconomics textbook uses geometric arguments accompanied by two-dimensional diagrams to present the principles of cost minimization and profit maximization (see Pindyck and Rubinfeld 1995). These tools are sufficient to explain the behavior of a firm with one choice variable-for example, the level of production of a single product from a single factory. More realistic models, where firms have multiple products, factories, and technologies, are usually not presented because they require the use of more sophisticated mathematical techniques, such as Lagrange multipliers. In this article, I develop a simple diagrammatic exposition of optimization principles for problems with two choice variables. In the basic model, the choice variables correspond to production levels in two different factories.' In addition to illustrating the familiar equal marginal value principle, the basic model makes more advanced topics, such as corner conditions and the role of convexity in optimization, accessible to a wide variety of students. An extension of the model illustrates the fundamental relationship between profit maximization and cost minimization. Other extensions are used to analyze important environmental issues, such as the cost effectiveness of policy proposals to reduce auto emissions, the economics of garbage disposal, and the economics of pollution abatement. THE MODEL Consider a firm that produces a single product in two different factories. Let q, be the quantity produced in the first factory and q2 be the quantity produced in the second. The factories have cost functions CI(qi) and C2(q2) and associated marginal cost functions MCI(ql) and MC2(q2), respectively.2 The firm wants to produce q, units of the product so that the total cost of production is minimized. The solution may be found graphically by placing the marginal costs curves back-to-back so that they share the same y axis, but quantity produced increases in opposite directions along the x axis. The total cost associated with any particular division Y, q q of production between the two factories is found by placing a ruler of length q, along the x axis so that the right edge is directly under qf and then calculating the area bounded by the marginal cost curves and the ruler.

Chapter 5 Duality, classification and slacks in DEA

This paper presents seven theorems which expand understanding of the theoretical structure of the Charnes-Cooper-Rhodes (CCR) model of Data Envelopment Analysis, especially with respect to slacks and the underlying structure of facets and faces. These theorems also serve as a basis for new algorithms which will provide optimal primal and dual solutions that satisfy the strong complementary slackness conditions (SCSC) for many (if not most) non-radially efficient DMUs; an improved procedure for identifying the setE of extreme efficient DMUs; and may, for many DEA domains, also settle in a single pass the existence or non-existence of input or output slacks in each of their DMUs. This paper also introduces the concept of a positivegoal vector G, which is applied to characterize the set of all possible maximal optimal slack vectors. The appendix C presents an example which illustrates the need for a new concept,face regular, which focuses on the role of convexity in the intersections of radial efficient facets with the efficient frontier FR. The same example also illustrates flaws in the popular “sum of the slacks” methodology.

The Role of Convexity in Equity Pricing

The Role of Convexity in Equity Pricing

Global uniqueness of homoclinic orbits for a class of fourth order equations

In this paper we show the global existence and uniqueness of certain orbits homoclinic to the zero stationary solution of the fourth order equation $$\alpha x\prime \prime \prime \prime + \beta x\prime \prime + yx + k(x) = 0,x > 0,$$ whenα, γ>0>β,dk/dx 0 andK(0)=0. The existence problem is approached via the general theory of [1] and uniqueness follows from the Maximum Principle and some geometrical observations about the role of convexity. There are no small amplitude assumptions.