Convexity Hypothesis Research Articles

Interest Rate Risk of Stock Prices in Nigeria

The article empirically examines the response of stock prices in the Nigerian Stock Market to interest rate risk, using the Duration and Convexity model. A simple non-linear stock price model that incorporates measure of interest rate duration and convexity and a vector containing a battery of other control variables is specified and estimated using annual secondary data that covered the 1981–2006 sample period. The empirical results largely corroborate the hypothesis of interest rate sensitivity of stock prices in the Nigerian Stock Market. Specifically, we find both measures of interest rate duration and convexity to exert strong but opposite effects on stock prices in Nigeria. The empirical evidence shows that the net effect of interest rate changes on stock prices is negative, an indication that stock prices fall with increases in stock risk. The results provide empirical support for the duration and convexity hypothesis of the existence of a non-linear relationship between interest rate risk and stock prices in Nigeria. JEL Classification: G1, G12

Duality for [formula omitted]-variational inequalities via the subdifferential calculus

Based on the properties of the (convex) ε-subdifferential calculus, we introduce to a general ε-variational inequality (formulated with the help of a set valued operator and a perturbation function) a dual one, expressed by making use of the (Fenchel) conjugate of the perturbation function. Under convexity hypotheses, we show that the fulfillment of a regularity condition guarantees that the primal ε-variational inequality is solvable if and only if its dual one is solvable. By particularizing the perturbation function, we obtain several dual statements and we succeed to generalize and improve a duality scheme recently given by Kum, Kim and Lee. An example justifying this generalization is also provided. Among the special instances of the general result, we rediscover also the duality scheme concerning variational inequalities due to Mosco.

SBV Regularity for Hamilton--Jacobi Equations with Hamiltonian Depending on $(t, x)$

In this paper we prove the special bounded variation regularity of the gradient of a viscosity solution of the Hamilton--Jacobi equation $\partial_t u+ H(t,x,D_x u)=0$ in $\Omega\subset [0,T]\times\mathbb{R}^n$ under the hypothesis of uniform convexity of the Hamiltonian $H$ in the last variable. This result extends the result of Bianchini, De Lellis, and Robyr obtained for a Hamiltonian $H=H(D_x u)$ which depends only on the spatial gradient of the solution.

Inexact block coordinate descent methods with application to non-negative matrix factorization

This work is concerned with the cyclic block coordinate descent method, or nonlinear Gauss-Seidel method, where the solution of an optimization problem is achieved by partitioning the variables in blocks and successively minimizing with respect to each block. The properties of the objective function that guarantee the convergence of such alternating scheme have been widely investigated in the literature and it is well known that, without suitable convexity hypotheses, the method may fail to locate the stationary points when more than two blocks of variables are employed. In this paper the general constrained non convex case is considered and three contributions are given. First, a general method allowing an approximate solution of each block minimization subproblem is devised and the related convergence analysis is developed, showing that the proposed inexact method has the same convergence properties of the standard nonlinear Gauss-Seidel method. Then, a cyclic block gradient projection method is analyzed, proving that it leads to stationary points for every number of blocks. Finally, the cyclic block gradient method is applied to large scale problems arising from the nonnegative matrix factorization approach. The results of a numerical experimentation on image recognition problems are also reported.

Modeling score distributions in information retrieval

We review the history of modeling score distributions, focusing on the mixture of normal-exponential by investigating the theoretical as well as the empirical evidence supporting its use. We discuss previously suggested conditions which valid binary mixture models should satisfy, such as the Recall-Fallout Convexity Hypothesis, and formulate two new hypotheses considering the component distributions, individually as well as in pairs, under some limiting conditions of parameter values. From all the mixtures suggested in the past, the current theoretical argument points to the two gamma as the most-likely universal model, with the normal-exponential being a usable approximation. Beyond the theoretical contribution, we provide new experimental evidence showing vector space or geometric models, and BM25, as being ‘friendly’ to the normal-exponential, and that the non-convexity problem that the mixture possesses is practically not severe. Furthermore, we review recent non-binary mixture models, speculate on graded relevance, and consider methods such as logistic regression for score calibration.

Convexity, Productivity Change and the Economic Performance of Countries

In this paper, we estimate indices of technological catch-up, technological change and variations in scale efficiency for a sample of 52 countries through the FDH approach with variable scaling parameters (VP-FDH) proposed in Kerstens and Vanden Eeckaut (1999). We compare the results with those obtained through the DEA-VRS approach and find little support for the hypothesis of convexity of the production set. We then argue that VP-FDH has significant advantages for the assessment of cross-country productivity changes.

Solvability of Uniformly Elliptic Fully Nonlinear PDE

We get existence, uniqueness and non-uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the Hamilton–Jacobi–Bellman–Isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness. Some of our results are new even for equations in divergence form.

Regularity Conditions in Differentiable Vector Optimization Revisited

This work is concerned with differentiable constrained vector optimization problems. It focus on the intrinsic connection between positive linearly dependent gradient sets and the distinct notions of regularity that come to play in this context. The main aspect of this contribution is the development of regularity conditions, based on the positive linear dependence or independence of gradient sets, for problems with general nonlinear constraints, without any convexity hypothesis. Being easy to verify, these conditions might be useful to define termination criteria in the development of algorithms.

Hedonic price function estimation in economics and marketing: revisiting Lancaster’s issue of “noncombinable” goods

Following Lancaster’s (J. Political Econ. 74(1):132–157, 1966; Variety, equity and efficiency, 1979) interpretation of his characteristics approach to consumer theory, this contribution focuses on theoretical and empirical arguments questioning the smoothness of traditional hedonic price estimation techniques. Lancaster argued strongly against “combinability”, i.e., that any efficient combination of characteristics is feasible and sensible. We therefore explicitly test the impact of convexity using a set of recent non-parametric estimators. The test is carried out on a sample of 114 digital cameras whose price evolution is followed over 6 months. The hypothesis of convexity is rejected using the Li (Econ. Rev. 15(3):261–274, 1996) test. The conclusions point out implications for economics and marketing.

An inverse problem for Schrödinger equations with discontinuous main coefficient

This article concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of ℝ N with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and Lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the Schrödinger equation with discontinuous coefficients, result also interesting by itself.

Non-commutative partial matrix convexity

Let p be a polynomial in the non-commuting variables (a, x) = (a1, . . . , ag, x1, . . . , xg). If p is convex in the variables x, then p has degree two in x and moreover, p has the form p = L + Λ Λ, where L has degree at most one in x and Λ is a (column) vector which is linear in x, so that Λ Λ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the x and a variables separately are presented.

A multiplicity result for the problem [formula omitted

In this paper we consider the nonlinear equation involving differential forms on a compact Riemannian manifold δ d ξ = f ′ ( 〈 ξ , ξ 〉 ) ξ . This equation is a generalization of the semilinear Maxwell equations recently introduced in a paper by Benci and Fortunato. We obtain a multiplicity result both in the positive mass case (i.e. f ′ ( t ) ≥ ε > 0 uniformly) and in the zero mass case ( f ′ ( t ) ≥ 0 and f ′ ( 0 ) = 0 ) where a strong convexity hypothesis on the nonlinearity is assumed.

Poincaré’s Variational Problem in Potential Theory

One of the earliest attempts to rigorously prove the solvability of Dirichlet’s boundary value problem was based on seeking the solution in the form of a “potential of double layer”, and this leads to an integral equation whose kernel is (in general) both singular and non-symmetric. C. Neumann succeeded with this approach for smoothly bounded convex domains, and H. Poincare, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was (according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Unfortunately, a rigorous treatment of this problem was not possible with the tools available at that time (as Poincare was well aware). So far as we know, the only one to propose a rigorous treatment of Poincare’s problem was T. Carleman (in the two-dimensional case) in his doctoral dissertation. Thanks to later developments (especially concerning Sobolev spaces, and spectral theory of operators on Hilbert space) we can now give a complete, general and rigorous account of Poincare’s variational problem, and that is the main object of the present paper. As a by-product, we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar Bergman–Schiffer singular integral equation. We interpret Poincare’s variational principle as a non-selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials. We also prove a series of novel spectral analysis facts (some of them conjectured by Poincare) related to the Poincare–Neumann integral operator.

Continuité lipschitzienne des solutions d'un problème en calcul des variations

In this Note, we describe some recent developments concerning the regularity of the minimizers u of ∫ Ω F ( ∇ u ) + G ( x , u ) , over the functions u ∈ W 1 , 1 ( Ω ) that assume given boundary values ϕ on ∂ Ω. The classical Hilbert–Haar theory derives regularity of u from an assumption on ϕ, the well-known bounded slope condition. Instead of this, we impose the less restrictive lower (or upper) bounded slope condition, which is satisfied if ϕ is the restriction to ∂ Ω of a convex (or even semiconvex) function. Under this new assumption and some convexity hypotheses on F and Ω, we show that any minimizer u is locally Lipschitz in Ω. In some cases we are also able to assert that u is continuous on Ω ¯ . To cite this article: P. Bousquet, F. Clarke, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

A Unified Approach and Optimality Conditions for Approximate Solutions of Vector Optimization Problems

This paper deals with approximate (V-efficient) solutions of vector optimization problems. We introduce a new V-efficiency concept which extends and unifies different approximate solution notions introduced in the literature. We obtain necessary and sufficient conditions via nonlinear scalarization, which allow us to study this new class of approximate solutions in a general framework, since any convexity hypothesis is required. Several examples are proposed to show the concepts introduced and the results attained.

A Viro theorem without convexity hypothesis for trigonal curves

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coherent way we prove that no convexity hypothesis is required to patchwork such curves.

Saddle points and Lagrangian-type duality for discrete minmax fractional subset programming problems with generalized convex functions

In this paper we present four sets of saddle-point-type optimality conditions, construct two Lagrangian-type dual problems, and prove weak and strong duality theorems for a discrete minmax fractional subset programming problem. We establish these optimality and duality results under appropriate ( b , ϕ , ρ , θ ) -convexity hypotheses.

On Brown's Method with Convexity Hypotheses

Given two initial points generating monotone convergent Brown iterations in the context of the monotone Newton theorem (MNT), it is proved that if one of them is an upper bound of the other, then the same holds for each pair of respective terms in the Brown sequences they generate. This comparison result is carried over to the corresponding Brown-Fourier iterations. An illustration is discussed.

Existence and Relaxation for Finite-Dimensional Optimal Control Problems Driven by Maximal Monotone Operators

We study the optimal control of a class of nonlinear finite-dimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesari. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the "relaxed problem". We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Caratheodory's theorem for convex sets in RN and the fourth uses lower semicontinuity arguments and Γ-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the "closure" of the original one.

Sharp rate of average decay of the Fourier transform of a bounded set

Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L 2-average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary bounded convex set in $\mathbb{R}^{d}$ is $${\bigg(\int_{S^{d-1}}{\big|\widehat{\chi}_B(R\omega)\big|}^2d\omega \bigg)}^{{1}/{2}} \lesssim R^{-\frac{d+1}{2}}.\eqno(*)$$ This estimate is optimal for any convex body and in particular it agrees with the familiar estimate for the ball. The above estimate was proved in two dimensions by Podkorytov, and in all dimensions by Varchenko under additional smoothness assumptions. The main result of this paper proves (*) in all dimensions under the convexity hypothesis alone. We also prove that the same result holds if the boundary of ∂Ω is C3/2.