Quasiconvex Research Articles

On extension of uniformly continuous quasiconvex functions

We show that each uniformly continuous quasiconvex function defined on a subspace of a normed space X X admits a uniformly continuous quasiconvex extension to the whole X X with the same “invertible modulus of continuity”. This implies an analogous extension result for Lipschitz quasiconvex functions, preserving the Lipschitz constant. We also show that each uniformly continuous quasiconvex function defined on a uniformly convex set A ⊂ X A\subset X admits a uniformly continuous quasiconvex extension to the whole X X . However, our extension need not preserve moduli of continuity in this case, and a Lipschitz quasiconvex function on A A may admit no Lipschitz quasiconvex extension to X X at all.

A limiting case in partial regularity for quasiconvex functionals

<abstract><p>Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $ p $-growth of the type</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ w\mapsto \int \left[F(Dw)-f\cdot w\right]{\,{{\rm{d}}}x} $\end{document} </tex-math></disp-formula></p> <p>feature almost everywhere $ \mbox{BMO} $-regular gradient provided that $ f $ belongs to the borderline Marcinkiewicz space $ L(n, \infty) $.</p></abstract>

Coefficient Inequalities for Biholomorphic Mappings on the Unit Ball of a Complex Banach Space

In the first part of this paper, we give generalizations of the Fekete–Szegö inequalities for quasiconvex mappings F of type B and the first elements F of g-Loewner chains on the unit ball of a complex Banach space, recently obtained by H. Hamada, G. Kohr and M. Kohr. We obtain the Fekete–Szegö inequalities using the norm under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F. In the second part of this paper, we give the estimation of the difference of the moduli of successive coefficients for the first elements of g-Loewner chains on the unit disc. We also give the estimation of the difference of the moduli of successive coefficients for the first elements F of g-Loewner chains on the unit ball of a complex Banach space under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F.

Highlight removal for endoscopic images based on accelerated adaptive non-convex RPCA decomposition

Objective:Highlights always occur in endoscopic images due to their special imaging environment. It not only increases the difficulty of observation and diagnosis from surgeons, but also influences the performance of Mixed/Augmented Reality techniques in surgery navigation.Methods:In this paper, we propose a novel accelerated adaptive non-convex robust principal component analysis method (AANC-RPCA) to remove highlights in endoscopic images. We first detect absolute highlight pixels of the enhanced endoscopic images with adaptive threshold segmentation. The quasi-convex function is proposed to approximate a new non-convex objective function. With detected highlight pixels and quasi-convex function, it introduces gradient to shrink sparse matrix and obtains a faster speed of convergence. Then we divide the image into multiple blocks and perform the parallel computation to enhance the efficiency. Finally, we design a weighted template that decays outward with dilation and linear filtering to reconstruct the endoscopic images. Our approach is almost independent of hyper-parameters and can achieve adaptive decomposition.Results:It has been verified on multiple types of endoscopic images through experiments and clinical blind tests. The results demonstrate that our method can obtain the best performance for the recovered images with more details in a shorter time (about 3–5 times).Conclusion:Coupled with the user study, both the quantitative and qualitative results indicate that our approach has the potential to be highly useful in endoscopy images. Compared with the existing highlight removal approaches, our method obtains the SOTA results and has the potential to be applied in the various medical processing processes.

A natural correspondence between quasiconcave functions and fuzzy norms

In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm N:X×R→[0,1] defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function f:X→[0,1]. We explore the minimum requirements that we need to impose to some quasiconcave function f:X→[0,1] in order to define a fuzzy norm N:X×R→[0,1]. Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.

Isomorphisms on interpolation spaces generated by the method of means

We investigate the stability of isomorphisms acting between interpolation spaces generated by the method of means. We focus on the methods which are determined by balanced sequences for non-degenerate quasi-concave functions. The key point for our investigation is that these methods have orbital description by a single element generated by a special ideal of operators between Banach couples. We prove that if an operator is invertible in one orbit it is also invertible by nearby orbits provided that the corresponding indices of quasi-concave functions generated these orbits are close to each other. In particular, these results apply to the real method of interpolation.

A Novel 2-D Phase Unwrapping Method Based on PUMA for Aerospace Topographic Mapping Systems

Two-dimensional phase unwrapping (PU) is the key step of interferometric synthetic aperture radar technology. The accuracy of the PU results directly affects the quality of the final product. Due to the limitation of the phase continuity assumption, the results obtained by the single-baseline PU in large gradient regions are not ideal. To solve this problem, this article proposes an improved PU max-flow/min-cut algorithm (PUMA) which can efficiently deal with the phase of large gradient regions. First, we construct a quasiconvex function which is the combination of logarithmic and quadratic functions as the potential function in the energy function model of the PUMA method. Then, we use the gradient information of the external digital elevation model to assist in setting the weight of the potential function. Finally, the unwrapped phase is obtained by minimizing the new energy function model. The experimental results on the TanDEM-X dataset show that the improved approach in this article can achieve higher accuracy PU results in the region of large gradient variation than the existing PU methods.

The saturation number of c-bounded stable monomial ideals and their powers

Let S=K[x1,…,xn] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine the saturation numbers of strongly stable ideals and equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese-type ideals I. Using this formula, we show that sat(Ik) is quasilinear from the beginning, and we determine the quasilinear function explicitly.

Relaxed-inertial proximal point type algorithms for quasiconvex minimization

We propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. A relaxed version of the method where the constraint set is only closed and convex is also discussed, and so is the case of a quasiconvex objective function. Numerical experiments illustrate the theoretical results.

Bregman proximal point type algorithms for quasiconvex minimization

We discuss a Bregman proximal point type algorithm for dealing with quasiconvex minimization. In particular, we prove that the Bregman proximal point type algorithm converges to a minimal point for the minimization problem of a certain class of quasiconvex functions without neither differentiability nor Lipschitz continuity assumptions, this class of nonconvex functions is known as strongly quasiconvex functions and, as a consequence, we revisited the general case of quasiconvex functions.

Partial regularity for omega -minimizers of quasiconvex functionals

We establish partial regularity for the omega -minimizers of quasiconvex functionals of power growth. A first-order partial regularity result of BVomega -minimizers is obtained in the linear growth case under a Dini-type condition on omega . Only assuming the smallness of omega near the origin, we show partial Hölder continuity in the subquadratic case by considering a normalised excess.

On Surrogate Duality and Competition between Entropy and Potential Energy in the Finite Element Model of Elastic Contact Mechanics

The competition between entropy and energy that usually occurs in thermodynamics — an increase in one usually leads to a decrease in the other — was also shown to hold in static–elastic contact mechanics in our previous work on solving contact problems with an iterative algorithm. In this paper, we first present a theoretical analysis of the surrogate duality of the optimization model of contact problems, propose several propositions characterizing the surrogate duality, and identify the condition under which the fractional objective function of the surrogate dual problem is quasi concave. Second, we further clarify the correspondence between the concepts of statistical physics and the finite element model of contact problems so that the concepts of statistical physics can be more cleanly used to solve contact problems. Third, we provide examples to calculate the contact force with an improved iterative algorithm based on the quasi concavity that more clearly verifies the competition between entropy and potential energy, and the competition shows strong negentropy behavior: potential energy increases while entropy decreases throughout the iterative process.

Verifying Measures of Quantum Entropy

This paper introduces a new measure of quantum entropy, called the effective quantum entropy (EQE). The EQE is an extension, to the quantum setting, of a recently derived classical generalized entropy. We present a thorough verification of its properties. As its predecessor, the EQE is a semi-strict quasi-concave function; it would be capable of generating many of the various measures of quantum entropy that are useful in practice. Thereafter, we construct a consistent estimator for our proposed measure and empirically test its estimation error, under different system dimensions and number of measurements. Overall, we build the grounds of the EQE, which will facilitate the analyses and verification of the next innovative quantum technologies.

A condition for the identification of multivariate models with binary instruments

This article introduces an empirical condition for the nonparametric point-identification of multivariate instrumental variable models with continuous endogenous variables using binary instruments. Verifying this condition can confirm point-identification in settings in which traditional approaches are not applicable. In particular, it shows that nonlinear instrumental variable models with general heterogeneity can be point-identified with only a binary instrument. This generalizes existing identification results which either restrict the unobserved heterogeneity substantially or require the instrument to have a large support. The main assumption on the instrumental variable model is cyclic monotonicity of its first stage, a multivariate generalization of the classical rank-invariance assumption for univariate models. Asymptotic convergence results for the empirical observable distributions are derived that allow to check the condition in practice. The identification rests on a fixed-set convergence result of cyclically monotone maps between quasi-concave functions.

Quasiconvexity and partial regularity via nonlinear potentials

We show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from Nonlinear Potential Theory. In particular, in the setting of functionals with (p,q)-growth - according to the terminology of Marcellini [52] - we derive optimal local regularity criteria under minimal assumptions on the data.

A note on generalized subclasses of multivalent quasi-convex functions

Abstract This paper is concerned with certain generalized subclasses of multivalent quasi-convex functions defined with subordination. Various properties of these classes in regard to the coefficient estimates, distortion theorems, growth theorems, argument theorems and inclusion relations are discussed. Also, the relations with the earlier known results are established.

Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity

We establish general “collapse to the mean” principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the “collapse to the mean” for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the “collapse to the mean” to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.

Optimal pricing and inventory control strategy for a continuous-review system with product return

We consider joint pricing and inventory control strategies for a stochastic inventory system with product return. Customer's demand and product return follow independent point processes with demand rate being price-dependent. Using a general verification theorem, we prove that an (s⁎,S⁎,p⁎) policy is optimal, where the optimal price p⁎(x) is a quasi-concave function of inventory level with a nonpositive changeover point. Furthermore, we propose an algorithm to search for the optimal policy parameters and numerically study the effect of product return.

Quasiconvex functions: how to separate, if you must!

"Since quasiconvex functions have convex lower level sets it is possible to minimize them by means of separating hyperplanes. An example of such a procedure, well-known for convex functions, is the subgradient method. However, to nd the normal vector of a separating hyperplane is in general not easy for the quasiconvex case. This paper attempts to gain some insight into the computational aspects of determining such a normal vector and the geometry of lower level sets of quasiconvex functions. In order to do so, the directional di erentiability of quasiconvex functions is thoroughly studied. As a consequence of that study, it is shown that an important subset of quasiconvex functions belongs to the class of quasidifferentiable functions. The main emphasis is, however, on computing actual separators. Some important examples are worked out for illustration."

Continuity and maximal quasimonotonicity of normal cone operators

In this paper we study some properties of the adjusted normal cone operator of quasiconvex functions. In particular, we introduce a new notion of maximal quasimotonicity for set-valued maps different from similar ones recently appeared in the literature, and we show that it is enjoyed by this operator. Moreover, we prove the $s\times w^*$ cone upper semicontinuity of the normal cone operator in the domain of $f$ in case the set of global minima has non empty interior.