Class Of Convex Research Articles

Convergence Results of a Nested Decentralized Gradient Method for Non-strongly Convex Problems

We are concerned with the convergence of NEAR-DGD\(^+\) (Nested Exact Alternating Recursion Distributed Gradient Descent) method introduced to solve the distributed optimization problems. Under the assumption of the strong convexity of local objective functions and the Lipschitz continuity of their gradients, the linear convergence is established in Berahas et al. (IEEE Trans Autom Control 64:3141-3155, 2019). In this paper, we investigate the convergence property of NEAR-DGD\(^+\) in the absence of strong convexity. More precisely, we establish the convergence results in the following two cases: (1) When only the convexity is assumed on the objective function. (2) When the objective function is represented as a composite function of a strongly convex function and a rank deficient matrix, which falls into the class of convex and quasi-strongly convex functions. The numerical results are provided to support the convergence results.

Convergence Rates of the Heavy Ball Method for Quasi-strongly Convex Optimization

In this paper, we study the behavior of solutions of the ODE associated to the Heavy Ball method. Since the pioneering work of B.T. Polyak [25], it is well known that such a scheme is very efficient for C2 strongly convex functions with Lipschitz gradient. But much less is known when the C2 assumption is dropped. Depending on the geometry of the function to minimize, we obtain optimal convergence rates for the class of convex functions with some additional regularity such as quasi-strong convexity or strong convexity. We perform this analysis in continuous time for the ODE, and then we transpose these results for discrete optimization schemes. In particular, we propose a variant of the Heavy Ball algorithm which has the best state of the art convergence rate for first order methods to minimize strongly, composite non smooth convex functions.

Midpoint type Fractional Integral Inequalities for convex and positive symmetric Increasing functions

: In this study, some midpoint type Hermite-Hadamard fractional integral inequalities and related results for a class of convex functions with respect to an increasing function incorporating a positive-weighted symmetric function generalizing some classical results are discussed.

First-Order Convex Fitting and Its Application to Economics and Optimization

This paper studies a function fitting problem which we coin first-order convex fitting (FCF): given any two vector sequences x1, ..., xT and p1, ..., pT, when is it possible to efficiently construct a convex function f(x) that ``fits'' the two sequences in the first-order sense, i.e, the (sub)gradient of f(xi) equals precisely pi, for all i = 1, ..., T? Despite a basic question of convex analysis, FCF has surprisingly been overlooked in the past literature. With an efficient constructive proof, we provide a clean answer to this question: FCF is possible if and only if the two sequences are permutation stable: p1 * x1 + ... + pT * xT is greater than or equal to p1 * x’1 + ... + pT * x’T where x’1, ..., x’T is any permutation of x1, ..., xT. We demonstrate the usefulness of FCF in two applications. First, we study how it can be used as an empirical risk minimization procedure to learn the original convex function. We provide efficient PAC-learnability bounds for special classes of convex functions learned via FCF, and demonstrate its application to multiple economic problems where only function gradients (as opposed to function values) can be observed. Second, we empirically show how it can be used as a surrogate to significantly accelerate the minimization of the original convex function.

On convolution, convex, and starlike mappings

"Let $C$ and $S^*$ stand for the classes of convex and starlike mapping in $\D$, and let $\cc$, $\cs$ denote the closures of the respective convex hulls. We derive characterizations for when the convolution of mappings in $\cc$ is convex, as well as when the convolution of mappings in $\cs$ is starlike. Several characterizations in terms of convolution are given for convexity within $\cc$ and for starlikeness within $\cs$. We also obtain a correspondence via convolution between $C$ and $S^*$, as well as correspondences between the subclasses of convex and starlike mappings that have $n$-fold symmetry."

Shearlet-based regularization in statistical inverse learning with an application to x-ray tomography

Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, p-homogeneous regularizers with p ∈ (1, 2], in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with ℓ p -norm regularization, with p ∈ (1, 2), for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The p = 1 case is approached as the limit case (1, 2) ∋ p → 1, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from Γ-convergence theory. We numerically validate our theoretical results in the context of x-ray tomography, under random sampling of the imaging angles, using both simulated and measured data. This application allows to effectively verify the theoretical decay, in addition to providing a motivation for the extension to shearlet-based regularization.

Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions

<abstract><p>Interval analysis distinguishes between inclusion relation and order relation. Under the inclusion relation, convexity and nonconvexity contribute to different kinds of inequalities. The construction and refinement of classical inequalities have received a great deal of attention for many classes of convex as well as nonconvex functions. Convex theory, however, is commonly known to rely on Godunova-Levin functions because their properties enable us to determine inequality terms more precisely than those obtained from convex functions. The purpose of this study was to introduce a ($ \subseteq $) relation to established Jensen-type and Hermite-Hadamard inequalities using $ (h_1, h_2) $-Godunova-Levin interval-valued functions. To strengthen the validity of our results, we provide several examples and obtain some new and previously unknown results.</p></abstract>

Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

<abstract> <p>The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

Jensen-Type Inequalities for (h, g; m)-Convex Functions

Jensen-type inequalities for the recently introduced new class of (h,g;m)-convex functions are obtained, and certain special results are indicated. These results generalize and extend corresponding inequalities for the classes of convex functions that already exist in the literature. Schur-type inequalities are given.

A Statistical Physics approach to a multi-channel Wigner spiked model

In this letter we present a finite temperature approach to a high-dimensional inference problem, the Wigner spiked model, with group-dependent signal-to-noise ratios. For two classes of convex and non-convex network architectures the error in the reconstruction is described in terms of the solution of a mean-field spin-glass on the Nishimori line. In the cases studied the order parameters do not fluctuate and are the solution of finite dimensional variational problems. The deep architecture is optimized in order to confine the high temperature phase where reconstruction fails.

Some Inequalities of Generalized p-Convex Functions concerning Raina’s Fractional Integral Operators

Convex functions play an important role in pure and applied mathematics specially in optimization theory. In this paper, we will deal with well-known class of convex functions named as generalized p-convex functions. We develop Hermite–Hadamard-type inequalities for this class of convex function via Raina’s fractional integral operator.

Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx whose coefficients constitute a convex slowly varying sequence b. The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g(b,x) from the main term of its asymptotics bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(x)/x in some neighborhood of zero admits a two-sided estimate in terms of the function σ(b,x) with absolute constants independent of b. Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g(b,x) on the class of convex slowly varying sequences in the regular case is obtained.

New Hermite\u2013Hadamard and Jensen Inequalities for Log-h-Convex Fuzzy-Interval-Valued Functions

In the preset study, we introduce the new class of convex fuzzy-interval-valued functions which is called log-h-convex fuzzy-interval-valued functions (log-h-convex FIVFs) by means of fuzzy order relation. We have also investigated some properties of log-h-convex FIVFs. Using this class, we present Jensen and Hermite–Hadamard inequalities (HH-inequalities). Moreover, some useful examples are presented to verify HH-inequalities for log-h-convex FIVFs. Several new and known special results are also discussed which can be viewed as an application of this concept.

Some Classes of Co-Schwarzian Functions and Its Coefficient Inequality that Is Sharp

In our present work, we defined an inequality called Fekete – Szegö Inequality for functions f(z) in the classes of starlike functions and convex functions along with subclasses of these classes.

Ostrowski-type inequalities for n-polynomial mathscr{P}-convex function for k-fractional Hilfer\u2013Katugampola derivative

In this article, we develop a novel framework to study a new class of convex functions known as n-polynomial mathscr{P} -convex functions. The purpose of this article is to establish a new generalization of Ostrowski-type integral inequalities by using a generalized k-fractional Hilfer–Katugampola derivative. We employ this technique by using the Hölder and power-mean integral inequalities. We present analogs of the Ostrowski-type integrals inequalities connected with the n-polynomial mathscr{P}-convex function. Some new exceptional cases from the main results are obtained, and some known results are recaptured. In the end, an application to special means is given as well. The article seeks to create an exciting combination of a convex function and special functions in fractional calculus. It is supposed that this investigation will provide new directions in fractional calculus.

Decay of correlations and memory loss for Lasota–Yorke convex maps

For a class of piecewise convex maps f on the interval , we show that f has a unique absolutely continuous invariant probability measure μ with exponential decay of correlations, and we also present the explicit upper bounds on the rate. Moreover, we show the exponential loss of memory for a sequential dynamical system consisting of piecewise convex maps.

Subordination factor sequence results for starlike and convex classes defined by q-Cătaş operator

In this paper, we investigate and introduce new two subclasses of analytic functions by using q-Cătas operator and definition of subordination. Furthermore, we obtain coefficient estimate theorems. Also, we present new results for subordination factor sequence.

Invariant densities for intermittent maps with critical points

ABSTRACT For a class of piecewise convex maps T with an indifferent fixed point and critical points on the interval , we show that T has a unique absolutely continuous invariant probability measure μ, and the invariant density has a upper bound and a lower bound. The Frobenius–Perron operator of T is asymptotically stable. We also obtain the polynomial decay rate of correlations with respect to μ by using the probabilistic method proposed by Liverani, Saussol and Vaienti.

Fuzzy-interval inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals

<abstract> <p>The objective of the authors is to introduce the new class of convex fuzzy-interval-valued functions (convex-FIVFs), which is known as $ p $-convex fuzzy-interval-valued functions ($ p $-convex-FIVFs). Some of the basic properties of the proposed fuzzy-interval-valued functions are also studied. With the help of $ p $-convex FIVFs, we have presented some Hermite-Hadamard type inequalities ($ H-H $ type inequalities), where the integrands are FIVFs. Moreover, we have also proved the Hermite-Hadamard-Fejér type inequality ($ H-H $ Fejér type inequality) for $ p $-convex-FIVFs. To prove the validity of main results, we have provided some useful examples. We have also established some discrete form of Jense's type inequality and Schur's type inequality for $ p $-convex-FIVFs. The outcomes of this paper are generalizations and refinements of different results which are proved in literature. These results and different approaches may open new direction for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions.

In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR (≼) and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) inequalities. With the support of this relation, we also derive some related HH-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.