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Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of Polyak (USSR Comput Math Math Phys 4(5):1–17, 1964), it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in Aujol et al. (Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization, 2020). Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Łojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Łojasiewicz property. [ABSTRACT FROM AUTHOR]

The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems. [ABSTRACT FROM AUTHOR]

In this paper, we proved two new Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann–Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given by İşcan (Hacet J Math Stat 46(6):935–942, 2014). [ABSTRACT FROM AUTHOR]

This paper is concerned with strictly k-convex large solutions to Hessian equations Sk(D2u(x)) = b(x)f(u(x)), x ∈ Ω, where Ω is a strictly (k − 1)-convex and bounded smooth domain in ℝn, b ∈ C ∞ ( Ω ¯) is positive in Ω, but may be vanishing on the boundary. Under a new structure condition on f at infinity, the author studies the refined boundary behavior of such solutions. The results are obtained in a more general setting than those in [Huang, Y., Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math., 244, 2010, 85–98], where f is regularly varying at infinity with index p > k. [ABSTRACT FROM AUTHOR]

In previous work (Alves et al. in Z Angew Math Phys 69:106, 2018), by using the linear semigroup theory, Alves et al. investigated the existence and exponential stability results for a Moore–Gibson–Thompson model encompassing memory of type 1, 2 or 3 in a history space framework. In this paper, we continue to consider the similar problem with type 1 and establish explicit and general decay results of energy for system in both the subcritical and critical cases, by introducing suitable energy and perturbed Lyapunov functionals and following convex functions ideas presented in Guesmia (J Math Anal Appl 382:748–760, 2011). Our results allow a much larger class of the convolution kernels which improves the earlier related results. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of structured nonsmooth difference-of-convex (DC) minimization in which the first convex component is the sum of a smooth and nonsmooth functions while the second convex component is the supremum of possibly infinitely many convex smooth functions. We first propose an inexact enhanced DC algorithm for solving this problem in which the second convex component is the supremum of finitely many convex smooth functions, and show that every accumulation point of the generated sequence is an (α , η) -D-stationary point of the problem, which is generally stronger than an ordinary D-stationary point. In addition, inspired by the recent work (Pang et al. in Math Oper Res 42(1):95–118, 2017; Wen et al. in Comput Optim Appl 69(2):297–324, 2018), we propose two proximal DC algorithms with extrapolation for solving this problem. We show that every accumulation point of the solution sequence generated by them is an (α , η) -D-stationary point of the problem, and establish the convergence of the entire sequence under some suitable assumption. We also introduce a concept of approximate (α , η) -D-stationary point and derive iteration complexity of the proposed algorithms for finding an approximate (α , η) -D-stationary point. In contrast with the DC algorithm (Pang et al. 2017), our proximal DC algorithms have much simpler subproblems and also incorporate the extrapolation for possible acceleration. Moreover, one of our proximal DC algorithms is potentially applicable to the DC problem in which the second convex component is the supremum of infinitely many convex smooth functions. In addition, our algorithms have stronger convergence results than the proximal DC algorithm in Wen et al. (2018). [ABSTRACT FROM AUTHOR]

We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth $$\ell _1$$ regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function. [ABSTRACT FROM AUTHOR]

In this paper we investigate continuity properties of functions $${f : \mathbb {R}_+ \to \mathbb {R}_+}$$ that satisfy the ( p, q)-Jensen convexity inequality where H stands for the pth power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are ( p, p)-Jensen convex for all positive rational numbers p. A counterpart of this result states that if f is ( p, p)-Jensen convex for all $${p \in P \subseteq \mathbb {R}_+}$$ , where P is a set of positive Lebesgue measure, then f must be continuous. [ABSTRACT FROM AUTHOR]

In this paper a(x)-convex functions are defined and Jensen, Hermite-Hadamard, Lah-Ribarić and other inequalities for them are derived. Related analogous of the Lagrange and the Cauchy mean value theorems are also obtained and means of the Cauchy type are generated. [ABSTRACT FROM AUTHOR]

The dynamical systems approach to sizing and SIMP topology optimization, introduced in a previous paper, is extended to the case of time-varying loads. A general dynamical system, satisfying a Lyaponov-type descent condition, is derived and specialized to a goal function combining stiffness and mass. For a cyclic time-dependent load it is indicated how, in the limit of short cycles compared to the overall time scale, this can be handled by multiple load cases. Numerical examples, both for a convex and a non-convex case, illustrates the theory. [ABSTRACT FROM AUTHOR]

Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J , ( t). In this paper, We can compute the constant J , (1) under the absolute normalized norms on ℝ by means of their corresponding continuous convex functions on [0, 1]. Moreover, some sufficient conditions which imply uniform normal structure are presented. 2000 Mathematics Subject Classification: 46B20. [ABSTRACT FROM AUTHOR]

Hubert, Mauduit and Sárközy introduced the pseudorandom measure of order ℓ of binary lattices. This measure studies the pseudorandomness only on box lattices of very special type. In certain applications one may need measures covering a more general situation. In this paper the line measure and the convex measure are introduced. [ABSTRACT FROM AUTHOR]

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Vélez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Vélez. [ABSTRACT FROM AUTHOR]

In this paper a new hybrid conjugate gradient algorithm is proposed and analyzed. The parameter β k is computed as a convex combination of the Polak-Ribière-Polyak and the Dai-Yuan conjugate gradient algorithms, i.e. β=(1− θ k) β+ θ k β. The parameter θ k in the convex combination is computed in such a way that the conjugacy condition is satisfied, independently of the line search. The line search uses the standard Wolfe conditions. The algorithm generates descent directions and when the iterates jam the directions satisfy the sufficient descent condition. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that this hybrid computational scheme outperforms the known hybrid conjugate gradient algorithms. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

The article provides an overview of the study which investigates the Jensen inequality for convex-concave antisymmetric functions and applications. It pointed out that a class of functions which is between of convex functions and the class of Jensen-convex functions is the class of Wright-convex functions. It presented several theorems which were the main motivations of the paper. It noted that it has proven that without loss of generality, it can suppose that (a,b) = (-1,1), thus, f is an odd function.

We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations. [ABSTRACT FROM AUTHOR]

This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from -Lagrangian theory and from Riemannian geometry. The Hessian based on the -Lagrangian depends on the selection of a dual parameter g; by revealing the connection to Riemannian geometry, a natural choice of g emerges for which the two Newton directions coincide. This choice of g is also shown to be related to the least-squares multiplier estimate from a sequential quadratic programming (SQP) approach, and with this multiplier, SQP gives the same search direction as the Newton methods. [ABSTRACT FROM AUTHOR]

In this paper we consider the collection of convex programming problems with inequality and equality constraints, in which every problem of the collection is obtained by linear perturbations of the cost function and right-hand side perturbation of the constraints, while the ``core'' cost function and the left-hand side constraint functions are kept fixed. The main result shows that the set of the problems which are not well-posed is σ-porous in a certain strong sense. Our results concern both the infinite and finite dimensional case. In the last case the conclusions are significantly sharper. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of uniformly starlike functions defined by certain integral operator. We determine a sufficient condition for a function f to be uniformly starlike function that is also necessary when f has negative coefficients. Similar results for corresponding subclasses of uniformly convex functions are also obtained. [ABSTRACT FROM AUTHOR]

We consider the following problem: minimize the functional ∫ Ω f (∇ u (x)) d x in the class of concave functions u : Ω → [ 0 , M ] , where Ω ⊂ R 2 is a convex body and M > 0 . If f (x) = 1 / (1 + | x | 2) and Ω is a circle, the problem is called Newton's problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of ∂ Ω are regular and (1 + | x |) f (x) / (| y | f (y)) → + ∞ as (1 + | x |) / | y | → 0 then a solution u to the problem satisfies u ⌋ ∂ Ω = 0 . This result proves the conjecture stated in 1993 in the paper by Buttazzo and Kawohl (Math Intell 15:7–12, 1993) for Newton's problem. [ABSTRACT FROM AUTHOR]

This research investigates bounds of the sum of left-sided and right-sided fractional integrals in a compact form. These bounds are established by using (α , m) -convex functions. The results of this paper also have connection with some known and already published results (Dragomir and Agarwal in Appl. Math. Lett. 11(5):91–95, 1998; Farid in J. Anal. 2018, https://doi.org/10.1007/s41478-0079-4; Farid et al. in Mathematics 2018(6):248, 2018). Moreover, the investigated results are applied to the end points as well as the mid point of the domain and some interesting consequences have been obtained. [ABSTRACT FROM AUTHOR]