In the frame of fractional calculus, term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus objective this review paper present Hermite–Hadamard (H-H)-type inequalities involving a variety classes convexities pertaining integral operators. Included various are classical convex functions, m-convex r-convex (α,m)-convex (α,m)-geometric...

the rational cubic function with three parameters has been extended to rational bi-cubic function to visualize the shape of regular convex surface data. the rational bi-cubic function involves six parameters in each rectangular patch. data dependent constraints are derived on four of these parameters to visualize the shape of convex surface data while other two are free to refine the shape of s...

The concept of M-convex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress–Wenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete op...

M-convex functions have various desirable properties as convexity in discrete optimization. We can find a global minimum of an M-convex function by a greedy algorithm, i.e., so-called descent algorithms work for the minimization. In this paper, we apply a scaling technique to a greedy algorithm and propose an efficient algorithm for the minimization of an M-convex function. Computational result...

We study the minimization of an M-convex function introduced by Murota. It is shown that any vector in the domain can be easily separated from a minimizer of the function. Based on this property, we develop a polynomial time algorithm.

This supplement provides the proof for all the theorems and corollaries in the main paper. We provide the proofs as well as the theorem statements for the ease of understanding while borrowing the notation from the main draft. Proof. Assume that we are given an > 0 for which the condition mentioned in the theorem statement holds true (for the sake of simplicity we assume the same for all w ∈ R ...

We discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and, more generally,M-estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set S. We give a counterexample showing that regularity of S in the sense of Clarke is not sufficient for asymptotic equivalence of √ n-consistent locally optimal M-estimato...

Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

M-convex functions, introduced by Murota (1996, 1998), enjoy various desirable properties as “discrete convex functions.” In this paper, we propose two new polynomial-time scaling algorithms for the minimization of an M-convex function. Both algorithms apply a scaling technique to a greedy algorithm for M-convex function minimization, and run as fast as the previous minimization algorithms. We ...

In this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite n × n matrices. We show that the condition number is a Clarke regular strongly pseudoconvex function. We prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program. This asymptotic ana...