Class Of Nonconvex Functions Research Articles

New Class Up and Down Pre-Invex Fuzzy Number Valued Mappings and Related Inequalities via Fuzzy Riemann Integrals

Numerous applications of the theory of convex and nonconvex mapping exist in the fields of applied mathematics and engineering. In this paper, we have defined a new class of nonconvex functions which is known as up and down pre-invex (pre-incave) fuzzy number valued mappings (F-N-V∙Ms). The well-known fuzzy Hermite–Hadamard (

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.

Fractional Versions of Hermite-Hadamard, Fejér, and Schur Type Inequalities for Strongly Nonconvex Functions

In modern world, most of the optimization problems are nonconvex which are neither convex nor concave. The objective of this research is to study a class of nonconvex functions, namely, strongly nonconvex functions. We establish inequalities of Hermite-Hadamard and Fejér type for strongly nonconvex functions in generalized sense. Moreover, we establish some fractional integral inequalities for strongly nonconvex functions in generalized sense in the setting of Riemann-Liouville integral operators.

Sifat Kemonotonan Barisan Trapezoid Sum dari Kelas Fungsi Nonkonveks dan Nonkonkaf

The objective of this research is to show the monotonicity properties of the trapezoid sum sequence in general of nonconvex or nonconcave real valued continuous functions on interval corresponding to partitions of obtained by dividing into equal length subintervals. The decreasing monotony of the trapezoid sum generically does not happen in class of nonconcave functions. The same thing happens when restricted to the monotone nonconcave functions, namely in class of nonconcave increasing or nonconcave decreasing functions. Furthermore, in class of nonconvex functions, the trapezoid sum sequence generically does not increasing, as well as in class of increasing nonconvex or decreasing nonconvex functions.

Some Fractional Integral Inequalities for a Generalized Class of Nonconvex Functions

Fractional integral inequalities help to solve many difference equations. In this paper, we present some fractional integral inequalities for generalized harmonic nonconvex functions. Moreover, we also present applications of developed inequalities.

Some New Fractional Estimates of Inequalities for LR-p-Convex Interval-Valued Functions by Means of Pseudo Order Relation

It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis, both the inclusion relation (⊆) and pseudo order relation (≤p) are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-p-convex interval-valued functions (LR-p-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-p-convex-IVFs and Hermite-Hadamard type inequalities (HH-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-p-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area.

Integral inequalities for hyperbolic type preinvex functions

<abstract> In this work, we establish the concept of a new class of non-convex functions, namely hyperbolic type preinvex functions. Secondly few algebraic properties of this class are obtained. Further Hermite-Hadamard type integral inequalities are established for this class. We also derive several new inequalities for the functions for which absolute value of first derivative, with exponent greater or equal to one is hyperbolic type preinvexity. The results are obtained by using both the Hölder's inequality and Hölder-Iscan inequality and compared at the end. Several special cases are discussed as applications of the results. </abstract>

Gradient Surfing: A New Deterministic Approach for Low-Dimensional Global Optimization

We describe a novel global optimization technique which utilizes global minima basins of attraction in order to quickly converge to a global minima. A key to the proposed method is the “steeper goes deeper” heuristic: coupling between magnitudes of gradients on different level sets of a basin of attraction and the depth of its minima. Local minima are avoided with a combination of local optimization and a heuristic-based leaping step. Gradient surfing performance is evaluated across a set of small-scale problems from the literature, and results are compared to those of 12 previously published methods. A practical six-dimensional non-convex image registration application is presented as well, where GS performance exceeds that of classic global optimization methods in both speed and accuracy. Additionally, we validate the optimization method by applying a new Gaussian mixture model benchmark for non-convex function. Finally, the “steeper goes deeper” heuristic is validated empirically on five different classes of non-convex functions using two different evaluation approaches. In all cases, steeper gradients are shown to lead to deeper optima with a high probability.

The q-asymptotic function in c-convex analysis

We establish several connections between generalized asymptotic functions and different areas of convexity theory, without coercivity assumptions. Properties and characterizations of abstract subdifferentials, normal cones, conjugates, support functions and optimality conditions for the minimization problem are given. We provide a new result on existence of minimizers for a class of nonconvex functions which is strictly larger than the class of quasiconvex ones.

A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering

$\ell_1$ mean filtering is a conventional, optimization-based method to estimate the positions of jumps in a piecewise constant signal perturbed by additive noise. In this method, the $\ell_1$ norm penalizes sparsity of the first-order derivative of the signal. Theoretical results, however, show that in some situations, which can occur frequently in practice, even when the jump amplitudes tend to $\infty$, the conventional method identifies false change points. This issue is referred to as stair-casing problem and restricts practical importance of $\ell_1$ mean filtering. In this paper, sparsity is penalized more tightly than the $\ell_1$ norm by exploiting a certain class of nonconvex functions, while the strict convexity of the consequent optimization problem is preserved. This results in a higher performance in detecting change points. To theoretically justify the performance improvements over $\ell_1$ mean filtering, deterministic and stochastic sufficient conditions for exact change point recovery are derived. In particular, theoretical results show that in the stair-casing problem, our approach might be able to exclude the false change points, while $\ell_1$ mean filtering may fail. A number of numerical simulations assist to show superiority of our method over $\ell_1$ mean filtering and another state-of-the-art algorithm that promotes sparsity tighter than the $\ell_1$ norm. Specifically, it is shown that our approach can consistently detect change points when the jump amplitudes become sufficiently large, while the two other competitors cannot.

Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-convex Functionals: A Simple Approach Via Quadratic Lower Bounds

We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the constant state. In doing so, we numerically derive an almost optimal lower bound for both the order–disorder transition curve of the Ohta–Kawasaki energy and the liquid–solid interface of the phase-field crystal energy. We discuss how this strategy extends to non-constant computed metastable states, and the resulting symmetry issues that one must overcome. We give a preliminary analysis of these symmetry issues by addressing the global optimality of a computed lamellar structure for the Ohta–Kawasaki energy in one (1D) and two (2D) space dimensions. We also consider global optimality of a non-constant state for a spatially in-homogenous perturbation of the 2D Ohta–Kawasaki energy. Finally we use one of our simple quadratic lower bounds to rigorously prove that for certain values of the Ohta–Kawasaki parameter and aspect ratio of an asymmetric torus, any global minimizer $$v(x)$$ for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus.

Differentiable non-convex functions and general variational inequalities

In this paper, we introduce a new class of non-convex functions, which is called the g-convex functions. We prove that the minimum of the differentiable non-convex functions on the non-convex sets can be characterized by a class of variational inequalities, which is called the general variational inequality. Essentially using the projection technique, we establish the equivalence between the general variational inequalities and the fixed-point problems as well as with the Wiener–Hopf equations. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general variational inequalities. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.

γ-preinvexity and γ-invexity in mathematical programming

In the paper, we introduce and characterize a new class of nonconvex functions in mathematical programming, named γ-preinvex functions, which are called γ-invex functions in the case of differentiability. The class of γ-reinvex functions is wider than that of preinvex functions and thus, in the case of differentiability, than the class of invex functions. Some optimality results are obtained under γ-preinvexity assumption for a constrained optimization problem with not necessarily differentiable functions. Further, a number of sufficiency conditions and Wolfe duality theorems are established under γ-invexity assumption. An alternative approach with a modified γ-invexity notion to optimality conditions and duality results is also considered.

A class of B-( p, r)-invex functions and mathematical programming

Invexity of a function is generalized. The new class of nonconvex functions, called B-( p, r)-invex functions with respect to η and b, being introduced, includes many well-known classes of generalized invex functions as its subclasses. Some properties of the introduced class of B-( p, r)-invex functions with respect to η and b are studied. Further, mathematical programming problems involving B-( p, r)-invex functions with respect to η and b are considered. The equivalence between saddle points and optima, and different type duality theorems are established for this type of optimization problems.

Submonotone Mappings in Banach Spaces and Applications

The notions ‘submonotone’ and ‘strictly submonotone’ mapping, introduced by J. Spingarn in ℝn, are extended in a natural way to arbitrary Banach spaces. Several results about monotone operators are proved for submonotone and strictly submonotone ones: Rockafellar's result about local boundedness of monotone operators; Kenderov's result about single-valuedness and upper-semicontinuity almost everywhere of monotone operators in Asplund spaces; minimality (w*-cusco mappings) of maximal strictly submonotone mappings, etc. It is shown that subdifferentials of various classes of nonconvex functions defined as pointwise suprema of quasi-differentiable functions possess submonotone properties. Results about generic differentiability of such functions are obtained (among them are new generalizations of an Ekeland and Lebourg's theorem). Applications are given to the properties of the distance function in a Banach space with a uniformly Gateaux differentiable norm.

On lower perturbations of a class of non-convex integrals*

Given h : R→R in a certain class of non-convex functions, we determine lower continuous perturbations to the (non-convex) integral of the Laplacian associated to h. This allows us to formulate a perturbed minimizing problem without any solution although the unperturbed problem always admits a solution. When some additional conditions on h are imposed the perturbation can be chosen as small as one may desire.

Existence theorems for a class of nonconvex problems in the calculus of variations

We give some existence results of minima for a class of nonconvex functionals depending on the Laplacian. We minimize these functionals on the set of functionsu inW2,p (Ω) ∩W01,p (Ω) such that ∂u/∂n=0 on ∂Ω,p>1, with Ω either an annulus or the whole space ℝn. Our approach allows us to deal with integrands without any regularity conditions. The results are obtained first by showing that the corresponding convexified problem has at least one radially symmetric solution via a rotation; then, by using a Liapunov's theorem on the range of a vector-valued measure, we construct a function that is a solution to our problem.

A class of nonconvex functions and pre-variational inequalities

A class of nonconvex functions is introduced, called semi-preinvex function, which includes the classes of preinvex functions and arc-connected convex functions. The Fritz-John conditions of the mathematical programming problem are derived for these kinds of functions. The pre-variational inequality is given as a necessary condition and also a sufficient condition for a mathematical programming for invex functions. The Type I function related to unconstrained problems is given as an equivalent form of the pre-variational inequality. Existence theorems for the solution of the pre-variational inequality are also proved.

Convergence of the constrained gradient method for a class of nonconvex functions

Convergence of the constrained gradient method for a class of nonconvex functions

A class of nonconvex functions and mathematical programming

A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.