arXiv subjects

Akhlad Iqbal

Publications and source records attributed to Akhlad Iqbal.

13 recordsLinked to original sources

arXiv2026

Interval-Valued Optimization Problems for Strongly LU-E-Invex and Strongly LU-E-Preinvex Functions

In this paper, we introduce and explore the concepts of strongly LU-E-preinvex (SLUEP), pseudo strongly LU-E-preinvex (PSLUEP) and strongly LU-E-invex (SLUEI) functions. To illustrate and validate these definitions, we provide several non-trivial examples. Additionally, we extend the idea of strongly-G invex sets to the context of interval-valued functions. The epigraph of a SLUEP function is derived, and a relationship between SLUEP and PSLUEP functions have been explored. A key contribution of this work is the identification of a significant connection between weakly-strongly E-invex functions and SLUEP functions. As an application, we consider a nonlinear programming problem involving SLUEP functions. Under certain conditions, we prove that a local minimum of the problem is also a global minimum. Moreover, the sufficiency of Karush-Kuhn-Tucker (KKT) optimality conditions by considering the objective and constraint functions are SLUEI and SEI respectively. The theoretical results are validated through illustrative examples and counterexamples.

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arXiv2025

On the Characterization of gH-partial derivatives and gH-Product for Interval-Valued Functions

In this paper, we show by a counterexample that the gH-partial derivative of interval-valued functions (IVFs) may exist even when the partial derivative of the end point functions do not. Next, we introduce the gH-partial derivative in terms of gH-derivative and discuss its complete characterization. Furthermore, we introduce the gH-product of a vector with an n-tuples of intervals and illustrate by a suitable example that our definition refines the definition existing in the literature. To illustrate and validate these definitions, we provide several non-trivial examples.

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arXiv2025

Optimality Conditions for Interval-Valued Optimization Problems on Riemannian Manifolds Under a Total Order Relation

This article explores fundamental properties of convex interval-valued functions defined on Riemannian manifolds. The study employs generalized Hukuhara directional differentiability to derive KKT-type optimality conditions for an interval-valued optimization problem on Riemannian manifolds. Based on type of functions involved in optimization problems, we consider the following cases: 1. objective function as well as constraints are real-valued; 2. objective function is interval-valued, and constraints are real-valued; 3. objective function as well as constraints are interval-valued. The whole theory is justified with the help of examples. The order relation that we use throughout the paper is a total order relation defined on the collection of all closed and bounded intervals in $\mathbb{R}$.

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arXiv2025

Generalized Hukuhara directional differentiability of interval-valued functions on Riemannian manifolds

In this paper, we show that generalized Hukuhara directional differentiability of an interval-valued function (IVF) defined on Riemannian manifolds is not equivalent to the directional differentiability of its center and half-width functions and hence not to its end point functions. This contrasts with S.-L. Chen's \cite{chen} assertion which says the equivalence holds in terms of endpoint functions of an IVF which is defined on a Hadamard manifold. Additionally, the paper addresses some other inaccuracies which arise when assuming the convexity of a function at a single point in its domain. In light of these arguments, the paper presents some basic results that relate to both the convexity and directional differentiability of an IVF.

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arXiv2024

The Karush-Kuhn-Tucker Optimality Conditions for Multi-Objective Interval-Valued Optimization Problem on Hadamard Manifolds

The KKT optimality conditions for multi-objective interval-valued optimization problem on Hadamard manifold are studied in this paper. Several concepts of Pareto optimal solutions, considered under LU and CW ordering on the class of all closed intervals in $\mathbb{R}$, are given. The KKT conditions are presented under the notions of convexity, pseudo-convexity and generalized Hukuhara difference. We show, with the help of an example, that the results done in this paper for solving multi-objective interval-valued optimization problems on Hadamard spaces are more general than the existing ones on Euclidean spaces. The main results are supported by examples.

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arXiv2024

On relationships between vector variational inequalities and optimization problems using convexificators on Hadamard manifold

An important concept of convexificators has been extended to Hadamard manifolds in this paper. The mean value theorem for convexificators on the Hadamard manifold has also been derived. Monotonicity of the bounded convexificators has been discussed and an important characterization for the bounded convexificators to be $\partial_{*}^{*}$-geodesic convexity has been derived. Furthermore, a vector variational inequalities problem using convexificators on Hadamard manifold has been considered. In addition, the necessary and sufficient conditions for vector optimization problems in terms of Stampacchia and Minty type partial vector variational inequality problem ($\partial_{*}^{*}$-VVIP) have been derived.

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arXiv2024

Relations between nonsmooth vector variational inequalities and nonsmooth vector optimization problems on Hadamard manifold in terms of bifunction

In this paper, we discuss the concepts of bifunction and geodesic convexity for vector valued functions on Hadamard manifold. The Hadamard manifold is a particular type of Riemannian manifold with non-positive sectional curvature. Using bifunction, we introduce a definition of generalized geodesic convexity in the context of the Hadamard manifold. To support the definition, we construct a non-trivial example that demonstrates the property of geodesic convexity on Hadamard manifold. Additionally, we define the geodesic $h$-convexity, geodesic $h$-pseudoconvexity and geodesic $h$-quasiconvexity for vector valued function using bifunction and study their several properties. Furthermore, we demonstrate the uniqueness of the solution for nonsmooth vector variational inequality problem (NVVIP) and prove the characterization property for the solution of NVVIP and the Minty type NVVIP (MNVVIP) on Hadamard manifold in terms of bifunction. Afterward, we consider a nonsmooth vector optimization problem (NVOP) and investigate the relationships among the solutions of NVOP, NVVIP, and MNVVIP.

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arXiv2023

Strongly geodesic preinvexity and Strongly Invariant η-Monotonicity on Riemannian Manifolds and its Application

In this paper, we present strongly geodesic preinvexity on Riemannian manifolds (RM) and strongly η-invexity of order m on RM. Furthermore, we define strongly invariant η-monotonicity of order m on RM. Under Condition C, an important characterization of these functions are studied. We construct several non-trivial examples in support of these definitions. Afterwords, an important and significant characterization of a strict η-minimizers (η-minimizers)of order m for MOP and a solution to the variational like-inequality problem (VVLIP) has been derived.

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arXiv2023

Quasi Strongly $E$-preinvexity and its Relationships with Nonlinear Programming

In this paper, we extend the class of strongly $E$-preinvex and strongly $E$-invex functions to quasi strongly $E$-preinvex, quasi strongly $E$-invex and pseudo strongly $E$-invex functions. Some nontrivial suitable examples have been constructed in support of our definitions. Several interesting properties and relationships of these functions are discussed. Furthermore, to show the application of our results, we consider a nonlinear programming problem and show that the local minimum point is also a strictly global minimum.

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arXiv2023

Non-linear programming problem for semi strongly $E$-preinvexity

In this article, we present semi strongly $E$-preinvexity and semi strongly $E$-invexity. To demonstrate the existence of these functions, certain nontrivial examples have been developed. Several significant relationships and characterizations of these functions on strongly $E$-invex sets are discussed. Furthermore, we consider a non-linear programming problem for semi strongly $E$-preinvex functions and investigate relationships between the set of optimal solutions and these functions.

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arXiv2017

A note on $p^λ$-convex set in a complete Riemannian manifold

In this paper we have generalized the notion of $λ$-radial contraction in complete Riemannian manifold and developed the concept of $p^λ$-convex function. We have also given a counter example proving the fact that in general $λ$-radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some relations between geodesic convex sets and $p^λ$-convex sets and showed that under certain conditions they are equivalent.

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arXiv2017

Some results on $φ$--convex functions and geodesic $φ$-convex functions

As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic $φ$-convex function and deduce some basic properties of $φ$-convex function and geodesic $φ$-convex function. We also introduce the concept of geodesic $φ$-convex set and $φ$-epigraph and investigate a characterization of geodesic $φ$-convex functions in terms of their $φ$-epigraphs.

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arXiv2017

Strong geodesic convex functions of order m

Strong geodesic convex function and strong monotone vector field of order $m$ on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order $m$ for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequality problem and the strict minimizers of order $m$ for a multiobjective programming problem has also been established.

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