Balendu Bhooshan Upadhyay

This article investigates a class of nonsmooth interval-valued multiobjective programming problems with vanishing constraints (NIMPPVCs) on Hadamard manifolds, under locally Lipschitz continuity assumptions on objective and constraint functions. We establish that the various standard constraint qualifications, namely Cottle-type, Slater-type, Mangasarian-Fromovitz-type, and linearly independent constraint qualifications, usually violate at any feasible point of NIMPPVC. Moreover, we introduce several NIMPPVC-tailored constraint qualifications for NIMPPVC, in particular, Abadie constraint qualification (ACQ-VC), generalized Abadie constraint qualification (GACQ-VC), generalized Guignard constraint qualification (GGCQ-VC), Cottle-type constraint qualification (CCQ-VC), Slater-type constraint qualification (SCQ-VC), Mangasarian-Fromovitz-type constraint qualification (MFCQ-VC), and linearly independent constraint qualification (LICQ-VC), and further establish interrelations among them. In addition to this, by employing GGCQ-VC, we establish the Karush-Kuhn-Tucker (KKT)-type necessary optimality conditions for LR-efficient solutions of NIMPPVC via Clarke subdifferentials. Moreover, sufficient criteria of optimality for NIMPPVC are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Furthermore, the sufficient optimality conditions established in this paper are applied to propose an algorithm for identifying the LR-efficient solutions of NIMPPVC. Various illustrative examples on Hadamard manifolds are furnished to highlight the significance of the results derived in this paper. To the best of our knowledge, constraint qualifications and optimality conditions for NIMPPVC have been investigated for the first time in the Hadamard manifold framework.

In this paper, we consider the classes of generalized Stampacchia mixed weak vector variational–hemivariational inequalities (in short, GSMWVVHIs) and generalized Minty mixed weak vector variational–hemivariational inequalities (in short, GMMWVVHIs) formulated in terms of bifunctions. By employing the Knaster–Kuratowski–Mazurkiewicz–Fan (in short, KKM-Fan) lemma, we deduce an existence result for the solutions of GSMWVVHIs without imposing any monotonicity assumptions on bifunctions and relaxed compactness hypotheses. Moreover, under a generalized, stable pseudomonotonicity hypothesis on bifunctions, we derive an existence theorem for the solutions of GMMWVVHIs and establish an equivalence relation between the solutions of the considered vector variational–hemivariational inequalities. Furthermore, uniqueness results for the solutions of the considered vector variational–hemivariational inequalities are established under suitable assumptions. In addition, we establish the stability of the solution of GSMWVVHIs under some appropriate conditions. We furnish several illustrative examples to demonstrate the significance of the results established in this paper. Finally, we demonstrate that the sustainable electricity generation planning problem can be naturally formulated as a GSMWVVHI.

In this paper, using the properties of Clarke subdifferential, we derive some characterizations of locally Lipschitz geodesic pseudolinear functions on Hadamard manifolds. Moreover, employing these characterizations, we derive some properties of the solution sets of pseudolinear programming problems on the Hadamard manifold. The results established in this paper generalize and extend several known results in the literature. We furnish several examples to illustrate these results. Furthermore, to demonstrate the practical applications of our outcomes, we solve a specific form of the Karcher mean problem by employing the results established in this paper.

In this paper, we consider the classes of generalized Stampacchia mixed weak vector variational-hemivariational inequalities (in short, GSMWVVHI) and generalized Minty mixed weak vector variational-hemivariational inequalities (in short, GMMWVVHI), formulated in terms of bifunctions. By employing the Knaster-Kuratowski-Mazurkiewicz-Fan (in short, KKM-Fan) lemma, we deduce an existence result for the solutions of GSMWVVHI without imposing any monotonicity assumptions on bifunctions and relaxed compactness hypotheses. Moreover, under generalized stable pseudomonotonicity hypotheses on bifunctions, we derive an existence theorem for the solutions of GMMWVVHI and establish an equivalence relation between the solutions of GSMWVVHI and GMMWVVHI. Furthermore, uniqueness results for the solutions of GSMWVVHI and GMMWVVHI are established under suitable assumptions. Several illustrative examples have been furnished to highlight the applicability and relevance of the established results. To the best of our knowledge, existence and uniqueness results for solutions of GSMWVVHI and GMMWVVHI have been established for the first time in this paper on the Euclidean spaces.