symmetry S S Article Multiobjective Convex Optimization in Real Banach Space Kin Keung Lai 1, * ,† , Mohd Hassan 2,† , Jitendra Kumar Maurya 3,† , Sanjeev Kumar Singh 2,† and Shashi Kant Mishra 2,†   Citation: Lai, K.K.; Hassan, M.; Maurya, J.K.; Singh, S.K.; Mishra, S.K. Multiobjective Convex Optimization in Real Banach Space. Symmetry 2021, 13, 2148. https://doi.org/10.3390/ sym13112148 Academic Editors: Octav Olteanu and Savin Treanta Received: 8 October 2021 Accepted: 27 October 2021 Published: 10 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 International Business School, Shaanxi Normal University, Xi’an 710119, China 2 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India; mohd.hassan10@bhu.ac.in (M.H.); sanjeevk.singh1@bhu.ac.in (S.K.S.); shashikant.mishra@bhu.ac.in (S.K.M.) 3 Kashi Naresh Government Postgraduate College, Bhadohi 221304, India; jitendrak.maurya1@bhu.ac.in * Correspondence: mskklai@outlook.com † These authors contributed equally to this work. Abstract: In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn– Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019. Keywords: multiobjective programming; nonlinear programming; convex optimization; saddle point 1. Introduction Consider the general multiobjective optimization problem (MOP) min f ( x)=( f 1 ( x), ··· , f p ( x)), subject to g( x) ≦ 0, h( x)= 0, (1) where the functions f : X → R p , g : X → R q , and h : X → R r are real vector valued functions and X is real Banach space. Multiobjective optimization problem (MOP) arises when two or more objective func- tions are simultaneously optimized over a feasible region. The multiobjective optimization has been considerably analyzed and studied by many researchers, see for instance [1–6]. Multiobjective optimization problems play a crucial role in various fields like economics, engineering, management sciences [2,7–11], and many more places in daily life. To deal with the multiobjective optimization problems, we have to find Pareto optimal solutions. These solutions are non-dominated by one another. A solution is called non- dominated or Pareto optimal if none of the objective functions can be improved in value without reducing one or more objective values. One of the best techniques to deal with multiobjective optimization problems is scalarization. Wendell and Lee [12] developed the scalarization technique to deal with multiobjective optimization problems. Wendell and Lee [12] generalized the results on efficient points for multiobjective optimization problems to nonlinear optimization problems. The multiobjective problem is converted into a single objective problem in the scalarization technique. The saddle point optimality conditions are briefly explained in [13], Rooyen et al. [14] constructed a Langrangian function for the convex multiobjective problem and established a relationship between saddle point optimality conditions and Pareto optimal solutions. Symmetry 2021, 13, 2148. https://doi.org/10.3390/sym13112148 https://www.mdpi.com/journal/symmetry