In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces are characterized, and necessary and sufficient conditions are presented to prove that the subdifferential set in Hadamard spaces is nonempty. Proximal subdifferential in Hadamard spaces is addressed and some basic properties are highlighted. Finally, a density theorem for subdifferential set is established.

In this paper, we investigate relation between weak subdifferential and augmented normal cone. We define augmented normal cone via weak subdifferential and vice versa. The necessary conditions for the global maximum are also stated. We produce preliminary properties of augmented normal cones and discuss them via the distance function. Then we obtain the augmented normal cone for the indicator function. Relation between weak subifferential and augmented normal cone and epigraph is also explored. We also obtain optimality conditions via weak subdifferential and augmented normal cone. Finally, we define the Stampacchia and Minty solution via weak subdifferential and investigate the relation between Stampacchia and Minty solution and the minimal point.

The purpose of this paper is to develop nonsmooth optimization problems (P) in which all emerging functions are assumed to be real-valued quasiconvex functions that are defined on a finite-dimensional Euclidean space. First, we introduce two linear optimization problems with the same optimal value of the considered problem. Then, we introduce a real-valued non-negative gap function for (P), and we provide some conditions which ensure that its null points are the same as the optimal solution of problem (P). The results are based on incident subdifferential, which is an important concept in the analysis of quasiconvex functions.

In this paper, some properties of  pseudoinvex functions, defined by means of  Limiting subdifferential, are discussed. Furthermore, the Minty vector variational-like inequality,  the Stampacchia vector variational-like inequality, and the  weak formulations of these two inequalities  defined by means of Limiting subdifferential are studied. Moreover, some relationships  between the vector variational-like inequalities and vector optimization problems are established.

This article is concerned with non-smooth/nonconvex robust multi-objective optimization problems involving uncertain inequality and equality constraints. Employing some advanced tools of variational analysis such as the approximate extremal principle and the weak fuzzy sum rule for the Frechet subdifferential, we first drive a fuzzy necessary optimality condition of a non-smooth/nonconvex multi-objective optimization problem without any constrained qualification in the sense of the Frechet subdifferential. Then by exploiting the obtained fuzzy optimality condition, the non-smooth version of Fermat’, s rule and formulae for the Limiting subdifferential of an infinite family of non-smooth functions, we establish a necessary optimality condition in terms of the Limiting subdifferential for weakly robust efficient solutions of the reference problem. Further, we present an example to illustrate this condition for an uncertain multi-objective optimization problem involving equality and inequality constraints. Finally sufficient conditions for weakly robust efficient solutions and robust efficient solutions of the problems are provided by presenting new concepts of generalized convexity.

In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are locally Lipschitz.Necessary optimality conditions and regularity conditions are given. Our approach are based on the Michel-Penot subdifferential.

Purpose: To compare the anatomical and functional results of deep vitrectomy with and without ILM peeling for full thickness macular hole.Materials & Methods: 37 eyes of 37 patients with full thickness macular hole (FTMH), confirmed by clinical examination and optical coherence tomography (OCT), enrolled in this clinical trial. Based on the stage of FTMH (stage 2, 3, 4) as described by Gass and the techniques of surgery (with or without ILM peeling) the patients were randomized into two different groups. All patients underwent preoperative Act and at least one OCT image was obtained during the postoperative follow up.After obtaining informed consent, deep vitrectomy and 20% SF6 injection was performed. The patients were unaware of their allocation status. Patients data & surgical outcome were collected and statistically analyzed. Anatomical success was defined as hole closure at postoperative OCT and functional success was referred to any improvement in visual acuity. Macular hole index (MHI) was defined as a ratio of the greatest height to the base diameter of the hole on cross-sectional OCT images.Results: 37 eyes of 37 patients (15 men & 22 women) were enrolled. The mean age of the patients was 56.4±21.94 years and FTMH were traumatic in 13 cases (35%) and Idiopathic in 24 (65%) cases. Twenty cases underwent deep vitrectomy with ILM peeling (ILM group) and the remaining cases underwent vitrectomy without ILM peeling (no ILM group).The mean preoperative 10gMAR visual acuity was 1.38±1 in the ILM group and 1.22±0.13 in the no ILM group, these improved to 0.95±1 and 0.77±0.31 postoperatively in the ILM and no ILM group, respectively (P=0.871). The overall anatomical success rate was 64%. The hole was closed in 70% of ILM peeling and 47% in the no ILM group (Chi-square test, P=0.15). In anatomically successful cases the mean of postoperative VA was significantly improved (P=0.001).In traumatic subgroup, ILM peeling seems did not affect the anatomical (P=0.2) and visual Success rates (P=0.5). There was no significant differences in hole closure rates between the traumatic and idiopathic FTMH (P=0.968). Visual acuity significantly improved after operation In MHI ³ 0.5 group compared with the MHk 0.5 group. The stage of the hole did not affect the anatomical success rate (P=0.52) or visual acuity improvement (P=0.741). Macular hole index ³ 0.5 had a prognostic value for postoperative visual acuity improvement. The hole duration was not related to anatomical success rate but there is an inverse relation between duration of the symptoms and improvement in VA.Conclusion: Deep vitrectomy and SF6 injection was a safe & effective surgical technique for full thickness macular hole and peeling of the ILM seems did not affect the anatomical or functional success rates. The hole duration, stage and origin of the hole (traumatic or idiopathic) does not related to the anatomical success or visual acuity improvement. MHI  ³ 0.5 seems to be a prognostic factor for postoperative visual improvement but not for closure rate.

We are interested in proposing the convex fuzzy functions from R into E to describe the concepts of left and right generalized derivatives. In this way, several properties of the concepts are discussed. As well, the notions of generalized subdifferential for these functions are studied and calculated. Finally, the properties of the concepts are illustrated and applied through some examples.

Several voting systems are utilized to allocate both political and non-political positions across countries worldwide. Plurality voting and approval voting are among the most widely implemented electoral systems. Established statistical methods are employedto ensure the accuracy of vote counting and the validation of election results. Two fundamental approaches that significantly enhance the likelihood of identifying errors in election outcomes are Risk-Limiting audit (RLA) and Bayesian audit (BA). These audit methods assess the security of elections using statistical tools, based on the random selection of castvotes and their interpretation as evidence supporting or contradicting the reported results. In this paper, we first examine the advantages of a specific form of approval voting, referred to as score voting, and then describe two types of risk-Limiting audits to evaluate the accuracy of vote counting and the results. The proposed auditing method for score voting is adapted from the Ballot-polling risk-Limiting audits to verify outcomes (BRAVO). Simulation results confirm the effectiveness and accuracy of our approach.