The aim of this article is to develop an approximate ()-conjugate duality theory for a general vector optimization problem with set-valued maps on the basis of -weak efficiency. We first introduce the concepts of -conjugate maps and -weak subgradients for set-valued maps and establish several properties of them. Then we introduce an -conjugate dual problem for the vector set-valued optimization problem. Finally, we establish a weak duality theorem and a strong duality theorem for the relationship between the primal and the dual problems.
IntroductionIn the last two decades, much attention has been paid to approximate solutions of optimization problems. There are two reasons for this. First, optimization models are simplified representations instead of complete copies of real problems. Second, these models are solved usually by numerical methods (iterative procedures or heuristic algorithms) which produce approximations to the theoretical solutions.On multiobjective programming or vector optimization, a significant portion of the literature deals with approximate solution concepts, approximate optimality conditions, approximate multiplier rules, approximate saddle-point theorems, approximate duality theorems, properties of approximate solutions, etc. For example, Loridan [27] originally introduced the concept of approximate (-)efficiency in general vector optimization problems. White [49] introduced six alternative concepts of approximate solutions. Examinations or relations of these concepts as well as new definitions of approximate solutions were given by Va´lyi [46], Ne´meth [32], Tammer [38-40], Helbig and Pateva [13], Tanaka [41,42], Yokoyama [51,52], Li and Wang [2], Liu [25] and Gutie´rrez, Jime´nez and Novo [12]. Optimality conditions of approximate solutions were investigated by Yokoyama [50], Liu [24], Deng [6] and Dutta and Vetrivel [7]. Approximate multiplier
This paper addresses the distributed finite-time tracking problem for multiple uncertain mechanical systems with dead-zone input and external disturbances. An observer-based adaptive finite-time consensus protocol is designed, which consists of two steps. Firstly, distributed observers are developed such that all the mechanical systems can obtain the leader’s state in finite settling time. Then, based on backstepping method and adding a power integrator technique, the finite-time consensus protocol and appropriate adaptive laws are designed to track the estimated leader’s state. Rigorous proofs show that the tracking errors between each mechanical system and the leader can converge to a small neighborhood of origin in finite time despite the presence of dead-zone nonlinearity and external disturbances. Finally, simulation example is provided to demonstrate the effectiveness of the proposed scheme.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.