Multi-objective Stochastic Linear bandit (MOSLB) plays a critical role in the sequential decision-making paradigm, however, most existing methods focus on the Pareto dominance among different objectives without considering any priority. In this paper, we study bandit algorithms under mixed Pareto-lexicographic orders, which can reflect decision makers' preferences. We adopt the Grossone approach to deal with these orders and develop the notion of Pareto-lexicographic optimality to evaluate the learners' performance. Our work represents a first attempt to address these important and realistic orders in bandit algorithms. To design algorithms under these orders, the upper confidence bound (UCB) policy and the prior free lexicographical filter are adapted to approximate the optimal arms at each round. Moreover, the framework of the algorithms involves two stages in pursuit of the balance between exploration and exploitation. Theoretical analysis as well as numerical experiments demonstrate the effectiveness of our algorithms.

In this work, an adaptive constraint-handling approach is developed to improve the efficiency of surrogate-based optimization (SBO). Similar to other SBO methods, the proposed approach is a sequential updating process, whereas two candidate points considering the significance of objective and constraints are generated respectively in each cycle. In detail, the candidate point of objective is obtained through the penalized lower confidence bounding (PLCB) infill criterion. Additionally, an infill criterion of the constraints (called MLCB) which can accurately characterize the boundaries of the constraints is developed to determine the candidate point of constraints. Then, a selection algorithm is developed to select one or two candidate point(s) as the new training point(s) adaptively according to the current optimal value and the accuracy of the constraint boundaries. The selection algorithm is composed of three phases. In the first phase, the candidate point of constraints is selected to find a feasible solution. Two candidate points of objective and constraints are added to speed up the convergence in the second phase. In the third phase, the candidate point of objective is chosen to improve the quality of the feasible optimal solution. The proposed approach is tested on seven numerical functions and compared with state-of-the-art methods. Results indicate that the proposed approach has excellent global optimization ability, meanwhile, it reduces significantly computational resources.