MIN-MAX SOLUTIONS FOR PARAMETRIC CONTINUOUS STATIC GAME UNDER ROUGHNESS (PARAMETERS IN THE COST FUNCTION AND FEASIBLE REGION IS A ROUGH SET)

Any simple perturbation in a part of the game whether in the cost function and/or conditions is a big problem because it will require a game re-solution to obtain the perturbed optimal solution. This is a waste of time because there are methods required several steps to obtain the optimal solution, then at the end we may find that there is no solution. Therefore, it was necessary to find a method to ensure that the game optimal solution exists in the case of a change in the game data. This is the aim of this paper. We first provided a continuous static game rough treatment with Min-Max solutions, then a parametric study for the processing game and called a parametric rough continuous static game (PRCSG). In a Parametric study, a solution approach is provided based on the parameter existence in the cost function that reflects the perturbation that may occur to it to determine the parameter range in which the optimal solution point keeps in the surely region that is called the stability set of the $1^{st}$ kind. Also the sets of possible upper and lower stability to which the optimal solution belongs are characterized. Finally, numerical examples are given to clarify the solution algorithm

Procrastinated Tree Search: Black-box Optimization with Delayed, Noisy, and Multi-fidelity Feedback

In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), called ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and noise-tolerant bandit algorithm. We provide a generic proof technique to quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks. Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1) and delayed-UCB-V (DUCBV) algorithms. Given a horizon $T$, PCTS retains the regret bound of non-delayed HOO for expected delay of $O(\log T)$ and worsens by $O(T^{\frac{1-\alpha}{d+2}})$ for expected delays of $O(T^{1-\alpha})$ for $\alpha \in (0,1]$. We experimentally validate on multiple synthetic functions and hyperparameter tuning problems that PCTS outperforms the state-of-the-art black-box optimization methods for feedbacks with different noise levels, delays, and fidelity