Non-Smooth, H\"older-Smooth, and Robust Submodular Maximization

We study the problem of maximizing a continuous DR-submodular function that is not necessarily smooth. We prove that the continuous greedy algorithm achieves an [(1-1/e)\OPT-\epsilon] guarantee when the function is monotone and H\"older-smooth, meaning that it admits a H\"older-continuous gradient. For functions that are non-differentiable or non-smooth, we propose a variant of the mirror-prox algorithm that attains an [(1/2)\OPT-\epsilon] guarantee. We apply our algorithmic frameworks to robust submodular maximization and distributionally robust submodular maximization under Wasserstein ambiguity. In particular, the mirror-prox method applies to robust submodular maximization to obtain a single feasible solution whose value is at least (1/2)\OPT-\epsilon. For distributionally robust maximization under Wasserstein ambiguity, we deduce and work over a submodular-convex maximin reformulation whose objective function is H\"older-smooth, for which we may apply both the continuous greedy and the mirror-prox algorithms

A New Approach of the Online Tuning Gain Scheduling Nonlinear PID Controller Using Neural Network

This chapter presents the design, development and implementation of a novel proposed online-tuning Gain Scheduling Dynamic Neural PID (DNN-PID) Controller using neural network suitable for real-time manipulator control applications. The unique feature of the novel DNN-PID controller is that it has highly simple and dynamic self-organizing structure, fast online-tuning speed, good generalization and flexibility in online-updating. The proposed adaptive algorithm focuses on fast and efficiently optimizing Gain Scheduling and PID weighting parameters of Neural MLPNN model used in DNN-PID controller. This approach is employed to implement the DNN-PID controller with a view of controlling the joint angle position of the highly nonlinear pneumatic artificial muscle (PAM) manipulator in real-time through Real-Time Windows Target run in MATLAB SIMULINK® environment. The performance of this novel proposed controller was found to be outperforming in comparison with conventional PID controller. These results can be applied to control other highly nonlinear SISO and MIMO systems. Keywords: highly nonlinear PAM manipulator, proposed online tuning Gain Scheduling Dynamic Nonlinear PID controller (DNN-PID), real-time joint angle position control, fast online tuning back propagation (BP) algorithm, pneumatic artificial muscle (PAM) actuator

Projection-Free Online Convex Optimization with Stochastic Constraints

This paper develops projection-free algorithms for online convex optimization with stochastic constraints. We design an online primal-dual projection-free framework that can take any projection-free algorithms developed for online convex optimization with no long-term constraint. With this general template, we deduce sublinear regret and constraint violation bounds for various settings. Moreover, for the case where the loss and constraint functions are smooth, we develop a primal-dual conditional gradient method that achieves $O(\sqrt{T})$ regret and $O(T^{3/4})$ constraint violations. Furthermore, for the setting where the loss and constraint functions are stochastic and strong duality holds for the associated offline stochastic optimization problem, we prove that the constraint violation can be reduced to have the same asymptotic growth as the regret