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1. Method for solving chance constrained optimal control problems using biased kernel density estimators

   https://doi.org/10.1002/oca.2675  

   Keil, Rachel_E; T Miller, Alexander; Kumar, Mrinal; Rao, Anil_V (October 2020, Optimal Control Applications and Methods)

   Summary A method is developed to numerically solve chance constrained optimal control problems. The chance constraints are reformulated as nonlinear constraints that retain the probability properties of the original constraint. The reformulation transforms the chance constrained optimal control problem into a deterministic optimal control problem that can be solved numerically. The new method developed in this paper approximates the chance constraints using Markov Chain Monte Carlo sampling and kernel density estimators whose kernels have integral functions that bound the indicator function. The nonlinear constraints resulting from the application of kernel density estimators are designed with bounds that do not violate the bounds of the original chance constraint. The method is tested on a nontrivial chance constrained modification of a soft lunar landing optimal control problem and the results are compared with results obtained using a conservative deterministic formulation of the optimal control problem. Additionally, the method is tested on a complex chance constrained unmanned aerial vehicle problem. The results show that this new method can be used to reliably solve chance constrained optimal control problems. 

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2. A Galerkin Approach to the Generalized Karush–Kuhn–Tucker Conditions for the Solution of an Elliptic Distributed Optimal Control Problem with Pointwise State and Control Constraints

   https://doi.org/10.1515/cmam-2025-0007  

   Brenner, Susanne C; Sung, Li-Yeng (February 2025, Computational Methods in Applied Mathematics)

   Abstract We develop a convergence analysis for the simplest finite element method for a model linear-quadratic elliptic distributed optimal control problem with pointwise control and state constraints under minimal assumptions on the constraint functions.We then derive the generalized Karush–Kuhn–Tucker conditions for the solution of the optimal control problem from the convergence results of the finite element method and the Karush–Kuhn–Tucker conditions for the solutions of the discrete problems. 

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3. Outer bounds for Gaussian multiple access channels with state known at one encoder

   https://doi.org/10.1109/ISIT.2017.8006652  

   Yang, Wei; Liang, Yingbin; Shitz, Shlomo Shamai; Poor, H. Vincent (June 2017, 2017 IEEE International Symposium on Information Theory (ISIT))

   This paper studies a two-user state-dependent Gaus- sian multiple-access channel with state noncausally known at one encoder. Two new outer bounds on the capacity region are derived, which improve uniformly over the best known (genie- aided) outer bound. The two corner points of the capacity region as well as the sum rate capacity are established, and it is shown that a single-letter solution is adequate to achieve both the corner points and the sum rate capacity. Furthermore, the full capacity region is characterized in situations in which the sum rate capacity is equal to the capacity of the helper problem. The proof exploits the optimal-transportation idea of Polyanskiy and Wu (which was used previously to establish an outer bound on the capacity region of the interference channel) and the worst- case Gaussian noise result for the case in which the input and the noise are dependent. 

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4. High Probability Convergence of Stochastic Gradient Methods

   Liu, Zijian; Nguyen, Ta Duy; Nguyen, Thien Hang; Ene, Alina; Nguyen, Huy (July 2023, International Conference on Machine Learning)

   In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. The method can be applied to the non-convex case. We demonstrate an $$O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$$ convergence rate when the number of iterations $$T$$ is known and an $$O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$$ convergence rate when $$T$$ is unknown for SGD, where $$1-\delta$$ is the desired success probability. These bounds improve over existing bounds in the literature. We also revisit AdaGrad-Norm (Ward et al., 2019) and show a new analysis to obtain a high probability bound that does not require the bounded gradient assumption made in previous works. The full version of our paper contains results for the standard per-coordinate AdaGrad. 

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5. High Probability Convergence of Stochastic Gradient Methods

   Liu, Zijian; Nguyen, Ta Duy; Nguyen, Thien Hang; Nguyen, Huy (July 2023, Proceedings of Machine Learning Research)

   In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. The method can be applied to the non-convex case. We demonstrate an $$O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$$ convergence rate when the number of iterations $$T$$ is known and an $$O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$$ convergence rate when $$T$$ is unknown for SGD, where $$1-\delta$$ is the desired success probability. These bounds improve over existing bounds in the literature. We also revisit AdaGrad-Norm \cite{ward2019adagrad} and show a new analysis to obtain a high probability bound that does not require the bounded gradient assumption made in previous works. The full version of our paper contains results for the standard per-coordinate AdaGrad. 

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