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1. Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting

   https://doi.org/10.1109/TAC.2026.3675545  

   Zheng, Yang; Pai, Chih-Fan Rich; Tang, Yujie (March 2026, IEEE Transactions on Automatic Control)

   Many optimal and robust control problems are nonconvex and potentially nonsmooth in their policy optimization forms. In Part II of this paper, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our optimization perspective offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. This ECL framework can cover many benchmark control problems, including LQR, LQG, and H∞ robust control. We also believe that the new ECL framework will be of independent interest for analyzing nonconvex problems beyond control. 

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2. Benign Nonconvex Landscapes in Optimal and Robust Control, Part I: Global Optimality

   https://doi.org/10.1109/TAC.2026.3675544  

   Zheng, Yang; Pai, Chih-Fan Rich; Tang, Yujie (March 2026, IEEE Transactions on Automatic Control)

   Direct policy search has achieved great empirical success in reinforcement learning. Many recent studies have revisited its theoretical foundation for continuous control, which reveals elegant nonconvex geometry in various benchmark problems. This paper considers two fundamental optimal and robust control problems with partial observability: Linear Quadratic Gaussian (LQG) control and H∞ robust control. In the policy space, the former problem is smooth but nonconvex, while the latter one is nonsmooth and nonconvex. We highlight some interesting and surprising “discontinuity” of LQG and H∞ cost functions around the boundary of their domains. Despite the lack of convexity (and possibly smoothness), we show that for a class of non-degenerate policies, all Clarke stationary points are globally optimal and there is no spurious local minimum for both LQG and H∞ control. The main results are established by a new and unified framework of Extended Convex Lifting (ECL), which reconciles the gap between nonconvex policy optimization and convex reformulations. This ECL framework is of independent interest, and we discuss its details in Part II of this paper. 

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3. Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

   https://doi.org/10.1287/moor.2021.1178  

   Blanchet, Jose; Murthy, Karthyek; Zhang, Fan (May 2022, Mathematics of Operations Research)

   We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent. 

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4. Optimal Control Strategies for Systems with Input Delay using the PIE Framework

   https://doi.org/10.1016/j.ifacol.2022.11.339  

   Peet, Matthew M. (January 2022, IFAC-PapersOnLine)

   The Partial Integral Equation (PIE) framework provides a unified algebraic representation for use in analysis, control, and estimation of infinite-dimensional systems. However, the presence of input delays results in a PIE representation with dependence on the derivative of the control input, u˙. This dependence complicates the problem of optimal state-feedback control for systems with input delay – resulting in a bilinear optimization problem. In this paper, we present two strategies for convexification of the H∞-optimal state-feedback control problem for systems with input delay. In the first strategy, we use a generalization of Young's inequality to formulate a convex optimization problem, albeit with some conservatism. In the second strategy, we filter the actuator signal – introducing additional dynamics, but resulting in a convex optimization problem without conservatism. We compare these two optimal control strategies on four example problems, solving the optimization problem using the latest release of the PIETOOLS software package for analysis, control and simulation of PIEs. 

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5. On tractability, complexity, and mixed-integer convex programming representability of distributionally favorable optimization

   https://doi.org/10.1007/s10107-025-02299-w  

   Jiang, Nan; Xie, Weijun (November 2025, Mathematical Programming)

   Abstract Distributionally Favorable Optimization (DFO) is a framework for decision-making under uncertainty, with applications spanning various fields, including reinforcement learning, online learning, robust statistics, chance-constrained programming, and two-stage stochastic optimization without complete recourse. In contrast to the traditional Distributionally Robust Optimization (DRO) paradigm, DFO presents a unique challenge– the application of the inner infimum operator often fails to retain the convexity. In light of this challenge, we study the tractability and complexity of DFO. We establish sufficient and necessary conditions for determining when DFO problems are tractable (i.e., solvable in polynomial time) or intractable (i.e., not solvable in polynomial time). Despite the typical nonconvex nature of DFO problems, our results show that they are mixed-integer convex programming representable (MICP-R), thereby enabling solutions via standard optimization solvers. Finally, we numerically validate the efficacy of our MICP-R formulations. 

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