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1. 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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2. On Robust Control of Partially Observed Uncertain Systems with Additive Costs

   Dave, A.; Venkatesh, N.; Malikopoulos, A.A. (May 2023, 2023 American Control Conference)

   In this paper, we consider the problem of optimizing the worst-case behavior of a partially observed system. All uncontrolled disturbances are modeled as finite-valued uncertain variables. Using the theory of cost distributions, we present a dynamic programming (DP) approach to compute a control strategy that minimizes the maximum possible total cost over a given time horizon. To improve the computational efficiency of the optimal DP, we introduce a general definition for information states and show that many information states constructed in previous research efforts are special cases of ours. Additionally, we define approximate information states and an approximate DP that can further improve computational tractability by conceding a bounded performance loss. We illustrate the utility of these results using a numerical example. 

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3. Online Allocation and Pricing: Constant Regret via Bellman Inequalities

   https://doi.org/10.1287/opre.2020.2061  

   Vera, Alberto; Banerjee, Siddhartha; Gurvich, Itai (July 2021, Operations Research)

   null (Ed.)

   We develop a framework for designing simple and efficient policies for a family of online allocation and pricing problems that includes online packing, budget-constrained probing, dynamic pricing, and online contextual bandits with knapsacks. In each case, we evaluate the performance of our policies in terms of their regret (i.e., additive gap) relative to an offline controller that is endowed with more information than the online controller. Our framework is based on Bellman inequalities, which decompose the loss of an algorithm into two distinct sources of error: (1) arising from computational tractability issues, and (2) arising from estimation/prediction of random trajectories. Balancing these errors guides the choice of benchmarks, and leads to policies that are both tractable and have strong performance guarantees. In particular, in all our examples, we demonstrate constant-regret policies that only require resolving a linear program in each period, followed by a simple greedy action-selection rule; thus, our policies are practical as well as provably near optimal. 

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4. Label consistency in overfitted generalized k-means

   Zhang, Linfan; Amini, Arash A. (January 2021, Advances in neural information processing systems)

   Ranzato, M.; Beygelzimer, A.; Dauphin, Y.; Liang, P.S.; Vaughan, J. W. (Ed.)

   We provide theoretical guarantees for label consistency in generalized k-means problems, with an emphasis on the overfitted case where the number of clusters used by the algorithm is more than the ground truth. We provide conditions under which the estimated labels are close to a refinement of the true cluster labels. We consider both exact and approximate recovery of the labels. Our results hold for any constant-factor approximation to the k-means problem. The results are also model-free and only based on bounds on the maximum or average distance of the data points to the true cluster centers. These centers themselves are loosely defined and can be taken to be any set of points for which the aforementioned distances can be controlled. We show the usefulness of the results with applications to some manifold clustering problems. 

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5. TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

   https://doi.org/10.1137/22M1526708  

   Nguyen, Hai V; Bui-Thanh, Tan (February 2024, SIAM Journal on Scientific Computing)

   Deep Learning (DL), in particular deep neural networks (DNN), by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, and hence promoting better generalization. This paper develops a model-constrained deep learning approach and its variant TNet—a Tikhonov neural network—that are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solution with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions in any desired accuracy for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for 1D deconvolution, 5 for inverse 2D heat conductivity problem, 100 for inverse initial conditions for time-dependent 2D Burgers’ equation, and 50 for inverse initial conditions for 2D Navier-Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. This is possible owing to the model-constrained term, replications, and randomization. 

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