We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the “larger number game.” This generalizes a recent theorem of Hanneke, Livni, and Moran. We also propose several applications of our results outside of game theory.

Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far. Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged.

We study the interplay between inventory replenishment policies and information sharing in the context of a two-tier supply chain with a single supplier and a single retailer serving an independent and identically distributed Gaussian market demand. We investigate how the retailer’s inventory policy impacts the supply chain’s cumulative expected long-term average inventory costs [Formula: see text] in two extreme information-sharing cases: (a) full information sharing and (b) no information sharing. To find the retailer’s inventory policy that minimizes [Formula: see text], we formulate an infinite-dimensional optimization problem whose decision variables are the MA([Formula: see text]) coefficients that characterize a stationary ordering policy. Under full information sharing, the optimization problem admits a simple solution and the optimal policy is given by an MA(1) process. On the other hand, to solve the optimization problem under no information sharing, we reformulate the optimization from its time domain formulation to an equivalent z-transform formulation in which the decision variables correspond to elements of the Hardy space H2. This alternative representation allows us to use a number of results from H2 theory to compute the optimal value of [Formula: see text] and characterize a sequence of ϵ-optimal inventory policies under some mild technical conditions. By comparing the optimal solution under full information sharing and no information sharing, we derive a number of important practical takeaways. For instance, we show that there is value in information sharing if and only if the retailer’s optimal policy under full information sharing is not invertible with respect to the sequence of demand shocks. Furthermore, we derive a fundamental mathematical identity that reveals the value of information sharing by exploiting the canonical Smirnov–Beurling inner–outer factorization of the retailer’s orders when viewed as an element of H2. We also show that the value of information sharing can grow unboundedly when the cumulative supply chain costs are dominated by the supplier’s inventory costs. Funding: R. Caldentey acknowledges the University of Chicago Booth School of Business for financial support. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0008 .

In this paper we consider a nonmonotone (mixed) variational inequality (VI) model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle point system for the VI model in question, we introduce an augmented Lagrangian primal-dual method, called ALAVI (Augmented Lagrangian Approach to Variational Inequality) in the paper, for solving a general constrained VI model. Under an assumption, called the primal-dual variational coherence condition in the paper, we prove the convergence of ALAVI. Next, we show that many existing generalized monotonicity properties are sufficient—though by no means necessary—to imply the abovementioned coherence condition and thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an [Formula: see text] global rate of convergence where k is the iteration count. By introducing a new gap function, this rate further improves to be [Formula: see text] if the mapping is monotone. Finally, we show that under a metric subregularity condition, even if the VI model may be nonmonotone, the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and nonmonotone VI problems show the practical efficacy of the newly proposed method. Funding: L. Zhao and D. Zhu were partially supported by the Major Project of the National Natural Science Foundation of China [Grant 72293582], the National Key R&D Program of China [Grant 2023YFA0915202], and the Fundamental Research Funds for the Central Universities (the Interdisciplinary Program of Shanghai Jiao Tong University) [Grant YG2024QNA36]. L. Zhao was partially supported by the Startup Fund for Young Faculty at SJTU (SFYF at SJTU) [Grant 22X010503839].

In practice, optimization models are often prone to unavoidable inaccuracies because of dubious assumptions and corrupted data. Traditionally, this placed special emphasis on risk-based and robust formulations, and their focus on “conservative” decisions. We develop, in contrast, an “optimistic” framework based on Rockafellian relaxations in which optimization is conducted not only over the original decision space but also jointly with a choice of model perturbation. The framework enables us to address challenging problems with ambiguous probability distributions from the areas of two-stage stochastic optimization without relatively complete recourse, probability functions lacking continuity properties, expectation constraints, and outlier analysis. We are also able to circumvent the fundamental difficulty in stochastic optimization that convergence of distributions fails to guarantee convergence of expectations. The framework centers on the novel concepts of exact and limit-exact Rockafellians, with interpretations of “negative” regularization emerging in certain settings. We illustrate the role of Phi-divergence, examine rates of convergence under changing distributions, and explore extensions to first-order optimality conditions. The main development is free of assumptions about convexity, smoothness, and even continuity of objective functions. Numerical results in the setting of computer vision and text analytics with label noise illustrate the framework. Funding: This work was supported by the Air Force Office of Scientific Research (Mathematical Optimization Program) under the grant: “Optimal Decision Making under Tight Performance Requirements in Adversarial and Uncertain Environments: Insight from Rockafellian Functions.”

We present three new pivot rules for the Simplex method for Linear Programs over 0/1-polytopes. We show that the number of nondegenerate steps taken using these three rules is strongly polynomial, linear in the number of variables, and linear in the dimension. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1-polytopes and novel modifications to the classical steepest-edge and shadow-vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization. Funding: A. E. Black and J. A. De Loera are grateful for the support received through the National Science Foundation [Grants DMS-1818969 and NSF GRFP]. L. Sanita is grateful for the support received from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant VI.Vidi.193.087].

We initiate the study of episodic reinforcement learning (RL) under adversarial corruptions in both the rewards and the transition probabilities of the underlying system, extending recent results for the special case of multiarmed bandits. We provide a framework that modifies the aggressive exploration enjoyed by existing reinforcement learning approaches based on optimism in the face of uncertainty by complementing them with principles from action elimination. Importantly, our framework circumvents the major challenges posed by naively applying action elimination in the RL setting, as formalized by a lower bound we demonstrate. Our framework yields efficient algorithms that (a) attain near-optimal regret in the absence of corruptions and (b) adapt to unknown levels of corruption, enjoying regret guarantees that degrade gracefully in the total corruption encountered. To showcase the generality of our approach, we derive results for both tabular settings (where states and actions are finite) and linear Markov decision process settings (where the dynamics and rewards admit a linear underlying representation). Notably, our work provides the first sublinear regret guarantee that accommodates any deviation from purely independent and identically distributed transitions in the bandit-feedback model for episodic reinforcement learning. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0202 .

We develop a method to solve, theoretically and numerically, general optimal stopping problems. Our general setting allows for multiple exercise rights—that is, optimal multiple stopping—for a robust evaluation that accounts for model uncertainty with a dominated family of priors and for general reward processes driven by multidimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem that satisfy appealing almost sure pathwise optimality properties. Next, we exploit these theoretical results to develop upper and lower bounds that, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine prelimiting upper and lower bounds. We illustrate the applicability of our approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies. Funding: R. J. A. Laeven received financial support from the Netherlands Organization for Scientific Research (NWO) [Grants NWO-Vidi and NWO-Vici]. J. G. M. Schoenmakers received financial support from the Deutsche Forschungsgemeinschaft Excellence Cluster Math+ Berlin [Project AA4-2]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0237 .

This paper deals with composite optimization problems having the objective function formed as the sum of two terms; one has a Lipschitz continuous gradient along random subspaces and may be nonconvex, and the second term is simple and differentiable but possibly nonconvex and nonseparable. Under these settings, we design a stochastic coordinate proximal gradient method that takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per iteration, also achieving fast convergence rates. We present a probabilistic worst case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings; in particular, we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm. Funding: This work was supported by Norway Grants 2014-2021 [Grant ELO-Hyp 24/2020]; Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii [Grants PN-III-P4-PCE-2021-0720, L2O-MOC, nr 70/2022]; and the ITN-ETN project TraDE-OPT funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement [Grant 861137].

The study of time-inhomogeneous Markov jump processes is a traditional topic within probability theory that has recently attracted substantial attention in various applications. However, their flexibility also incurs a substantial mathematical burden which is usually circumvented by using well-known generic distributional approximations or simulations. This article provides a novel approximation method that tailors the dynamics of a time-homogeneous Markov jump process to meet those of its time-inhomogeneous counterpart on an increasingly fine Poisson grid. Strong convergence of the processes in terms of the Skorokhod J1 metric is established, and convergence rates are provided. Under traditional regularity assumptions, distributional convergence is established for unconditional proxies, to the same limit. Special attention is devoted to the case where the target process has one absorbing state and the remaining ones transient, for which the absorption times also converge. Some applications are outlined, such as univariate hazard-rate density estimation, ruin probabilities, and multivariate phase-type density evaluation. Funding: M. Bladt and O. Peralta would like to acknowledge financial support from the Swiss National Science Foundation Project 200021_191984. O. Peralta acknowledges financial support from NSF Award #1653354 and AXA Research Fund Award on “Mitigating risk in the wake of the pandemic”.