Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a pacing multiplier between zero and one, and her bid on each item is multiplicatively scaled down by the pacing multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is known as a pacing equilibrium. In this work, we show that the problem of finding an approximate pacing equilibrium is PPAD-complete for second-price auctions. This resolves an open question of Conitzer et al. [Conitzer V, Kroer C, Sodomka E, Stier-Moses NE (2022a) Multiplicative pacing equilibria in auction markets. Oper. Res. 70(2):963–989]. As a consequence of our hardness result, we show that the tâtonnement-style budget-management dynamics introduced by Borgs et al. [Borgs C, Chayes J, Immorlica N, Jain K, Etesami O, Mahdian M (2007) Dynamics of bid optimization in online advertisement auctions. Proc. 16th Internat. Conf. World Wide Web (ACM, New York), 531–540] are unlikely to converge efficiently for repeated second-price auctions. This disproves a conjecture by Borgs et al. [Borgs C, Chayes J, Immorlica N, Jain K, Etesami O, Mahdian M (2007) Dynamics of bid optimization in online advertisement auctions. Proc. 16th Internat. Conf. World Wide Web (ACM, New York), 531–540], under the assumption that the complexity class PPAD is not equal to P. Our hardness result also implies the existence of a refinement of supply-aware market equilibria which is hard to compute with simple linear utilities. Funding: This work was supported by National Science Foundation (CCF-1703925, IIS-1838154).

Short spanning trees subject to additional constraints are important building blocks in various approximation algorithms, and moreover, they capture interesting problem settings on their own. Especially in the context of the traveling salesman problem (TSP), new techniques for finding spanning trees with well-defined properties have been crucial in recent progress. We consider the problem of finding a spanning tree subject to constraints on the edges in a family of cuts forming a laminar family of small width. Our main contribution is a new dynamic programming approach in which the value of a table entry does not only depend on the values of previous table entries, as is usually the case, but also on a specific representative solution saved together with each table entry. This allows for handling a broad range of constraint types. In combination with other techniques—including negatively correlated rounding and a polyhedral approach that, in the problems we consider, allows for avoiding potential losses in the objective through the randomized rounding—we obtain several new results. We first present a quasi-polynomial time algorithm for the minimum chain-constrained spanning tree problem with an essentially optimal guarantee. More precisely, each chain constraint is violated by a factor of at most [Formula: see text], and the cost is no larger than that of an optimal solution not violating any chain constraint. The best previous procedure is a bicriteria approximation violating each chain constraint by up to a constant factor and losing another factor in the objective. Moreover, our approach can naturally handle lower bounds on the chain constraints, and it can be extended to constraints on cuts forming a laminar family of constant width. Furthermore, we show how our approach can also handle parity constraints (or, more precisely, a proxy thereof) as used in the context of (path) TSP and one of its generalizations and discuss implications in this context. Funding: This project received funding through the Swiss National Science Foundation [Grants 200021_184622 and P500PT_206742], the European Research Council under the European Union’s Horizon 2020 research and innovation program [Grant 817750], and the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy – EXC 2047/1 [Grant 390685813].

We study the classic online bipartite matching problem with a twist: off-line vertices, called resources, are reusable. In particular, when a resource is matched to an online vertex, it is unavailable for a deterministic time duration d, after which it becomes available again for a rematch. Thus, a resource can be matched to many different online vertices over a period of time. Whereas recent work on the problem has resolved the asymptotic case in which we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of unit inventory and give the first algorithms that are provably better than the naïve greedy approach, which has a competitive ratio of (exactly) 0.5. Our first algorithm, which achieves a competitive ratio of 0.589, generalizes the classic RANKING algorithm for online bipartite matching of nonreusable resources (Karp et al. 1990) by reranking resources independently over time. Whereas reranking resources frequently has the same worst case performance as greedy, we show that reranking intermittently on a periodic schedule succeeds in addressing reusability of resources and performs significantly better than greedy in the worst case. Our second algorithm, which achieves a competitive ratio of 0.505, is a primal-dual randomized algorithm that works by suggesting up to two resources as candidate matches for every online vertex and then breaking the tie to make the final matching selection in a randomized correlated fashion over time. As a key component of our algorithm, we suitably adapt and extend the powerful technique of online correlated selection (Fahrbach et al. 2020) to reusable resources in order to induce negative correlation in our tie-breaking step and beat the competitive ratio of 0.5. Both of our results also extend to the case in which off-line vertices have weights. Funding: R. Niazadeh’s research is supported by the Asness Junior Faculty Fellowship at The University of Chicago Booth School of Business.

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss because of the mismatch between the true and assumed models. A robustness problem in this context is to show that the error because of the mismatch between a true and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we discuss continuity and robustness properties of finite and infinite horizon α-discounted/ergodic optimal control problems for a general class of nondegenerate controlled diffusion processes as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error because of the mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We see that, compared with related results in the discrete-time setup, the continuous-time theory lets us utilize the strong regularity properties of solutions to optimality (Hamilton–Jacobi–Bellman) equations, via the theory of uniformly elliptic partial differential equations, to arrive at strong continuity and robustness properties. Funding: The research of S. Yüksel was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

In a housing market of Shapley and Scarf, each agent is endowed with one indivisible object and has preferences over all objects. An allocation of the objects is in the (strong) core if there exists no (weakly) blocking coalition. We show that, for strict preferences, the unique strong core allocation “respects improvement”—if an agent’s object becomes more desirable for some other agents, then the agent’s allotment in the unique strong core allocation weakly improves. We extend this result to weak preferences for both the strong core (conditional on nonemptiness) and the set of competitive allocations (using probabilistic allocations and stochastic dominance). There are no counterparts of the latter two results in the two-sided matching literature. We provide examples to show how our results break down when there is a bound on the length of exchange cycles. Respecting improvements is an important property for applications of the housing markets model, such as kidney exchange: it incentivizes each patient to bring the best possible set of donors to the market. We conduct computer simulations using markets that resemble the pools of kidney exchange programs. We compare the game-theoretical solutions with current techniques (maximum size and maximum weight allocations) in terms of violations of the respecting improvement property. We find that game-theoretical solutions fare much better at respecting improvements even when exchange cycles are bounded, and they do so at a low efficiency cost. As a stepping stone for our simulations, we provide novel integer programming formulations for computing core, competitive, and strong core allocations. Funding: P. Biró gratefully acknowledges financial support from the Hungarian Scientific Research Fund, OTKA [Grant K143858] and the Hungarian Academy of Sciences [Grant LP2021-2]. F. Klijn gratefully acknowledges financial support from AGAUR–Generalitat de Catalunya [Grants 2017-SGR-1359 and 2021-SGR-00416] and the Spanish Agencia Estatal de Investigación [Grants ECO2017-88130-P and PID2020-114251GB-I00] (funded by MCIN/AEI/10.13039/501100011033) and the Severo Ochoa Programme for Centres of Excellence in R&D (Barcelona School of Economics) [Grant CEX2019-000915-S]. Research visits related to this work were financed by COST Action [Grant CA15210 ENCKEP], supported by COST (European Cooperation in Science and Technology), http://www.cost.eu/ .

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite-dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite-dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the literature.

Using prices induced by dual variables of a centralized optimization problem induces welfare-optimal equilibria among strategic drivers. We reveal a stark deficiency of such static pricing algorithms: it is possible for them to induce additional equilibria with arbitrarily low social welfare. Moreover, small perturbations to the marketplace, such as those caused by idiosyncratic randomness or model misspecification, can cause the welfare-optimal equilibrium to be Pareto-dominated (in terms of driver utility) by suboptimal equilibria. We show that dynamic pricing solves this problem. We describe a dynamic pricing algorithm that resolves the centralized optimization problem in each time period and show that it satisfies a new equilibrium robustness property, which guarantees that every induced (approximate) equilibrium is (approximately) welfare optimal. We also propose a novel two-level model of ridesharing networks with strategic drivers and spatiotemporal dynamics that lets us retain macroscopic uncertainty, such as correlated shocks caused by weather or other public events, when analyzing a large market limit in which idiosyncratic sources of uncertainty vanish. Funding: J. M. Cashore was supported by an NSERC PGS D Fellowship. P. Frazier was supported by AFOSR [Grant FA9550-19-1-0283]. É. Tardos was supported by AFOSR [Grant FA9550-19-1-0183] and [NSF Grants CCF-1408673 and CCF-1563714]. Supplemental Material: The online companion is available at https://doi.org/10.1287/moor.2022.0163 .

In this paper, we focus on the problem of minimizing the sum of a nonconvex differentiable function and a difference of convex (DC) function, where the differentiable function is not restricted to the global Lipschitz gradient continuity assumption. This problem covers a broad range of applications in machine learning and statistics, such as compressed sensing, signal recovery, sparse dictionary learning, matrix factorization, etc. We first take inspiration from the Nesterov acceleration technique and the DC algorithm to develop a novel algorithm for the considered problem. We then study the subsequential convergence of our algorithm to a critical point. Furthermore, we justify the global convergence of the whole sequence generated by our algorithm to a critical point and establish its convergence rate under the Kurdyka–Łojasiewicz condition. Numerical experiments on the nonnegative matrix completion problem are performed to demonstrate the efficiency of our algorithm and its superiority over well-known methods.

The progressive hedging algorithm (PHA) is an effective solution method for solving monotone stochastic variational inequalities (SVIs). However, this validity is based on the assumption of global maximal monotonicity. In this paper, we propose a localized PHA for solving nonmonotone SVIs and show that its validity is based on the weaker assumption of locally elicitable maximal monotonicity. Furthermore, we prove that such assumption holds when the mapping involved in the SVI is locally elicitable monotone or locally monotone. The local convergence of the proposed algorithm is established, and it is shown that the localized PHA has the rate of linear convergence under some mild assumptions. Some numerical experiments, including a two-stage orange market problem and randomly generated two-stage piecewise stochastic linear complementarity problems, indicate that the proposed algorithm is efficient. Funding: This work was supported by the National Natural Science Foundation of China [Grant 12171271]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0017 .

The problem of covering a two-dimensional bounded set with a fixed number of minimum-radius identical disks is studied in the present work. Bounds on the optimal radius are obtained for a certain class of nonsmooth domains, and an asymptotic expansion of the bounds as the number of disks goes to infinity is provided. The proof is based on the approximation of the set to be covered by hexagonal honeycombs and on the thinnest covering property of the regular hexagonal lattice arrangement in the whole plane. The dependence of the optimal radius on the number of disks is also investigated numerically using a shape-optimization approach, and theoretical and numerical convergence rates are compared. An initial point construction strategy is introduced, which, in the context of a multistart method, finds good-quality solutions to the problem under consideration. Extensive numerical experiments with a variety of polygonal regions and regular polygons illustrate the introduced approach. Funding: This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2019/25258-7] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 302682/2019-8, 303243/2021-0, 304258/2018-0, and 408175/2018-4].