We investigate the portfolio selection problem for an agent with rank-dependent utility, for which the investment strategy is constrained to take values in a closed convex set. For a constant coefficient market and constant relative risk-aversion utilities, we characterize the deterministic strict equilibrium strategies. For the case of time-invariant probability weighting functions, we provide a comprehensive characterization of the deterministic strict equilibrium strategy. The unique nonzero equilibrium, if it exists, can be determined by solving an autonomous ordinary differential equation (ODE). In the case of time-varying probability weighting functions, we observe that there may be infinitely many nonzero deterministic strict equilibrium strategies, which are derived from the positive solutions to a nonlinear singular ODE. By specifying the maximal solution to the singular ODE, we are able to identify all the positive solutions. In addition, we address the issue of selecting an optimal strategy from the numerous equilibrium strategies available. Funding: This research was supported by the National Key R&D Program of China [Grant 2020YFA0712700] and the National Natural Science Foundation of China [Grants 12431017, 12471447, 11971301].

Min-max optimization of a function f from [Formula: see text] to [Formula: see text] is an important framework for modeling robustness in adversarial settings with applications to optimization, economics, and deep learning. Oftentimes, f is nonconvex–nonconcave, and finding a global min-max point is computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max optimization formulation. However, many of these alternative solution concepts guarantee the existence of solution points only under strong assumptions on f, such as convexity or monotonicity of its gradient. We propose a new solution concept, the [Formula: see text]-greedy adversarial equilibrium, and show that it can serve as a computationally tractable alternative to min-max optimization. We prove the existence of such a point for any smooth bounded function with Lipschitz Hessian and give an algorithm that converges to an [Formula: see text]-greedy adversarial equilibrium in a number of evaluations of f, [Formula: see text], and [Formula: see text] that is polynomial in d, [Formula: see text], and the bounds of f and its Lipschitz constant. Funding: This work was supported by the National Science Foundation [Grant CCF-1908347]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0262 .

Bandits with knapsacks (BwK), the generalization of the multiarmed bandits problem under global budget constraints, has received a lot of attention in recent years. It has numerous applications, including dynamic pricing, repeated auctions, ad allocation, network scheduling, etc. Previous work focuses on one of the two extremes: stochastic BwK in which the rewards and consumptions of the resources of each round are sampled from an independent and identical distribution and adversarial BwK in which these parameters are picked by an adversary. Achievable guarantees in the two cases exhibit a massive gap: no-regret learning is achievable in the stochastic case, but in the adversarial case, only competitive ratio style guarantees are achievable, in which the competitive ratio depends on either the budget or both the time and the number of resources. What makes this gap so vast is that, in adversarial BwK, the guarantees get worse in the typical case when the budget is more binding. Whereas best of both worlds–type algorithms are known (single algorithms that provide the best achievable guarantee in each extreme case), their known theoretical bounds degrade to the adversarial case as soon as the environment is not fully stochastic. Our work aims to bridge this gap, offering guarantees for a workload that is not exactly stochastic but is also not worst case. We define a condition, approximately stationary BwK, that parameterizes how close to stochastic or adversarial an instance is. Based on these parameters, we explore what is the best competitive ratio attainable in BwK. We explore two algorithms that are oblivious to the values of the parameters but guarantee competitive ratios that smoothly transition between the best possible guarantees in the two extreme cases, depending on the values of the parameters. Our guarantees offer great improvement over the adversarial guarantee, especially when the available budget is small. We also prove bounds on the achievable guarantee, showing that our results are approximately tight when the budget is small. Funding: This work was supported by the Air Force Office of Scientific Research [Grants FA9550-19-1-0183, FA9550-23-1-0068], the U.S. Department of Defense (National Defense Science & Engineering Graduate (N), and the Alexander S. Onassis Public Benefit Foundation [Grant F ZS 068-1/2022-2023].

This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum-of-squares (SOS) relaxations that is dual to the generalized moment problem approach proposed by Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: (1) it allows us to conduct a convergence rate analysis for the hierarchy; (2) it leads to a sign symmetry–adapted hierarchy consisting of block-diagonal semidefinite relaxations. When the problem possesses correlative sparsity as well as sign symmetries, we propose sparse semidefinite relaxations by exploiting both structures. Various numerical experiments are performed to demonstrate the efficiency of our approach. Finally, an application to maximizing sums of generalized Rayleigh quotients is presented. Funding: F. Guo was supported by the Chinese National Natural Science Foundation [Grant 12471478]. J. Wang was funded by the National Key R&D Program of China [Grant 2023YFA1009401] and the Natural Science Foundation of China [Grants 12201618 and 12171324].

Banerjee, Budhiraja and Estevez (2025) studied a randomized load balancing model in a heavy traffic asymptotic regime where the load balancing stream is thin compared to the total arrival stream. It was shown that the limit is given by a system of rank-based Brownian particles on the half-line. In this paper we extend this result from the case of exponential service time to an invariance principle, where service times have finite second moment. The main tool is a new notion of rank-based stochastic differential inclusion, which may be of interest in its own right. Funding: R. Atar is partially supported by the Israel Science Foundation [Grant 1035/20]. T. Ichiba was supported in part by the National Science Foundation [Grant NSF-DMS-2008427].

Matroid intersection is a classical optimization problem where given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain “sparsifer”: a subset of elements of size [Formula: see text], where [Formula: see text] denotes the optimal solution, that is guaranteed to contain a [Formula: see text] approximation while guaranteeing certain robustness properties. We call such a small subset a density constrained subset, which is inspired by the edge-degree constrained subgraph, originally designed for the maximum cardinality matching problem in a graph. Our proof is constructive and hinges on a greedy decomposition of matroids, which we call the density-based decomposition. We show that this sparsifier has certain robustness properties that can be used in one-way communication and random-order streaming models. Funding: This research was supported by the Agence Nationale de la Recherche [Grant ANR-19-CE48-0016].

Bilevel optimization has gained significant attention in recent years because of its broad applications in machine learning. This paper focuses on bilevel optimization in decentralized networks and proposes a novel single-loop algorithm for solving decentralized bilevel optimization with a strongly convex lower-level problem. Our approach is built on the basis of the SOBA framework, and it is a fully single-loop method that approximates the hypergradient by using merely two matrix-vector multiplications per iteration. Importantly, by incorporating the gradient tracking and projection techniques, our algorithm does not require any gradient heterogeneity assumption, which distinguishes it from existing methods for decentralized bilevel optimization and federated bilevel optimization. We establish the convergence rate of the proposed algorithm. Moreover, we present experimental results on hyperparameter optimization and data hyper-cleaning problems, which demonstrate the efficiency of our proposed algorithm. Funding: This work was supported by the National Science Foundation Division of Mathematical Sciences [Grant 2243650]; the Division of Computing and Communication Foundations [Grants 2308597 and 2311275]; the Division of Electrical, Communications and Cyber Systems [Grant 2326591]; the National Natural Science Foundation of China [Grants 12431011, 12371301 and 12501425]; the Key Laboratory of NSLSCS, Ministry of Education of China; and the Natural Science Foundation of Jiangsu Province [Grant SBK20250404650].

We consider a classical scheduling problem on m identical machines. For an arbitrary constant [Formula: see text], the aim is to assign jobs to machines such that [Formula: see text] is minimized, where [Formula: see text] is the total processing time of jobs assigned to machine i. It is well known that this problem is strongly NP-hard. Under mild assumptions, the running time of a [Formula: see text]-approximation algorithm for a strongly NP-hard problem cannot be polynomial on [Formula: see text], unless [Formula: see text]. For most problems in the literature, this translates into algorithms with running time at least as large as [Formula: see text]. For the natural scheduling problem above, we establish the existence of an algorithm that violates this threshold. More precisely, we design a PTAS that runs in [Formula: see text] time. This result is in sharp contrast to the closely related minimum makespan variant, where an exponential lower bound is known under the exponential time hypothesis (ETH). We complement our result with an essentially matching lower bound on the running time, showing that our algorithm is best possible under ETH. The lower bound proof exploits new number-theoretical constructions for variants of progression-free sets, which might be of independent interest. Furthermore, we provide a fine-grained characterization on the running time of a PTAS for this problem depending on the relation between [Formula: see text] and the number of machines m. More precisely, our lower bound only holds when [Formula: see text]. Better algorithms that go beyond the lower bound exist for other values of m. In particular, there even exists an algorithm with running time polynomial in [Formula: see text] if we restrict ourselves to instances with [Formula: see text]. Funding: This research was supported by the Center for Mathematical Modeling BASAL [Fund FB210005], ANID-Chile, and ANID/Fondecyt [Regular 1221460]. National Science Foundation China [12271477], National Science Foundation [1756014].

In this paper, we address the effective degree bound problem for Lasserre’s hierarchy of moment-sum-of-squares (SOS) relaxations in polynomial optimization involving n variables. We assume that the first n equality constraint polynomials [Formula: see text] do not share any nontrivial common complex zero locus at infinity and that the optimal solutions are nonsingular. Under these conditions, we derive an effective degree bound for the exactness of Lasserre’s hierarchy. Importantly, the assumption of no solutions at infinity holds on a Zariski open set within the space of polynomials of fixed degrees. As a direct consequence, we provide the first explicit degree bound for gradient-type SOS relaxation under a generic condition. Funding: Z. Hua is supported by the Research Grants Council, University Grants Committee [Grant 17305123] . Z. Qu is supported by the National Natural Science Foundation of China [Grant 12001458].

In the literature on risk measures, cash subadditivity was proposed to replace cash additivity, motivated by the presence of stochastic or ambiguous interest rates and defaultable contingent claims. Cash subadditivity has been traditionally studied together with quasi-convexity, in a way similar to cash additivity with convexity. In this paper, we study cash-subadditive risk measures without quasi-convexity. One of our major results is that a general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notion of quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced, and we obtain a corresponding representation result via the lower envelope of normalized, quasi-convex, and cash-subadditive risk measures. Funding: J. Xia received financial support of the National Key R&D Program of China [Grant 2018YFA0703900] and the National Natural Science Foundation of China [Grants 12071146, 12431017, and 12471447]. X. Han, Q. Wang, and R. Wang received financial support from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2024-03728 and CRC-2022-00141]. X. Han also received financial support from the National Natural Science Foundation of China [Grants 12301604, 12371471, and 12471449]. Q. Wang also received financial support from the Society of Actuaries through the James C. Hickman Scholar Doctoral Stipend.