Papers

We study two-player zero-sum stochastic games in which a stream of stage payoffs is evaluated by a translation-invariant charge on the set of stages. Every patient valuation, that is, every diffuse charge that obeys the time value of money principle, is translation-invariant, and by Mertens and Neyman each player has, for every $\varepsilon>0$, a strategy that is $\varepsilon$-optimal for all patient valuations simultaneously. We show that this robustness breaks down for translation-invariant charges. In the Big Match with $0$–$1$ payoffs, whose uniform value is $1/2$, every strategy of Player 1 is worthless: for every strategy of Player 1 and every $\delta>0$, Player 2 has a pure reply, and there is a translation-invariant charge under which Player 1's payoff is less than $\delta$. Consequently, the Big Match has no value with respect to all translation-invariant charges. This contrasts with finite Markov decision processes, where every stationary Blackwell optimal strategy is optimal for every translation-invariant charge. We pose questions on robust guarantees in general stochastic games and on optimality with respect to a fixed translation-invariant charge.

We study a PDE-constrained variational problem for the optimal spatial distribution of a catalytic material reacting with a substrate in a biological or materials system. With a given catalyst concentration, the substrate concentration is determined by a reaction-diffusion equation. The reaction functional for the catalyst concentrations is defined to be the total reaction output, which is the product of catalyst and substrate concentrations. Both the linear and Michaelis-Menten reaction kinetics are considered. We construct reaction-maximizing sequences through the localization of catalysts using barrier functions and show the non-existence of a global maximizer. We also derive the first and second variations of the reaction functional. By constructing a competing catalyst concentration, we show the non-existence of a local maximizer. Based on our analysis, we propose a regularized variational model and prove the existence of its global maximizer.

We study factorized gradient descent (FGD) for overparameterized asymmetric matrix sensing with the balancing regularizer $\frac18\|L^\top L-R^\top R\|_F^2$. An exact indefinite lifting of the population objective decomposes the dynamics of FGD into positive, negative, and null spectral subspaces. The first two components have linear restoring terms, whereas the null component contracts only quadratically. For independent isotropic sub-Gaussian sensing matrices and independent centered sub-Gaussian noise with scale proxy $\sigma$, we prove the local bound \[ \|L_tR_t^\top-X^\star\|_2 \lesssim\max\left\{\frac{1}{\eta t+1/\sigma_r}, \kappa\sigma\sqrt{\frac{d_1+d_2}{n}}\right\}, \] under a sufficiently small initialization error, the number of linear measurements $n\gtrsim\kappa^2(d_1+d_2)\max\{k^2,\sigma^2/\sigma_r^2\}$, and step size $\eta\lesssim\sigma_1^{-1}$. Here $\sigma_1$ and $\sigma_r$ are the largest and smallest nonzero singular values of the target $X^\star\in \mathbb{R}^{d_1\times d_2}$ respectively, and $\kappa=\sigma_1/\sigma_r$. Furthermore, a population trajectory was constructed to have tight product error $\Theta(1/t)$. These results provide a local rate characterization for ordinary gradient descent for overparameterized asymmetric matrix sensing with balancing regularizer.

Random reshuffling stochastic gradient descent (RR-SGD) is widely used for finite-sum optimization in machine learning. For convex $L$-smooth components, recent work establishes convergence guarantees for an averaged iterate with component stepsizes as large as order $1/L$. However, the final iterate is a natural output in practice, and existing last-iterate guarantees in this setting require stepsizes that scale inversely with the number of components $n$. We establish an expected objective-gap bound for the last iterate of standard RR-SGD on unconstrained problems with a constant component stepsize satisfying $0<\eta\lesssim 1/(L(1 + \log(nK)))$, where $K$ is the number of epochs. By retaining the sampling bias at random comparison points, we remove the explicit $1/n$ factor from the stepsize restriction. Combined with existing small-step bounds, our result matches the corresponding averaged-iterate convergence rate up to logarithmic factors.

Reduced-order models (ROMs) make model predictive control (MPC) of high-dimensional nonlinear systems tractable, but the resulting modeling error can lead to constraint violations on the real system. We introduce CoRE-MPC, a robust MPC formulation that plans with a nonlinear ROM and guarantees recursive feasibility and constraint satisfaction for the full-order system. It can be applied with a broad class of nonlinear model reduction techniques and the resulting MPC does not require evaluating the full-order model online, thus ensuring computational efficiency. The key insight is in how the ROM affects the dynamics of the prediction error. We account for this with two error radii that are propagated alongside the ROM and provide a guaranteed bound on the prediction error. The approach is validated in simulation on an 80-state oscillator chain, a Burgers equation, and a 9,768-state finite-element soft robot, where a two-state ROM with data-driven bounds keeps all 1,628 nodes clear of a wall at a median solve time below 1 ms for a 20 ms control period. Across these examples, CoRE-MPC is 13 to 61 times faster than full-order robust MPC at only 27 to 35% higher closed-loop cost, whereas existing certified schemes are either infeasible or at least twice as conservative. Code: [released upon publication].

Dantzig-Wolfe reformulation applied to two-stage stochastic mixed-integer programs (SMIP) has been shown empirically to yield a formulation with strong bounds that can be solved via column generation. However, the time to solve the restricted master problem grows significantly with the number of scenarios due to the increasing number of nonanticipativity constraints (NACs). Moreover, the associated high-dimensional dual space can cause the dual multipliers to oscillate between iterations, further slowing convergence. We propose an aggregated column generation framework for solving a Dantzig-Wolfe reformulation. Applied to SMIP, this method partitions the NACs into groups and replaces each group with a weighted surrogate constraint, leading to a reduced size and providing implicit stabilization by restricting the dual space. While aggregation leads to a relaxation, we develop two strategies for recovering the standard Dantzig-Wolfe reformulation bound. The first iteratively refines the partition by splitting groups that contain violated constraints. The second keeps the partition fixed and adaptively updates the aggregation weights. We prove convergence of the two approaches and study their computational performance on two benchmark problem classes.

This paper studies several first-order methods for low-rank matrix minimization in the Burer–Monteiro form, where the target matrix is represented as the product of a nonnegative factor matrix and its transpose. Such a factorization destroys convexity of the objective function and global Lipschitz continuity of the gradient. We prove that for a quadratic objective function that is strongly convex only on the subspace of symmetric matrices, the Burer-Monteiro factorized objective possesses generalized smoothness with a fractional exponent 2/3. For a quadratic objective, the generalized smoothness property requires positive definiteness of the operator A only on the subspace of symmetric matrices, which relaxes the assumptions of Dragomir et al. On this basis, three methods are proposed: an adaptive gradient descent method with Armijo backtracking; an unconstrained SafeAdam method combining the Adam direction with a descent-angle certificate and a fallback to the normalized gradient; and a certified projected version for the nonnegativity constraint. Global convergence and explicit convergence rates for the proposed methods have been proved. The obtained convergence rate is better than the convergence rate of ordinary unaccelerated gradient descent in the case of a sufficiently large gradient. In the small-gradient regime, the convergence guarantee is of square-root type, and the corresponding complexity scales as the inverse square of the desired accuracy. In the large-gradient regime, the convergence guarantee is of three-quarters type, and the corresponding complexity is proportional to the inverse of the accuracy raised to the power of 4/3. Experiments on SymNMF using the TDT2, Reuters and CBCL datasets, which are standard benchmarks for document and image clustering, show that the methods match or outperform Dyn-NoLips without tuning the relative-smoothness kernel parameters.

Pediatric vaccine procurement for low- and lower-middle-income countries (LICs and LMICs) requires long-term coordination across heterogeneous vaccine markets, competing stakeholder objectives, and uncertain demand, yet tenders are currently scheduled as a series of independent renewals. We extend the Hypothetically Coordinated Vaccine Market (HCVM) framework to schedule multi-year tenders with flexible lengths and all-unit quantity discounts, and compare three objectives: a balanced coordinatingentity model (UG), a social-benefit model that minimizes missed doses (SB), and a producer profit-maximizing model (MP). Using UNICEF ten-year demand forecasts and 30 demand replications across six discount scenarios, we find that the procurement objective, not demand variability, drives cost-to-tender and schedule complexity. Quantity discounts favor large producers, concentrating the market and increasing supply chain vulnerability. Discount effectiveness depends on market maturity rather than valency: pentavalent and hexavalent vaccines are already near their price floor, and among monovalent vaccines only rotavirus responds to discounts, whereas HPV and PCV remain too expensive to compete. Removing pentavalent and hexavalent vaccines from the procurement pool sharply increases missed doses. These results give planners practical guidance on objective selection, discount design, and protecting access to combination vaccines.

Because its binary optimization landscape contains exponentially many local optima, MAX-CUT is highly sensitive to initialization. We study Particle Polynomial Relaxation (PPR), a moment-based nonconvex continuous reformulation possessing an advantageous optimization landscape. We warm-start PPR with Combinatorial Multigrid (CMG), a spectral graph-coarsening method that produces structure-aware initial solutions in near-linear time without training data. On t2g20 ($n=400$), benchmark instance with known exact optimum, the combined CMG+PPR pipeline reaches $97.5\%$ of optimum on average and $98.4\%$ on the best run. An extended variant with discrete local-search refinement reaches the published optimum on all sixty Biq Mac instances evaluated by a recent neural branch-and-bound method. Using a sparse CSR representation, the same approach scales to G-set instances with up to $20{,}000$ vertices reducing memory requirements by up to $3333\times$ while maintaining a consistent improvement over the CMG warm start. Gains are largest on sparse weighted graphs; on dense graphs we complement the pipeline with spectral sparsification.

We study online task assignment with reusable resources. This problem arises in peer reviewer assignment, skill-based call routing, rideshare matching, emergency dispatch, and crowdsourcing: Tasks arrive according to a geometric process and must be immediately assigned to servers that become temporarily busy, yielding rewards that depend on the task-server pair. The goal is to maximize long-run average reward. We cast the problem as a contextual restless multi-armed bandit (CRMAB). We derive an occupancy-measure LP (OCC-LP) that upper-bounds the reward of any work-conserving policy. From its dual we obtain Lag, a closed-form index policy. We introduce a scaling regime in which the number of copies of each server and the number of arrivals per step grow together. This regime yields the first deterministic fluid limit for a contextual bandit. In it, Lag is asymptotically optimal. Finally, we evaluate our algorithms on peer review datasets and call center routing. The Lagrangian index achieves 83-99% of the LP upper bound across all load levels. Moreover, the Lagrangian index policy significantly outperforms the natural greedy assignment scheme, improving reward by up to 12.3% on journal data and 36% on structured synthetic instances.

Chance-constrained stochastic programs (CCSPs) require solutions that satisfy uncertain constraints with high probability. Standard approaches either embed scenarios directly into optimization by considering sample average approximation (SAA)-based reformulations, growing prohibitively expensive and often failing to find feasible solutions; or employ conservative reformulations that guarantee feasibility but sacrifice objective quality by design. We propose ACCESS (Adaptive Chance-Constrained Exploration via Structured Sampling), a structured sampling framework that decouples optimization from feasibility evaluation. First, high-quality candidate solutions are generated by solving many inexpensive single-scenario subproblems, avoiding sampling directly in the constrained high-dimensional solution space. Second, feasibility is assessed separately via large-scale batched simulation over a massive number of scenarios, which is naturally amenable to GPU parallelism. That reduces the search from a high-dimensional solution space to a two-dimensional parameter space controlling conservatism and scenario diversity; while concentrating on candidates near the feasibility boundary, where high-quality feasible solutions lie. We also introduce a variant, ACCESS($\delta$), which provides certified feasibility guarantees with probability $1-\delta$. Experiments across three benchmark classes and five scenario distributions show that ACCESS matches or exceeds the best objective found by any method in nearly every setting, while ACCESS($\delta$) is empirically feasible on every instance. The conservative reformulations remain several percent worse in objective, and SAA is frequently infeasible while taking an order of magnitude longer on the harder benchmarks.

Sensitivity analysis of coupled systems typically propagates derivatives of single-valued maps, whereas multiobjective optimization generally treats sensitivity at the system level rather than through interconnected subsystems. We develop a mathematical framework for compositional sensitivity analysis in multiobjective system design, building on monotone co-design. Subsystems are represented as design problems (DPs), interconnected through series, parallel, union, intersection, and feedback composition. We introduce tangential DPs (TDPs) to describe first-order variations around an operating point, show that they preserve the structure of DPs, and characterize their queries through graphical derivatives. To account for multiple internal realizations of a system-level operating point, tangential co-design problems compose families of local TDPs by taking a union over these realizations while retaining the original interconnection structure. We show that these compositions yield families of TDPs and establish conditions under which sensitivity analysis commutes with individual composition operations. Beyond infinitesimal sensitivity, we show that bounded tangential models of convex DPs provide tighter outer approximations of feasible finite changes, which can be propagated through co-design problems without union compositions. We conclude with several convex and nonconvex numerical examples that showcase the framework's accuracy and computational benefits.

We develop a rate-transfer method that converts finite-time Lyapunov estimates for an associated stochastic differential equation (SDE) into finite-time convergence bounds for stochastic gradient algorithms. Under generic moment and smoothness conditions, the convergence analysis of the stochastic gradient method reduces to a calculus argument verifying a Lyapunov generator inequality for the SDE. The method recovers established rates for classical stochastic gradient methods and provides a way to establish convergence rates for new optimization algorithms through calculus arguments applied to their associated SDEs.

This paper extends the non-unique games framework introduced in Bandeira et al. (2020) to address symmetric structures. Precisely, we consider the problem \[ \begin{aligned} \underset{g_1,\ldots,g_n\in \mathcal{G}}{\operatorname{min}}\,\, \frac{1}{|S|}\sum_{i,j=1}^n \sum_{s\in S} f_{ij}(g_i s g_j^{-1}), \end{aligned} \] where $\mathcal{G}$ is a compact group, $S\subset \mathcal{G}$ is a finite subgroup of symmetry elements, and $f_{ij}:\mathcal{G}\rightarrow \mathbb{R}$ is a loss function. A notable example is common lines based orientation estimation of cryo-EM images for molecules with point group symmetry $S$, where $\mathcal{G}$ is the rotation group $\operatorname{SO}(3)$. With a semidefinite programming relaxation, we turn the problem into a convex program that is solved with the alternating direction method of multipliers. Unlike existing approaches which are tailored to specific symmetry groups, ours provides a unified computational framework for addressing all point group symmetries. We demonstrate the applicability of the proposed algorithm on the cryo-EM orientation estimation problem on both simulated and experimental datasets. In particular, our algorithm is effective in generating high-quality ab initio models from only a few class averages computed from experimental images.

Many safety requirements, such as keeping a manipulator away from kinematic singularities, a network connected, or a system observable, amount to keeping a state-dependent matrix positive definite. Matrix control barrier functions (MCBFs) enforce such requirements with a semidefinite program that minimally modifies a nominal control input. The MCBF constraint bounds how fast the safety margin, the smallest eigenvalue of the matrix, may decay. Yet a fixed decay rule is conservative, rejecting safe controls that drive the state quickly toward, but never across, the boundary of the safe set. Scalar optimal decay (SOD) addresses this issue by scaling the decay term by a scalar weight that is optimized together with the control. For scalar barrier functions, SOD then rejects a control at the boundary only if the control would decrease the margin. For MCBFs, however, SOD can reject controls that leave the margin unchanged, and near the safety boundary SOD accepts such controls only at a high cost. We introduce matrix optimal decay (MOD), which replaces the scalar weight with a matrix multiplier while keeping the semidefinite program convex. MOD is provably sound and complete, certifying at the boundary exactly the controls that do not decrease the margin. If some control can strictly increase the margin at every boundary point, the MOD-MCBF filtered controller is continuous and renders the safe set forward invariant. We also add an optional recovery condition that drives the system toward the safe set after a violation without affecting soundness or completeness. On a planar two-link arm avoiding singularities, MOD-MCBF certifies safe motions that both a fixed decay rule and SOD reject, and tracks an end-effector target more closely than SOD.

We consider an infinite-horizon discounted dynamic programming (DP) problem for a countable-innovation belief-state model. The model is a surrogate for a linear-Gaussian dual control problem involving an unknown parameter vector and Gaussian observation noise. The surrogate preserves the first two moments of the Gaussian innovation and the resulting posterior second-moment structure. We allow unbounded belief and control spaces and nonnegative stage costs with quadratic growth in the posterior mean and covariance. Using a posterior-moment weight, we show that the Bellman operator is a contraction in a weighted sup-norm space. We characterize the optimal cost as the unique solution of the Bellman equation in this space, establish the existence of optimal stationary policies. We further prove weighted sup-norm convergence of value iteration, policy iteration, and optimistic policy iteration under vanishing approximation errors. The convergence results under vanishing approximation errors rely on general weighted sup-norm contraction properties and are applicable beyond the particular dual control problem considered here.

In this paper, we introduce a new class of data-driven ambiguity sets for distributionally robust optimization (DRO) that aim to enhance the local inference accuracy of an unknown distribution. Building on optimal transport-based discrepancy measures, we depart from the conventional paradigm of monolithic ambiguity balls, which rely on a single global transport budget to capture distributional uncertainty. Instead, we introduce local-discrepancy ambiguity sets, where the support of the distribution is partitioned into predefined regions and transport cost constraints are imposed locally to each region. This added flexibility allows for a more refined characterization of plausible distributions by preventing highly imbalanced transport across regions. As a result, the proposed approach can yield less conservative and more robust decisions, particularly in settings where decisions depend on the local properties of the sample space. We further derive tractable reformulations of the resulting DRO problems and demonstrate through simulations that our method can outperform traditional Wasserstein ambiguity sets in terms of decision quality.

In this paper, we study the last-iterate convergence of clipped stochastic gradient descent under heavy-tailed noise. We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\le2$. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(LR^2\log(1/\delta)/T+\sigma R(\log(1/\delta)/T)^{(p-1)/p})$. We further establish last-iterate guarantees for nonsmooth composite objectives and smooth composite objectives with proximal updates. In addition, when a causal adversary replaces at most $q$ gradients, the same clipped update achieves a bound with $\log(1/\delta)$ replaced by $\log(1/\delta)+q$. The corresponding lower bounds for the clean and corrupted oracle models are also provided.

Finite-horizon optimal control becomes difficult when a dynamical model can be simulated reliably, but its derivatives are unavailable, discontinuous, or unreliable. Hamiltonian Primal-Dual Policy Optimization (HPDPO) is introduced as a derivative-free, model-based reinforcement learning framework for this setting. The control is represented as a time-indexed finite-horizon policy. Candidate actions are evaluated by deterministic suffix rollouts of the supplied transition model, while a regularized Hamiltonian dual formulation supplies structured future-cost information through block-coordinate multiplier updates. No differentiation of the state dynamics or learned critic is required. The method is validated on nonsmooth problems with exact solutions, an analytical and learned ReLU biomarker cascade, an amyloid Neural ODE trained from longitudinal ADNI data, and three multidimensional-control problems with up to five simultaneous controls. Across these examples, HPDPO closely reproduces exact or reference objective values, trajectories, and control structures. The results show that forward simulation and Hamiltonian dual structure can be combined to solve deterministic finite-horizon control problems involving nonsmooth, learned, or black-box dynamics.

This work provides a framework to treat optimal control problems on separable Banach spaces in the viscosity sense. First, it extends the notion of $B$-viscosity solutions to the Banach setting and relates the extension to the notions of viscosity solutions used in metric space settings. As the main results, the paper proves a strict comparison principle and characterizes the value function of an optimal control problem as the unique viscosity solution to an infinite-dimensional HJB equation.

This paper deals with the optimal control problem for the general mean-field stochastic differential equations, in which the coefficients depend on both the state process as well as on its law. We focus on the connection between the general maximum principle and the dynamic programming principle for such a control problem. We establish the relationship between the value function of the mean-field optimal control problem and the adjoint equations in the maximum principle. We also give a general mean-field stochastic verification theorem. Some examples are solved explicitly to illustrate our results.

In this paper, we propose an adaptive variance-reduced stochastic sequential (quadratic constrained) quadratic programming (AVR-SS(QC)QP) method for nonlinear constrained optimization. The framework integrates SPIDER-based variance reduction, stochastic quasi-Newton curvature information, merit-function globalization, and adaptive switching between stochastic SQP and SQCQP models. The switching mechanism is governed by a relative constraint model curvature indicator that measures the second-order constraint information neglected by the linearized SQP model. A smoothed hysteresis rule ac?tivates the SQCQP model only when the relative constraint curvature is significant. An exact L1 merit function with Armijo backtracking globalizes the resulting directions, while a coupled stochastic Lyapunov analysis controls the interaction between gradient-estimation error and optimization progress. Under standard smoothness, sampling, regularity, and second-order conditions, we establish almost-sure consistency of the variance-reduced gradient estimator and global convergence of the KKT residual. We further show that, near regular KKT points, the curvature indicator vanishes and the method eventually selects the computationally cheaper SQP model. Numerical experiments on a sparse principal component analysis problem illustrate the practical be?havior of the proposed framework.

The continuous-time mean-variance portfolio selection problem is investigated in this paper through a rigorous integration of backward stochastic differential equation (BSDE) and Kantorovich duality theory. We propose a novel analytical framework for addressing feedback control in McKean-Vlasov stochastic systems. By solving three coupled systems of ordinary differential equations (ODEs), we derive an explicit, closed-loop optimal feedback investment strategy.

Time-inconsistent stochastic control admits a natural game-theoretic interpretation in which successive decision dates are represented by different selves. In the Markovian framework of Björk-Khapko-Murgoci, the corresponding continuous-time equilibrium HJB system can be motivated by discrete subgame-perfect games, but the general discrete-to-continuous passage is not automatic. We establish a quantitative convergence theorem for a smooth uniformly elliptic class in which the control may enter both drift and diffusion. Using the exact sample-and-hold transition law on each time step, we construct, near a verified continuous equilibrium branch, a unique regular discrete Markov subgame-perfect equilibrium and prove first-order convergence of the continuation objects in an anisotropic $C^{2+\beta}$ topology and of the equilibrium policies in $C^0$. The key stability mechanism is structural: the discrete first-order condition forces the differential feedback correction in the frozen linearization to be nilpotent, which yields mesh-uniform block estimates through a two-norm dyadic Schauder argument despite the second-order coupling caused by controlled volatility. We also formulate a conditional transfer principle for structured polynomial-growth classes, with error-space stability stated as a separate hypothesis.

This paper proposes a Projection Anderson Acceleration (PAA) algorithm for variational inequality (VI) problems, where the underlying function is expressed as a finite sum of Hölder continuous functions and satisfies the $\sigma$-star strongly monotone condition that includes the star strong monotonicity with $\sigma = 1$ as a special case. In the absence of quasi-monotonicity and Lipschitz continuity, we establish the evaluation complexity of the PAA algorithm for finding an approximate solution. Specifically, for the star strongly monotone VI problems, the PAA algorithm needs at most $O (\log (\epsilon^{-1}) \epsilon^{2 (\underline{\alpha} - 1)})$ evaluations of the function and projection for finding an $\epsilon$-approximate solution, where $\underline{\alpha} \in (0, 1]$ is the smallest exponent of Hölder continuity of the component functions. We illustrate our theoretical results by numerical examples of non-quasi-monotone complementarity problems and non-monotone semilinear elliptic equations with non-Lipschitz terms.

In a recent breakthrough, Chen, Kyng, Liu, Peng, Probst Gutenberg, and Sachdeva (arXiv:2203.00671) designed the first almost-linear-time algorithm for minimum-cost flow. Their approach uses an Interior Point Method (IPM) to achieve complexity $m^{1+o(1)}\log U \log C$. However, the $m^{1+o(1)}$ factor hides intricate recursive data structures, large constants and multiple log factors. In this paper, we give an implementation following the core ideas of their IPM with practical data structures and empirically obtained decision rules. We compare our solver with the cost scaling and network simplex solvers of LEMON on the instance families of the DIMACS Flow Challenge 2.0 core benchmark. Our implementation scales competitively on five of twelve families and better on two. On grid_wide we observe close to linear scaling: $m^{1.04}$ against $m^{1.48}$ for LEMON. We read these results as first evidence that the almost-linear-time framework can be relevant in practice, discuss our insights gained during the implementation process, and motivate where future work should focus to bridge the theory-practice gap.

The increasing penetration of renewable energy makes power systems more vulnerable to low-probability extreme events, elevating system planning to a critical role in ensuring reliability. Distributionally robust optimization (DRO) based on the Wasserstein distance is widely used to address this challenge. However, we find that the conventional Wasserstein ambiguity set tends to excessively amplify the probabilities of low-probability extreme events, leading to over-conservative investment decisions. To mitigate this over-conservatism, this paper proposes a novel ambiguity set that modifies the Wasserstein distance with a Kullback-Leibler (KL) divergence regularization term. By leveraging the KL divergence's sensitivity to relative probability changes, the proposed set effectively curbs the unrealistic magnification of low-probability events. A two-stage distributionally robust planning model built upon this new ambiguity set is developed to optimize the installed capacity of photovoltaic, wind, and energy storage. The complex model is then reformulated as an equivalent exponential cone program with strict theoretical proofs, making it computationally tractable. Numerical results on the PJM 5-bus and IEEE 118-bus systems reveal the over-conservatism of standard Wasserstein DRO under low-probability extreme events and validate the effectiveness and efficiency of the proposed planning model and solution algorithm.

We study the task allocation problem, where our goal is to develop an online algorithm that efficiently distributes tasks among workers based on their computational performance. This task is challenging since computation times are random and non-stationary in practice. Recent works have introduced different online algorithms to address this problem. One of them is Adaptive Task Allocation (ATA), a bandit-style algorithm. However, previous theoretical guarantees leave substantial room for improvement, and existing methods are designed only for stationary computation times. Using new proof techniques, our work establishes an improved analysis of ATA with better gap-dependent and gap-independent regret guarantees, improving upon all previous approaches. We also introduce a new method, Sliding Window Adaptive Task Allocation (SW-ATA), designed for more practical non-stationary environments. Finally, we prove new lower bounds in both the stationary and non-stationary settings.

Smoothed decentralized Bregman proximal-gradient (SDBPG) methods combine relative-smoothness geometry with edge-dual consensus. We extend SDBPG to node-specific convex nonsmooth regularizers and unbiased stochastic local gradients while retaining its Bregman primal-dual smoothing mechanism. The analysis isolates temporal subgradient motion and aggregation mismatch as two defect sequences. For trajectories in a forward-invariant compact subset of the kernel domain and admissible parameters, finite accumulated defects yield an $O(T^{-1})$ bound on the squared best-iterate proximal residual over $T$ iterations. The stochastic bound separates accumulated oracle variance from expected defects and attains the same order when the variance bounds and expected defects are summable. Common active-structure identification, together with local subgradient conditions, gives sufficient finite-defect conditions for $\ell_1$ and group $\ell_{2,1}$ regularization. Experiments with common regularizers and heterogeneous sensing operators relate defect decay to observed support stabilization and compare reconstruction and communication with decentralized proximal and gradient-tracking methods. A separate phase-retrieval study evaluates the effect of a quartic Bregman kernel. Code and experiment scripts are available at https://github.com/shengze-xu/composite-stochastic-sdbpg .

We give a counterexample to the implication from generic local to generic global identifiability for linear networks with a fixed topology, independently unknown edge coefficients, and selected excited and measured nodes. A six-node network with nine unknown coefficients has exactly two compatible parameter vectors generically. Adding six excited and measured nodes preserves this ambiguity while making the added coefficients directly recoverable. The resulting twelve-node structure has forty-two independent unknown coefficients and zero diagonal. For every positive bound on the largest eigenvalue magnitude, a nonempty open set of parameters has two distinct compatible networks satisfying that bound. This holds for both real and complex coefficients. Other open regions have exactly one admissible solution. The proof uses a rational transformation exchanging the two solutions and a common limiting matrix whose fourth power vanishes. Exact rational-number certificates verify an explicit pair. The construction also extends algebraically when the unknown coordinates are whole rational transfer functions. Genericity in prescribed numerator and denominator coefficient families, and the requirement that every transfer function vanish at infinite frequency, remain separate questions.