Papers

In this work, we study a class of structured non-convex and non-smooth optimization problems arising in imaging applications, where the objective is the sum of three functions. We consider the Davis-Yin splitting method, a FISTA-type accelerated variant, along with a quasi-Newton line-search method and a plug-and-play (PnP) extension to solve the problem. We develop a unified convergence analysis based on the Davis-Yin envelope under mild assumptions, including the Kurdyka-Łojasiewicz property. Within this framework, we establish subsequential and global convergence of the iterates to stationary points, together with residual convergence rates. We demonstrate the performance of the proposed methods on image restoration and low-rank matrix completion problems. For low-rank matrix completion, we perform experiments on both synthetic data and public datasets to evaluate recovery accuracy and efficiency. In imaging tasks, including image deblurring, we compare the convergence behaviour and reconstruction quality of several algorithmic variants using peak signal-to-noise ratio (PSNR). The results show that FISTA-acceleration and line-search methods improve convergence, while PnP denoisers enhance image quality, and the proposed methods remain competitive on matrix completion tasks.

We study the complexity of finding $(\delta,\epsilon)$-Goldstein stationary points of nonsmooth nonconvex Lipschitz functions. By now, it is known that randomized first-order algorithms can solve this task with a dimension-free oracle complexity [Zhang et al., 2020], whereas deterministic algorithms cannot, as their complexity must scale at least linearly with the dimension $d$ [Jordan et al., 2023, Tian and So, 2024]. This leaves open whether deterministic algorithms can nevertheless solve the problem with oracle complexity polynomial in $d$. We answer this question negatively by proving a lower bound of order $(1/\epsilon)^{\Omega(d)}$ for deterministic algorithm, closing the exponential gap between the previously known lower and upper bounds and resolving an open problem posed by Jordan et al. [2023]. We further discuss several extensions and implications of this result to weaker stationarity notions, finding a descent direction and deterministic smoothing. Overall, our results establish an exponential computational advantage in nonsmooth nonconvex optimization offered by randomization.

Lagrangian duality is fundamental to optimization, from the development of solver techniques to the interpretation of solutions; yet its geometric intuition can be difficult to grasp through mathematical derivation alone. This tutorial explains duality and its implications in a highly visual way for a broad audience of students and practitioners. It develops the motivation for the Lagrangian, the two-player game interpretation, and the connection to Fenchel duality. Throughout, it emphasizes the geometric meaning of the solution from both the primal and dual

We study a contextual distributionally robust chance-constrained programming (C-DRCCP) model that controls violation probabilities conditional on a neighborhood of a target context. The ambiguity set is defined over the joint distribution of contexts and outcomes using the Wasserstein distance. For affine safety systems with right-hand-side uncertainty, we derive an exact mixed-integer programming (MIP) reformulation. The reformulation reveals that worst-case conditional risk is determined by two coupled uses of the transportation budget: moving empirical mass into the conditioning neighborhood and moving conditioned mass into the failure region. Equivalently, robust feasibility holds when the minimum transportation cost needed to reach the conditional risk boundary is at least the available budget. Calibrating the ambiguity set gives a finite-sample guarantee for the true conditional violation probability. This characterization leads to strong valid inequalities that exploit the contextual transport. Each contextual mass allocation determines both the conditional scenario weights and the budget available to move outcomes into failure, whereas conventional quantile cuts utilize only the induced weights. Accounting for the remaining budget generates strengthened quantile cuts that strictly dominate standard quantile bounds and yield tighter MIP coefficients. We also derive rank inequalities from an exchange property of contextual transportation costs, linking feasible failure patterns across contextual allocations. Extensive numerical experiments demonstrate that these inequalities close much of the linear-relaxation gap, speed up the solution process, and enable large-sample instances that are intractable for the baseline formulation.

We study second- and higher-order methods for solving smooth monotone variational inequalities (MVI). Monteiro and Svaiter (SIAM J. Optim., 2012) showed that a second-order method, NPE, converges at a rate of $\mathcal{O}(T^{-1.5})$. For convex-concave minimax optimization, a subclass of MVI problems, Chen, Liu, Luo, and Zhang (COLT 2025) recently improved this rate to $\tilde{\mathcal{O}}( T^{-1.75})$. However, the result has a substantial gap compared to the lower bound of $\Omega(T^{-2.5})$ established by Chen et al. (2026). In this paper, we propose a novel second-order method that achieves the optimal rate of $\mathcal{O}(T^{-2.5})$. Our algorithm also extends to higher-order methods: for any integer $ p \ge 1$, we obtain a $p$th-order method with a convergence rate of $\mathcal{O}(T^{-(3p-1)/2})$, matching the known lower bounds and therefore establishing optimal rates across all orders.

The proximal point algorithm for finding a zero of a maximal monotone mapping is the engine that drives augmented Lagrangian methods and various splitting schemes in convex optimization, where proximal terms that are merely positive semidefinite can be helpful in preconditioning. Here a semidefinite extension of that algorithm is developed with new features which guarantee convergence, typically even linear convergence, in broader territory than previously foreseen. Moreover, inexact minimization is allowed in the subproblems in a variable-metric implementation which might open up quasi-Newton-like approaches. The key to the extension is applying an advanced form of the positive definite proximal point algorithm to a projected monotone mapping derived from the given mapping, and combining that with a completion step. For this to work, the maximality of the projected mapping is essential, but maximality is not automatic. The circumstances that guarantee maximality are definitively analyzed and convenient criteria for maximality are thereby identified. Consequences of the extended algorithm are worked out for application to convex inf-projection, augmented Lagrangian steps, and bracketed progressive decoupling based on Spingarn's partial inverse as a scheme for problem decomposition.

Two-dimensional irregular bin packing combines combinatorial bin-assignment decisions with difficult geometric feasibility constraints, making exact optimization challenging. This paper develops an exact combinatorial Benders decomposition framework for the two-dimensional irregular bin packing problem with convex polygons, coupling a pattern-based master problem with an exact single-bin geometric feasibility oracle. The dynamically strengthened Benders master is solved by a tailored exact branch-and-price procedure that incorporates objective-layered search and adaptive exact pricing. Geometric information obtained from the oracle is further fed back to the master and pricing processes through dynamically generated Benders feasibility cuts, which progressively restrict the subsequent pricing problems. Computational experiments are conducted on 540 benchmark instances from 18 classes. The proposed method obtains the optimal solution for 318 instances across 12 classes within a 3600-second time limit. On these classes, the proposed method solves more instances than a direct mixed-integer programming formulation and a baseline combinatorial Benders decomposition approach.

Periodic resource-block (RB) scanning leaves most of the current wideband channel unobserved and mixes measurements of different ages. We develop an online canonical polyadic tracker with proximal block updates (CP-PBCD) that reconstructs the full channel after each narrow-RB acquisition. Age-weighted finite histories, frequency and temporal regularization, and bounded component management maintain an adaptive low-rank representation. Two warm-started conjugate-gradient iterations and parallel shifted frequency solves provide a fixed per-frame update budget. Training-user spatial projection strengthens noise suppression at low pilot SNR. Experiments cover ten test users, three speeds, four SNRs, and complete 1000-frame trajectories with 1-ms RB acquisition. Direct first-RB CP-PBCD achieves lower mean normalized mean squared error (NMSE) than all six baselines in all twelve conditions. At 3.6 km/h and 20 dB, it improves on periodic physical refitting by 3.91 dB and runs 35.1 times faster, averaging 0.717 ms per online update. Optional dispersed startup pilots improve early acquisition, while a controlled initialization study yields a last-100-frame NMSE range of 0.427 dB across pilot budgets. These results demonstrate accurate current-channel reconstruction through incremental updates with submillisecond average computation.

We analyze a counterexample concerning non-prox-regular functions provided by Rockafellar and Wets in their 1998 foundational monograph, Variational Analysis. By establishing a general theorem on the continuity of set-valued mappings, we show that the counterexample does not have its intended properties. We propose a slight adjustment to the construction and prove that the corrected version serves as the desired counterexample.

Mitigating drawdown, the decline in wealth from its running peak, presents a canonical problem in path-dependent risk control. In this paper, we develop a finite-horizon control framework that enforces a prescribed maximum percentage drawdown limit in multi-asset stochastic systems. Our first result is an exact robust-invariance theorem characterizing every control action that preserves a prescribed drawdown limit against all supported returns. We show that every robustly safe control admits a drawdown-modulated form: the product of the current drawdown cushion and a feasible normalized direction. This yields a complete parameterization of robustly drawdown-safe policies. Additionally, under stagewise-independent returns, we show that optimizing over all robustly safe causal policies reduces to a one-dimensional Bellman recursion and yields an optimal robustly safe state-feedback policy. Finally, we characterize the linear time-invariant (LTI) gains satisfying a prescribed drawdown limit and prove that optimal drawdown modulation achieves no lower expected return under the same limit. Strict expected-return improvement holds for horizons of at least two stages whenever the LTI policy has positive expected one-stage net return.

Blind image deconvolution (BID) is a prominent research topic in the field of imaging sciences, given its significant practical applications. Most existing model-based BID methods focus on natural images, incorporating appropriate prior knowledge about both the underlying image and the blur kernel. However, for certain classes of images, such as barcodes, text, and patterns, pixels can only take very limited values, a specific prior that is often overlooked in the literature. In this article, we introduce a novel pixel intensity constraint to leverage this important information, improving recovery performance for these specialized image classes. Specifically, we propose a unified framework for blind binary and pattern image deconvolution that incorporates both the pixel intensity constraint and a gradient sparsity regularizer. Numerical experiments demonstrate that our method outperforms many existing BID techniques, achieving superior results in terms of both visual quality and quantitative metrics.

The Deck-of-cards-based Ordinal Regression (DOR) infers a value function from a ranking of reference alternatives in which the Decision Maker (DM) inserts blank cards between consecutive levels to express preference intensity. DOR, and its stochastic extension (SMAA-DOR), treat these answers as hard constraints defining a set of compatible value functions. We propose B-DOR, a probabilistic reformulation of DOR in which each pair of adjacent levels yields an ordinal observation, the declared direction and the number of cards, modelled through a cumulative-link likelihood that relates the number of blank cards to the latent value difference between alternatives. Two Bayesian inference algorithms are proposed: BAYES-DOR samples the whole posterior distribution by Hamiltonian Monte Carlo; FTRL-DOR tracks the maximum a posteriori estimate by constrained convex optimization. Moreover, through a multi-step elicitation process, elicitation can be spread over several short sessions reducing the cognitive burden on the DM. Both algorithms enjoy logarithmic regret bounds for prediction that hold for any sequence of DM responses and that guide the choice of the prior hyperparameters. A Monte Carlo study over 768 configurations shows that accuracy grows with the number of sessions, that blank cards add significant information over preference directions alone, that both algorithms maintain good performance under inconsistent answers, and that both outperform DOR and SMAA-DOR. An illustrative application to Italian regional healthcare performance demonstrates the practical applicability of the approach for building composite indicators.

Resilience is a system's capability to prepare for, resist, absorb, and recover from adverse events. By definition, resilience therefore encompasses multiple criteria that can conflict and may prescribe different decisions. Importantly, a system must also maintain its effectiveness during routine operations and appropriately scale its preparedness for adverse events. Although resilience is inherently multicriteria, existing models often focus on a single criterion and do not explicitly analyze potential conflicts, the associated trade-offs, and their implications for decision-making. We formulate rebound, resistance, loss, and maximum performance degradation as separate resilience criteria in a general two-stage multicriteria model of network flows over time that integrates preparedness, effectiveness, and response. We show that, in general, such multicriteria optimization models are intractable and develop an enclosure-based heuristic for approximating the nondominated set. Our computational results show that different preparedness instruments and activation timings are associated with different resilience criteria and that the corresponding trade-offs are strongly instance-specific. Thus, resilience should be approached from an explicit multicriteria perspective rather than through a universal, preference-independent scalar index.

This paper introduces the two-stage stochastic quadratic knapsack problem (TSSQKP) with uncertainty in profits and weights. To overcome the computational difficulties arising from the combined stochastic, binary, and quadratic structure, we propose a solution framework that integrates an adaptation of the progressive hedging algorithm with clustering-based scenario reduction. The enhanced progressive hedging algorithm (EPHA) incorporates a rounding procedure and a dynamic penalty update strategy driven by the detection of oscillation and stagnation patterns. In parallel, we propose a clustering framework based on opportunity-cost distances between scenarios, from which we derive the medoid lower bound (MLB), the cluster lower bound (CLB) and the cluster upper bound (CUB). To further improve computational efficiency, EPHA is also used to solve the cluster subproblems heuristically; in this case, the resulting upper bound is referred to as the estimated cluster upper bound (ECUB), and the corresponding gaps are interpreted as estimated optimality gaps. Computational experiments on 800 generated TSSQKP instances show that the proposed framework yields smaller gaps than CPLEX under comparable computational conditions. In the EPHA-based clustering framework, the average estimated gaps associated with the CLB, EPHA and MLB 1.65%, 1.88% and 2.41% respectively, compared with an average optimality gap of 35.86% for CPLEX. MLB is fastest to compute, with an average execution time of 420.88s, while EPHA provides a favourable compromise between solution quality and computational effort, reducing the average computational time associated with the cluster bounds by approximately 52.7%, while increasing the average estimated gap by only 0.23 percentage points.

Modern prediction-driven decision systems often rely on black-box predictors, but a point forecast alone does not provide the uncertainty scale required for robust downstream decision-making. We build a score-calibrated robustness framework that converts any fixed point predictor into a decision-relevant uncertainty representation through distribution-free conformal calibration. We use the conformal score, rather than a particular uncertainty set, as the primitive unit of robustness. The same score determines coverage-calibrated uncertainty sets for reliability-based robust optimization and normalizes target violations in a target-oriented formulation, Conformal Robust Satisficing. This formulation induces a conformal fragility measure that quantifies how rapidly performance deteriorates as the realized parameter departs from the forecast on the conformal score scale. For objective-uncertainty problems under standard convexity and duality conditions, we show that the reliability-based and target-oriented formulations parameterize the same score-calibrated robust decision frontier. This equivalence yields a data-driven mapping between reliability levels and acceptable targets and characterizes the marginal cost of robustness. Synthetic experiments validate the theoretical guarantees and illustrate the reliability-target correspondence. A real-data online-grocery case study demonstrates how the interface combines deep-learning demand forecasts with tractable inventory optimization, thereby improving reliability and reducing operational costs. Overall, our work shows that conformal scores endow fixed black-box predictors with an interpretable uncertainty scale for downstream decision-making while enabling reliability guarantees, acceptable-target selection, and fragility analysis within a unified framework.

Small Island Developing States (SIDS) face simultaneous frequency nadir and rate-of-change-of-frequency (RoCoF) violations under high renewable penetration. This paper proposes a Composite Adaptive Higher-Order Control Barrier Function (CA-HOCBF) for BESS-coupled virtual synchronous generators that jointly enforces IEEE~1547 frequency ($\pm0.8$\,Hz) and ENTSO-E RoCoF ($\pm1$\,Hz/s) limits as hard safety constraints under parametric uncertainty. The method unifies three tools: (i)~a dual barrier architecture with a zeroing CBF for frequency and an algebraic RoCoF constraint with a robust inertia-floor fallback; (ii)~a composite energy function coupling logarithmic safety barriers with quadratic parameter and disturbance error terms, which drives safety-weighted adaptation; and (iii)~a disturbance observer integrated into the barrier dynamics. We prove that the closed-form QP is always feasible under a BESS capacity condition and, under explicitly stated assumptions, that the composite energy remains finite along trajectories, rendering the joint safe set forward invariant without requiring parameter convergence. Simulation on a 10\,MW Caribbean microgrid ($H=2$\,s, 70% renewables) shows the CA-HOCBF eliminates all frequency and RoCoF violations under a compound disturbance, achieving steady-state error of 0.001\,Hz and RoCoF of 0.50\,Hz/s, where fixed-gain VSGs suffer a 0.62\,Hz offset with 2.25\,Hz/s RoCoF violations.

This paper studies exact controllability of linear mean-field stochastic differential equations with backward-structure. We establish Hautus criteria and uncover a sharp dichotomy between the zero initial time and positive initial times. At time zero, exact controllability is characterized by a Hautus condition for the mean dynamics, with additional control directions generated by the centered stochastic dynamics. At any positive initial time, exact controllability further requires a stochastic Hautus condition for the centered system. Unlike the classical eigenvector-based condition, this stochastic criterion is formulated in terms of positive-semidefinite eigenmatrices of an associated Lyapunov-type operator. The dichotomy arises from the triviality of the initial sigma-field. We also clarify the controllability relations among the corresponding ODE, SDE, and mean-field SDE systems.

This paper concerns the variational inequality arising from a specific singular control problem: portfolio selection under proportional transaction costs. This variational inequality is a gradient-constrained partial differential equation (PDE). Generally, the literature only guarantees the W{2,\infty} regularity of the solution to such a PDE, and counterexamples exist for higher regularity. In this paper, by exploiting the concavity of the value function, we connect this gradient-constrained problem to an obstacle problem, and finally show that the solution is C^{2,\alpha}. This paper concerns a two-dimensional portfolio selection setup, but our approach can potentially be extended to more general multi-dimensional singular control problems when the value function is concave.

We deal with controlling a complex social network in which agents have actions and opinions that coevolve, mutually influencing one another. We consider an input consisting in broadcasting information to a target set of agents with the objective of steering the population, initially at a consensus, to a different consensus. For a constant input, we derive a monotone convergence result, building on which we design an algorithm that determines whether a target set is sufficient to achieve the objective and an effective heuristic to optimize the target set. Then, we introduce a feedback control law that, using information on the state of the system, dynamically revises the target set, reducing the effort needed to achieve the objective while guaranteeing convergence to the desired consensus state.

We study a one-player token game on an $N\times N$ board where tokens slide along diagonals or duplicate onto neighbouring ones to form a combinatorial rectangle $R\times S$. A conserved integer weight $W'$ and a strict monovariant guarantee $O(N^2)$-length solutions, placing the game in $\mathsf{NP}$. We prove that reaching a final position factors this $2N$-bit $W'$ into two $N$-bit factors $V,M < 2^N$ that encode the rectangle's rows and columns. Consequently, solving the game for a balanced-semiprime target is equivalent to integer factoring. However, if the target rectangle is known, the solution reduces to two polynomial-time steps: a forced downward chip-flow and a $0/1$-polynomial factorisation leveraging Cohn's theorem. The game's entire difficulty is thus isolated to the initial number-theoretic split. Supplying the popcounts of the factors as a promise preserves this asymptotic hardness but bounds the target search space. We exploit this constrained space using a learned policy/value network and an AlphaZero-style Monte-Carlo tree search, empirically probing the limits of neural look-ahead on a factoring-equivalent environment.

While compressed-sensing regularizers enable interpretable reconstruction of 4D Flow CMR through transparent variational objectives, their hand-crafted nature is too restrictive under high acceleration. State-of-the-art learning-based approaches mitigate this, but typically encode regularization implicitly through unrolled network modules, which limits their interpretability. To address this limitation, we propose CLEAR, designed to combine the interpretability of compressed sensing with the flexibility of learned models. To the best of our knowledge, it is the first learned regularizer for a 4D reconstruction task. In the ultra-accelerated \(10\times\)–\(50\times\) regime of the CMRx4DFlow2026 challenge, CLEAR outperforms compressed sensing locally low-rank (LLR) and the popular variational network FlowVN, while using less than 10k parameters and preserving an interpretable regularization structure.

We present reference implementations of four published primal heuristics for mixed-integer programming inside the open-source solver HiGHS (Huangfu and Hall, 2018): Feasibility Jump (Luteberget and Sartor, 2023), fix-propagate-repair (FPR; Salvagnin, Roberti and Fischetti, 2025) with its LP-guided dive-time variant, LocalMIP (Lin, Zou and Cai, 2024), and the first-order-LP feasibility pump Scylla (Mexi et al., 2025). They share one integration interface with per-heuristic budgets and patiences, and each runs in parallel on independently seeded workers. The evaluation is on the mipfeas benchmark (Bussieck and Dirkse, 2026), 233 instances at 600 s, against a separately built unpatched HiGHS on the same machine. The primal integral improves by 23.7% over the full set and by 12.9% on the instances not used for tuning: the patched solver reaches a good solution sooner. At the time limit it is slightly ahead. It finds a solution on 4 instances where the unpatched solver finds none, against 1 the other way, and its final gap is smaller, significantly so only over the full set. Scylla and the LP-guided dive produced no accepted incumbent in any full-limit run that carried them and ship disabled; their implementations remain. The implementation is released as open source.

In this paper, we establish the existence and uniqueness of solutions for mean-field stochastic differential equations (MF-SDEs in short) and backward stochastic differential equations (MF-BSDE) driven by anomalous sub-diffusions $\{B_{L_t}; t\geq 0\}$ with random coefficients, respectively. Here $B$ is a Brownian motion on ${\mathbb R}^d$ and $L$ is the inverse of a subordinator $S$ with drift $\kappa >0$ that is independent of $B$. We further study the stochastic maximum principles (SMPs) for control problems of the stochastic systems modelled by the MF-SDEs using a convex variational method. A linear quadratic control example is given in the last section of this paper, for which both the SMP and the sufficient SMP established in this paper are utilized to show explicitly that it admits a unique stochastic optimal control.

We study data-driven policy construction for dynamical systems using trajectories generated by surrogate finite-horizon Model Predictive Control (MPC), with the goal of approximating an infinite-horizon optimal control policy. We interpret these trajectories as non-expert demonstrations and propose a memory-based, nonparametric policy that constructs a Lipschitz-regularized upper envelope of the finite- horizon surrogate value function, which serves as an explicit upper bound on the value of the resulting policy, thereby providing a computable performance certificate while bypassing explicit optimization at runtime. We establish relative optimality guarantees with respect to the infinite-horizon optimal value function and derive sufficient MPC horizon conditions, together with sample complexity bounds, for achieving a prescribed relative error. Numerical experiments on a rocket landing task validate the theoretical predictions and demonstrate near-optimal performance.

We consider minimum-time control for the linear system $$ \dot{z}(t)=Az(t)+Bu(t),\qquad \|u\|_{L^\infty(0,\infty;\mathbb{R}^m)}\leq 1, $$ with $A\in\mathbb R^{n\times n}$ and $B\in\mathbb R^{n\times m}$. While the individual point-target and ball-target problems are classical, we study a different question: when are their optimal controls exactly compatible, in the sense that, for every nonzero initial state $x$ and all sufficiently small $\varepsilon>0$, the optimal control for the tolerance ball $\overline{B}_\varepsilon(0)$ is precisely the restriction of the point-target optimal control? We prove that this {\it{exact truncation}} property is equivalent to the rigidity conditions $$ B B^\top=\beta I_n,\qquad A+A^\top=2aI_n, \qquad \beta>0,\; a\leq 0, $$ and also to Euclidean radiality of the point-target minimum-time function. Thus exact truncation holds precisely when the sublevel sets of the point-target minimum-time function are Euclidean balls centered at the origin, matching the geometry of the tolerance targets. Moreover, the local property automatically extends to every $0<\varepsilon<|x|$, and the resulting optimal times and point-target optimal feedback are both explicit.

The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.

Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt–Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar $L_2$ equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.

Diffusion models have demonstrated strong power in generative modeling tasks across multiple domains, exhibiting a remarkable capability of learning complex distributions from samples. In this paper, we leverage such capability to design an efficient universal diffusion-based solver for parameterized black-box optimizations (BBO), where the optimizer has only black-box access to queries of the objective function at the learning stage, yet is able to reduce the additional computational cost at the inference stage for each BBO instance while also capturing the potential multi-modal landscape of non-convex objectives. To cast our formulation as a compatible generative modeling task, we introduce the notion of minimization policy as a new solution concept, which defines a sampling distribution over the solutions that should concentrate around the minimizer set for each BBO instance. We then propose Diffusion-based Iterative Minimization Policy Learning (D-IMPL), a practical generative-model-based solver for solving parameterized BBOs that employs diffusion models to learn a minimization policy, whose density is proportional to the exponential of the negated objective values, thereby amortizing the computational costs across different BBO instances. Furthermore, we demonstrate the performance of our D-IMPL algorithm by establishing a sample complexity guarantee showing that a $\delta$-approximate minimization policy can be effectively learned within $O(\log(1/\delta))$ iterations, and by extensive empirical evaluations over a range of constrained and unconstrained BBO tasks.

We develop an index policy for finite-horizon Bernoulli multi-armed bandits from minimax solutions to single-arm bandit (SAB) problems. Each SAB problem involves choosing between an unknown Bernoulli arm and a known reward. We show that minimizing worst-case regret of SAB problems over all non-anticipative policies admits an exact semi-infinite linear programming formulation. The resulting stopping policies offer a natural way to compare arms: the higher the known reward against which a policy continues sampling, the more promising the unknown arm. We turn this intuition into indices based on cumulative continuation probabilities, with a monotone adjustment and a reward-shortfall cap. By relating index errors to the regret of single-arm stopping policies, we establish a distribution-free regret bound of $4.45\sqrt{KT}+10.75K$ for $K$ arms and horizon $T$. This bound matches the minimax-optimal regret order established in the literature. The guarantee extends to rewards supported on $[0,1]$ through Bernoulli randomization. We also provide a finite-grid implementation with quantified approximation loss. In numerical experiments, the SAB-based index policy achieves lower worst-case regret than every tested benchmark policy across all evaluated numbers of arms and horizons, while closely matching the grid-based MAB minimax policy in the two-arm setting.

Density-Driven Optimal Control (D2OC) provides an effective framework for steering multi-agent systems toward prescribed spatial distributions. However, existing D2OC formulations are primarily developed for Euclidean domains and do not directly account for intrinsic manifold geometry. This paper extends D2OC to second-order multi-agent systems evolving on Riemannian manifolds. The proposed Riemannian D2OC (R-D2OC) constructs a local distribution objective in the tangent space of each agent through logarithmic maps and uses its weighted center as the reference for a finite-horizon LQR. The resulting control is executed on the manifold through intrinsic second-order dynamics and parallel transport within a receding-horizon scheme. We establish local curvature-dependent bounds that quantify the approximation introduced by the tangent-space reduction and characterize the resulting target bias. Furthermore, we derive a conditional discrete-descent result showing that the closed-loop objective decreases when a local velocity-alignment condition is satisfied. Numerical simulations on a 3D ellipsoidal manifold demonstrate distribution-level control and empirically support the proposed approximation and descent results.