Papers

Distributionally robust optimization (DRO) protects decisions against distributional uncertainty by optimizing over an ambiguity set, but poorly aligned set geometry can require large radii and yield overly conservative decisions. We introduce diagnostic-transport DRO (DT-DRO), which uses held-out calibration data to adapt the ambiguity-set geometry to observed predictive errors. DT-DRO uses the conditional probability integral transform cumulative distribution function to diagnose systematic probability misallocation and translates this information into an outcome-level transport that jointly adjusts the ambiguity-set center and ground cost. The resulting formulation admits a computationally tractable dual reformulation. Theoretically, we derive valid ambiguity radii and decision-risk guarantees that tighten as estimation and approximation errors vanish, and show that DT-DRO can eliminate the nonvanishing robustness floor caused by model misspecification. Synthetic experiments and a power-outage application demonstrate improved decision quality, particularly under structural and tail misspecification.

The Minimum Birkhoff Decomposition (MBD) seeks a sparse weighted sum of matchings and is a challenging optimization problem with applications in network scheduling and energy trading. We scale a quantum-assisted decomposition method by combining single-layer QAOA sampling with Extended Fully-Corrective Frank-Wolfe (E-FCFW) optimization, spectral graph partitioning, and greedy feasibility repair. We demonstrate the pipeline on the 1,354-node PEGASE bus test case (representing a 1,354-bus transmission grid with 1,710 transmission lines), whose 1,710 edges define a native 1,710-variable matching optimization problem requiring 1,710 qubits in the unpartitioned encoding. Partitioning enables distributed execution on IBM superconducting quantum processors. The best reported hardware result uses a 50-qubit partition bound, while reducing partitions to 20 qubits degrades convergence in the partition-size comparison. Larger partitions are more demanding for matrix product state (MPS) simulation, and bond-dimension tests show that truncating quantum correlations reduces candidate quality. The results therefore motivate retaining a substantive quantum sampling task within each partition as the overall problem scales. On the PEGASE-1354 benchmark, repaired QAOA samples achieve lower residual decomposition error than both simulated annealing and uniform random sampling baselines, while spatial circuit packing reduces hardware execution time by approximately a factor of three. These results demonstrate that combining spectral partitioning, spatial circuit packing, and classical feasibility repair provides an executable path for deploying gate-based quantum sampling on thousand-variable constrained optimization problems on current hardware.

We propose a continuous policy-value iteration framework for infinite horizon stochastic control problems, in which the policy evolves through Langevin type dynamics driven by the gradient of the inhomogeneous Hamiltonian, while the value function is updated simultaneously with optimal policy iterations, avoiding directly solving its corresponding PDE at each iteration. The iteration scheme enjoys policy improvement under suitable regularity and admissibility assumptions. We show the exponential contraction of the joint policy value dynamics in Wasserstein-2 distance under the Lipschitz condition in expectation via the inhomogeneous synchronous coupling and verify these conditions for linear quadratic problems. Numerical experiments on linear quadratic models and the optimal consumption-investment problems in an incomplete market factor model illustrate policy improvement, convergence toward optimal policies, and the effect of decaying exploration noise.

Understanding why deep neural networks make a given prediction is of great importance for their safe deployment. In computer vision, saliency maps, which highlight the image region most influential for a prediction, remain a widely-used form of explanation. An empirical observation is the apparent sparsity of gradient saliency maps of adversarially-trained neural networks. In this paper, we propose a theoretical explanation of this phenomenon for two-layer ReLU networks. We build on the established equivalence of adversarial training to the minimization of the empirical risk with weight-decay penalization and an added adversarial total variation term – valid for certain loss functions. As the number of data points and neurons grows and the regularization parameters are sent to zero at appropriate rates, we prove that minimizers converge to a Bayes classifier with minimal gradient and Barron norm. Sparsity appears since for adversarial training with $\ell_\infty$-attacks the gradient norm is anisotropic and favors axis-aligned / sparse gradients. We illustrate our theoretical findings experimentally by evaluating the gradient $\ell_1$-norm and thresholded sparsity of naturally versus adversarially trained models.

Heavy-tailed noise has been widely observed in modern machine learning, motivating the use of methods like gradient clipping and normalization. While these methods are well understood in centralized settings, much less is known in decentralized ones, where applying a nonlinearity to local gradients affects both optimization and consensus. Recent works on decentralized non-convex optimization have studied both clipping and normalization under heavy-tailed noise, with clipping yielding suboptimal rates and normalization needing local momentum or mini-batches to converge. This raises the question: can a baseline decentralized method using a nonlinearity achieve optimal convergence rates under heavy-tailed noise? We answer affirmatively with clipped decentralized SGD ($\mathtt{DSGD}$). For smooth non-convex costs under bounded $p$-th moment noise, $p \in (1,2]$, we show that clipped $\mathtt{DSGD}$ achieves order-optimal rates both with high probability and in expectation. Moreover, we establish a linear speed-up in the number of agents, which, to our knowledge, has not been shown for decentralized methods with clipping. The key technical ingredient is a sharp analysis of the consensus gap that exploits the structure of clipping, relegating network effects to higher-order terms. Our results highlight an important distinction between clipping and normalization in decentralized settings: while normalized $\mathtt{DSGD}$ can fail to converge, clipping retains magnitude information, enabling $\mathtt{DSGD}$ to be convergent and order-optimal. Numerical experiments validate our theory.

In this paper, we consider a finite-sum optimization problem, where the objective function is convex, the $n$ component functions are twice continuously differentiable and their Hessians are mean-square Lipschitz. Our contribution is twofold. First, we develop a stochastic algorithm, which requires $\tilde{\mathcal{O}}(n + n^{6/7}/\epsilon^{2/7})$ second-order oracle calls in expectation to find an approximate solution to the problem with the expected accuracy $\epsilon$. Second, we establish a lower complexity bound of $\Omega(n + n^{6/7}/\epsilon^{2/7})$ for any stochastic second-order algorithm satisfying a stochastic second-order version of the first-order linear span assumption, which is widely adopted in the optimization literature. Our lower and upper complexity bounds match up to logarithmic factors and thus resolve an important open question of establishing the optimal complexity of this problem class.

Procrustes-Wasserstein alignment jointly estimates a matching and rotation without supplied correspondences, but alternating minimization can stop at suboptimal solutions. Rubix solves the equally weighted planar problem globally under squared Euclidean loss. Each matching $\sigma$ of two centered $n$-point sets defines a complex correlation $z_\sigma=\sum_i\bar x_i y_{\sigma(i)}$. Their convex hull is the permutation polygon: supporting vertices give optimal matchings at fixed rotations, and the farthest vertex gives the global alignment. We prove the sharp bound of $n(n-1)$ vertices for $n\ge2$, answering Rote's rotation-assignment open problem. In exact arithmetic, assignment queries recover the polygon in $\mathcal O(n^5)$ operations. Assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound. On timed MPEG-7 shape pairs, Rubix attains every numerical reference value in 12 ms on average, 50 times faster than a rotation grid at the same accuracy. Its distances improve gravity-aligned matching of real 3D scans, shape retrieval and noisy crystal classification over alternating minimization.

Models can be retrained as new data arrive, but deploying every new version risks replacing a good policy with a worse one. We study how to plan policy updates (i.e., meta-policy) before future candidates are trained, balancing the benefits of improvement against the risk of performance regression. Our offline meta-policy maximizes expected cumulative value subject to a budget on the expected number of updates that perform worse than the policies they replace. We estimate the value and risk of possible switches from historical learning trajectories, represent an update schedule as a path in a directed acyclic graph, and select a schedule using dynamic programming. A leading-order analysis identifies the signal-to-noise ratio of policy improvement as a key driver of update frequency, waiting times, and risk allocation: clearer improvements support earlier, more frequent updates, while noisier improvements call for longer waits or greater risk expenditure. Their asymptotic rates also reveal a diminishing marginal cost of achieving greater safety over time. Experiments on synthetic and clinical trial data illustrate the performance–risk tradeoff and compare our method with alternative baselines.

The heavy-ball method can accelerate convergence near the minimizer, but it may also enter persistent oscillations on general strongly convex objectives. How should it be initialized to reliably realize this local advantage? We study the initialization neighborhood in which convergence can be guaranteed using only curvature and regularity bounds. This neighborhood shrinks as momentum approaches one; for Lipschitz-continuous Hessians, its radius is of the same order as damping. Matching upper and lower bounds show that this initialization requirement cannot be relaxed uniformly over the function class. We then derive a startup rule based on the current gradient: a finite gradient-descent warm-up enters the guaranteed region before switching to heavy-ball. With parameters optimal for strongly convex quadratics, the subsequent iterates satisfy an accelerated local asymptotic rate bound. Code is available at https://anonymous.4open.science/r/HeavyBall-811D/.

We study learning optimal policies in average-reward multichain Markov decision processes (MDPs), where the optimal gain may depend on the initial state and recurrence structures vary across policies, creating challenges for reinforcement learning (RL) methods. We propose an asynchronous value-iteration-based RL algorithm that requires no model knowledge beyond the MDP's transition graph and leverages Bather's decomposition to hierarchically partition the state space into communicating subsystems and transient states. This decomposition induces a recasting of the global decision problem into structured subproblems, which our algorithm exploits. We show that the algorithm converges to the optimal gain and produces gain-optimal policies after finite time. Building on this base algorithm, we develop two further algorithms: one approximately solves the multichain average optimality equations to obtain near gain-optimal policies, and another targets near bias-optimality by approximating the optimal bias function and solving an induced average-reward multichain MDP using the base algorithm. We provide almost-sure convergence guarantees for all three algorithms and empirically compare their tradeoffs, showing that the latter two also consistently improve transient performance relative to the base algorithm. To our knowledge, these are the first essentially model-free average-reward RL algorithms for general multichain MDPs without reductions to discounted problems.

Entropy-regularized optimal transport (EOT) has become a key tool in optimal transport, providing both theoretical insight and efficient computation via the Sinkhorn algorithm. We study the first-order \(\Gamma\)-expansion of EOT as the regularization parameter \(\varepsilon\) vanishes, by rewriting EOT as the minimization of a relative entropy with respect to a Gibbs measure \(\gibbs\) associated with the duality gap energy. Using a mild extension of the Laplace method, we prove the convergence of the Gibbs measures and derive this \(\Gamma\)-expansion for non-degenerate ground costs, under suitable assumptions on the duality gap, the contact set and the marginals. This yields at the same time an entropic selection principle and an expansion of the EOT cost up to \(o(\varepsilon)\) for this class of ground costs, generalizing existing results.

Building on a new regularized continuous formulation which avoids spurious solutions of the maximum $s$-plex problem, we develop a tailored block-coordinate first-order method that combines away-step Frank-Wolfe updates on the standard simplex with linear optimization over the fractional $b$-matching polytope. We establish finite stabilization of the auxiliary variables, convergence to stationary points, and finite support identification under suitable conditions. Numerical experiments on standard benchmark graphs show that, also in practice, regularization avoids the support-identification failures observed with the original cubic formulation and enables the proposed method to quickly find high-quality $s$-plexes.

We study a nonlocal stochastic optimal control problem in which the control acts through the elliptic smoothing operator $\mathcal{S}=(I-\Delta)^{-1}$ on $\mathbb{R}^d$. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on $\mathcal{S}u$. We prove the existence of an optimal control by the direct method in the calculus of variations. We then derive a Pontryagin-type maximum principle by spike variations, relying on sharp properties of the operator $\mathcal{S}$. The result yields a pointwise minimisation rule for the optimal control. We discuss the meaning of the optimal control problem in the case of population dynamics.

Mathematical programming formulations are routinely modified through algebraic reformulations, perturbations of constraint functions, and the addition or removal of constraints. Some of these modifications preserve the feasible set, whereas others reflect newly imposed modeling requirements and therefore change the optimization problem itself. This paper develops a metric framework for quantifying the robustness of formulation-dependent properties. Formulations are modeled as typed collections of constraint functions. Within each fixed typed-cardinality class, their distance is defined by optimally matching constraints of the same type and measuring the resulting discrepancies in a norm on the underlying function space. As a concrete demonstration of the framework, we study the Linear Independence Constraint Qualification (LICQ). We introduce the pointwise LICQ radius, defined as the distance from a feasible formulation to the set of formulations that remain feasible at a prescribed point but fail LICQ there. Under suitable assumptions, we derive an explicit formula for this radius and characterize the effect of adding a single equality or inequality constraint.

We study adaptive stabilization of known control-affine systems by treating the current parameterized critic itself as a Lyapunov candidate. Its total derivative contains a parameter-velocity term that can compensate unfavorable state evolution. A projection onto the intersection of a bounded admissible velocity set and one decay halfspace enforces parameter invariance and Lyapunov decrease simultaneously. We give an exact pointwise feasibility test, a conditional state-stability theorem, and a counterexample showing why a positive critic family containing a stabilizing member does not imply persistent feasibility. A convex endpoint constraint provides a distinct sampled implementation. A generic periodic polynomial-factor critic with a local Riccati anchor is evaluated on a force-limited cart-pole without mechanical-energy features, replay training, waypoints, or controller switching. The frozen controller passes stabilization and numerical decay checks on 600 sampled initial states across two draw sets; 400 matched continuous trajectories also pass. Independent reintegration audits and a conservative local feasibility certificate are distinguished from a global or machine-arithmetic guarantee. Compared with nonlinear model-predictive control, the measured implementation uses less CPU but has higher cost, settling time, and cart travel. The results support a conditional adaptive stabilization mechanism.

Hessian spectra at trained models in deep learning exhibit a persistent pattern: eigenvalues organize into distinct clusters, including a large bulk near zero and a few isolated outliers. This paper shows that a natural account of these spectral phenomena emerges when the original setting is understood as a departure from a nearby, otherwise hidden, highly symmetric reference. Modifications, including changes to the architecture, data distribution, or parameter metric, expose a nearby reference configuration whose Hessian exhibits rich invariances-ones not accounted for by weight symmetries. There, symmetry enables a precise description of the spectra, forcing high-dimensional kernels and eigenvalues of large multiplicity. Returning to the original configuration breaks the Hessian symmetry and thereby produces the observed hierarchy of clusters and outliers. The framework is developed in some generality, with a detailed analysis of three-layer ReLU networks and applications to convolutional, graph, and transformer models, as well as to the NTK. The same mechanism is further shown to yield analogous spectral structures in layerwise Hessians and the Gauss-Newton matrix.

We answer two questions about the heavy-ball method on smooth strongly convex functions. The first concerns Polyak's tuning. Its dimension-uniform worst-case exponential rate equals the largest growth factor of an interpolable mixture of rotating geometric sequences, and we determine this factor for every condition number. The optimal mixtures form four algebraic families; the classification combines exact polynomial certificates, a rational parametrization, an exact solution in a quintic number field at the critical three-cycle, and verified interval continuation. Two frequencies suffice, one function in dimension five attains the rate at every horizon, the upper and lower bounds differ by constant factors, and the method converges on the whole class if and only if $\kappa<9+4\sqrt5$, the threshold located by Badithela and Seiler. The second question concerns arbitrary step sizes and momenta, in particular the regions of the parameter plane between the computed Lyapunov and cycle regions. We describe the phase diagram through the angle $\Theta(\omega)$ between the characteristic polynomials of the two extreme quadratics. The method converges when $\Theta(\omega)>\omega/2$, has a $k$-periodic orbit when $\Theta(2\pi j/k)\le\pi/k$, and the remaining regions are rotation-number gaps organized by Farey fractions. A single strict certificate at one growth exponent bounds the rate, so convergence can be certified at the threshold exponent. In the first level of the gaps, fans of Farey fractions accumulating at a resonant endpoint carry periodic orbits with several harmonics, explicit fan-lag inequalities certify convergence, and both behaviors occur in the regions left open before.

Inverse optimization estimates the weights of an objective function that explain observed decisions as optimal solutions, and is used in a variety of fields. For mixed-integer linear programs (MILPs), existing methods aim to reproduce the observations as optimal solutions, and thus learn compromise weights when the observations are suboptimal. We propose outperformance inverse optimization, which instead seeks weights that induce, at each state, an optimal solution outperforming the observed action in every component. We give a loss function that can be evaluated with forward-problem oracles alone and is thus applicable to MILPs, together with gradient-based and DC optimization algorithms for minimizing it. For weights inducing a unique outperforming optimal solution at all observations, we prove that the probability of failing to induce such a solution at a new state (the generalization error) is bounded by a quantity inversely proportional to the number of observations, and that this bound is tight in the number of observations up to logarithmic factors. In experiments on synthetic and real data, the proposed methods improve the prediction of solutions outperforming the actions over existing methods.

Human learning is a dissipative dynamical process: mastery accumulates through practice, decays through forgetting, and propagates across interdependent concepts. We model it as a nonlinear dissipative system of ordinary differential equations whose parameters are mechanistically meaningful (a transfer-of-practice matrix, per-concept forgetting rates, and a saturating practice-response gain), and we study when those parameters can actually be recovered from data. For two concepts, we prove a constructive structural identifiability theorem for the associated inverse problem under explicit excitation conditions, together with monotonicity, robustness and L-stability results. We derive an implicit L-stable scheme and a batched solver numerically equivalent to the per-trajectory formulation yet two orders of magnitude faster. The empirical study is two-sided. Under the theorem's excitation conditions, synthetic recovery is exact. On large observational benchmarks it is not. An apparently strong recovery, with forgetting rates correlating with topic error rates at Spearman $\rho = 0.83$, is refuted by four controls: it survives destroying the temporal order of the data, is matched by a classical Bayesian baseline, and is unaffected by removing real timestamps. A single shared forgetting rate attains the same accuracy, and replaying the real schedule with identical true rates reproduces most of the correlation. A positive control shows that the controls detect forgetting where it is observable. We trace the effect to the stationary structure of the model and show that it is consistent with the degeneration anticipated by the theory in the absence of designed excitation.

This paper studies necessary optimality conditions for stochastic control problems governed by decoupled forward–backward stochastic differential equations with reflection constraints on the backward component. The cost functional is defined through the initial value of a reflected BSDE whose lower obstacle depends on the forward state. The nonsmooth nature of the reflection term prevents the direct application of classical variational methods. Using a penalization approach, spike variations, and duality arguments, we derive a penalized variational relation. A key feature of our method is to isolate the reflection contribution into a residual term, avoiding the differentiation of the nonsmooth penalization operator while preserving a Hamiltonian adjoint structure analogous to the classical stochastic maximum principle. Passing to the limit yields a reflected variational inequality of Pontryagin type, where the obstacle effect is explicitly represented through a residual contribution associated with the reflection term.

As in all fields of engineering, control engineers constantly face a dichotomy between achieving high performance and the limited availability of resources for sensing, actuation, and control. This tension often translates into formalizing design problems with sparsity constraints; that is, the design involves decision variables whose number, spatial reach, temporal spread, or other physical expressions are much smaller than in the ideal scenario of unconstrained optimization of a task-related performance metric. Even though this fundamental limitation underlines a plethora of methodological approaches and application domains, the control literature is fragmented into several lines of work tailored to specific scenarios. This survey fills the gap by bringing together several lines of work on handling sparsity in relevant control problems such as actuator scheduling, minimal-energy controllability, maximum hands-off control, and online convex optimization, and weaving a red thread through theoretical foundations and algorithms. Beyond establishing a coherent, structured narrative of problems and solutions for handling sparsity in control scenarios, we discuss open areas and relevant directions for future research, motivated by the growing complexity of control systems and their relations with key domains such as pervasive computing, distributed learning, cyberphysical security, and ecological footprint.

This paper derives necessary optimality conditions for free end-time optimal control problems governed by sweeping process dynamics. In contrast to classical optimal control problems, the presence of the normal cone term in the state equation introduces discontinuities with respect to the state variable and prevents the standard Lipschitz regularity assumptions typically used in optimal control theory. As a result, both the analysis of trajectories and the formulation of optimality conditions require refined tools. We address problems with general endpoint constraints and time-dependent data, considering both the case in which the dynamics are Lipschitz continuous in time and the more challenging setting of merely measurable time dependence. Special attention is devoted to the additional transversality conditions associated with the free end-time variable. In the Lipschitz case, these conditions are expressed in terms of a function of bounded variation that coincides almost everywhere with the maximized Hamiltonian. In the measurable case, we show that a suitable interpretation can be obtained via the notions of sub- and super-essential values of the maximized Hamiltonian, recently introduced in the literature, leading to strengthened transversality conditions. These conditions also involve an additional term reflecting the interaction between the costate arc and the normal cone to the moving set at the optimal endpoints. Our approach combines suitable perturbation techniques with a penalization method tailored to sweeping processes. The moving set is modelled as a finite intersection of inequality-defined sets satisfying suitable constraint qualifications, including a positive linear independence condition and a diagonal dominance property of the associated Gramian matrix. We also introduce a local version of the constraint qualification condition. Our results apply to a broad class of problems, including those involving unbounded moving sets such as polyhedra. The resulting optimality system extends and sharpens recent results for classical free end-time problems to the non-Lipschitz, discontinuous framework of sweeping processes. The relevance of the transversality conditions obtained in this article is illustrated through two simple examples.

We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order $r^4$ as $r\to0$ on squares of half-side length $r$ with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.

We develop a continuous-time deterministic policy gradient framework for reinforcement learning with reflected state dynamics, where the state process is governed by a controlled reflected stochastic differential equation on a hypercube. Under suitable regularity assumptions, we establish the connection between the value function and the Neumann Bellman equation, introduce an advantage-rate function that yields a deterministic policy gradient formula, and prove the martingale characterization theorem. Motivated by these theoretical results, we propose a continuous-time deep deterministic policy gradient algorithm for reflected stochastic systems, in which the Neumann boundary condition is imposed via either soft penalization or hard architectural constraint. We further quantify the discrepancy between the ideal continuous-time dynamics and the discretely sampled exploratory dynamics executed in practice, showing that the error decays as the time grid is refined and exploration noise vanishes. Our experiments on reservoir control problems illustrate the effectiveness of the RL framework, highlighting that boundary-aware methods substantially reduce Neumann boundary residuals and enhance learning stability.

We present a novel dual approach to characterizing the solution to optimal stopping problems for strong Feller processes on locally compact, separable metric spaces. Our characterization builds upon reinterpreting a standard result in optimal stopping theory as a dual formulation, which yields computationally tractable algorithms. Our framework is applicable to a broad class of Markov processes. First, in the one-dimensional case, while previous studies have managed to comprehensively solve only diffusion and one-sided jump Lévy processes over an infinite time horizon, our proposed method successfully extends to both infinite- and finite-horizon scenarios across all these processes, including two-sided jump Lévy processes. In addition, for multi-dimensional cases, our framework is applicable to diffusions over an infinite horizon with certain structural features such as convex stopping regions. To illustrate this versatility, we explicitly solve several challenging problems including a case where the optimal stopping region consists of two disjoint connected components – a structural feature that remains unaddressed in the literature.

Differential-algebraic equation (DAE) systems arise in power networks, multibody mechanics, and chemical processes, where algebraic constraints couple dynamic and algebraic states. Standard control barrier function (CBF) methods, designed for ordinary differential equations, fail for DAE systems because the algebraic constraints impose compatibility requirements that may render safety filters infeasible. This paper develops a safety-critical control framework for semi-explicit DAE systems using CBFs that consider compatibility with the algebraic constraints, termed DAE-aware CBFs. We construct projected vector fields on the constraint manifold to account for the hidden dynamics, while ensuring compatibility conditions that keep the algebraic constraints satisfied. We then present sum of squares (SOS) verification conditions that certify both correctness and feasibility of the resulting DAE-aware CBF. The framework is validated on power system case studies.

Optimal transport (OT) has found broad applications in machine learning, economics, and management. Classical OT models, however, are not directly suited to allocation problems with graph-structured connectivity and unbalanced mass. Distributed OT formulations on bipartite graphs address these requirements but are typically solved by ADMM, whose slow convergence limits communication efficiency. We adapt the accelerated preconditioned ADMM (AP-ADMM) to the quadratically regularized distributed OT model and develop DAP-ADMM, a fully distributed algorithm in which each node communicates only with its graph neighbors. For fixed local penalty parameters, we establish global convergence and non-ergodic $O(1/k)$ bounds for both the KKT-residual norm and the primal objective gap. We derive an exact solver-free routine for the interval-constrained projections arising in the local ADMM subproblems. We propose a distributed self-adaptive penalty mechanism that updates node-specific penalties using locally computable KKT-residual components without global aggregation. Numerical experiments confirm the efficiency of the solver-free routine and the practical acceleration of DAP-ADMM over standard distributed ADMM.

Restorative control seeks lasting recovery through an intervention that ends in finite time. We formulate this objective as a reach-avoid problem in which the input must bring the system safely into a desired basin of attraction before withdrawal. Two candidates share bistable baseline dynamics but accumulate irreversible damage through different allocations of drive-generated exposure. We derive a lower bound on the exposure required for basin transfer and a sufficient construction for safe restoration. When the lower bound exceeds the summed tolerances, every common open-loop input that restores either model causes failure under at least one. With sufficient tolerance, proportional allocation restores both without identification. For feedback from output samples with Gaussian noise, we bound safe-restoration probability in terms of failure risk under the alternative model. A numerical example illustrates the certificates under two sensing regimes.

Cancer is driven by an estimated two to nine gene mutations, known as multi-hit combinations. Due to the large number of genes, identifying these multi-hit combinations presents a computationally challenging problem, which has previously required supercomputers to enumerate all or most gene combinations. Recent work formulated this classification task as the Multi-Hit Cancer Driver Set Cover Problem (MHCDSCP) and solved it via column generation on a single commodity CPU, where a pricing problem identifies promising gene combinations (arXiv:2602.22551). The main bottleneck is the pricing problem that has to be solved repeatedly. To solve the pricing problem to optimality, we propose a dedicated depth first search method that enumerates all possible gene combinations. Although enumerating gene combinations with depth first search have been performed on supercomputers with around 150,000 CPU cores (arXiv:2603.16721), our depth first search algorithm solves the pricing problem to optimality within seconds on a single CPU by leveraging dual information to prune large parts of the search space. By eliminating this bottleneck, our approach yields the first exact method for the MHCDSCP, which is guaranteed to find a global optimal solution.

Newton-type algorithms are among the most effective methods for solving optimization problems because of their rapid local convergence. However, extending these methods to nonsmooth optimization is challenging due to the difficulty of incorporating second-order information. To address this issue, we propose a coderivative-based Newton framework for solving both unconstrained and constrained nonsmooth optimization problems using tools from variational analysis and generalized differentiation. The proposed method employs generalized Hessians, defined as coderivatives of the subgradient mapping, and is applicable to both $C ^{1,1}$ functions and convex composite optimization problems with extended-real-valued components. To enhance robustness, the algorithm incorporates an adaptive regularization strategy that enables it to handle problems whose generalized Hessians are positive semidefinite. In addition, a Hybrid Adaptive Nonmonotone (HAN) line search scheme is developed as a globalization technique to improve global convergence and practical performance. Under standard assumptions, both the exact and inexact versions of the algorithm are globally convergent and achieve local superlinear convergence when the associated subgradient mapping satisfies the semismooth* property. The proposed framework is further extended to convex composite optimization through the forward-backward envelope, allowing it to handle problems with or without strong convexity in the smooth component of the objective function. Numerical experiments on Lasso, logistic Lasso, and support vector machine (SVM) problems demonstrate the efficiency and robustness of the proposed algorithm. Comparisons with several well-established first-order and second-order methods for nonsmooth optimization show that the proposed approach is computationally competitive while maintaining strong convergence properties.