Papers

In a predictor-corrector arc-search method, a point on the predictor arc is first selected and a corrector is then computed at that point. Since the Karush-Kuhn-Tucker (KKT) matrix changes with the selected point, this correction may require a second factorization in each iteration. We propose a one-factorization arc-search method (OFAS) for semidefinite programming (SDP) that avoids this second factorization, retains the corrector after the arc step, and uses only one KKT factorization per iteration. The key is a three-term decomposition of the complementarity mismatch at the selected point on the arc, which allows the correction to be assembled from linear-system solutions using the factorization at the current iterate. We also propose a curvature-amplified variant (CA-OFAS) that amplifies the second-order arc term. The proposed methods are formulated with a homogeneous self-dual embedding. For a wide long-step neighborhood, we prove that both methods reduce the homogeneous complementarity and residuals below $\varepsilon$ in $O(n \log(1/\varepsilon))$ iterations for an $n$-by-$n$ SDP matrix variable. Numerical experiments show that the one-factorization structure and the correction after the arc step are associated with reductions in computation time. Compared with the Mehrotra-type method, CA-OFAS reduces the geometric-mean computation time by 17% on SDPLIB and 13% on neural network verification SDP problems.

The concept of tensor eigenpairs has attracted increasing research attention in the past decades. Recent works have focused on a special class termed regular simplex tensors, which are constructed from an equiangular tight frame of n vectors in (n-1) dimensional space for n >= 3 and order m >= 3. Existing works focus on analyzing the robustness of eigenpairs obtained by the tensor power method. At the end of that work, a conjecture was made that if they exist, the only robust eigenvectors of a regular simplex tensor, up to sign equivalence, are the vectors in the regular simplex frame. A subsequent study theoretically proved that this holds for the simplest triangle case where n = 3. However, for cases with higher n, the process becomes complicated in both checking all eigenpairs and determining the explicit formula for the robustness criterion. In this paper, to deal with this issue, a connection between robust and locally optimal eigenpairs is built, recognizing the latter as another pivotal concept in the field of optimization. Then, we turn to checking the local optimality of all eigenpairs, for which we have developed an efficient model with a favorable structure that facilitates the enumeration of all eigenpairs and delineates the optimization landscape for the model. Then, integrating the two advances enables us to narrow the scope of the robust eigenpairs to those locally maximized ones, which are exactly the vectors in the regular simplex frame. Finally, the proof of the conjecture reduces to only examining the vectors in the frame, whose robustness can be easily checked for any higher n and m. This work shows that, up to sign equivalence, excluding the exceptional cases (m, n) = (3, 3)/(3, 4)/(4, 3) where no robust eigenpairs exist, the only robust eigenvectors of a regular simplex tensor are the vectors in the regular simplex frame.

Large Reasoning Models (LRMs) have shown remarkable reasoning capabilities, yet they still suffer from overthinking, generating redundant reasoning steps which incur substantial token consumption. Existing methods, such as suppressing reflective keywords or forcing shorter reasoning lengths, attempt to mitigate this issue but inevitably truncate necessary steps and induce underthinking, thereby compromising performance. To address this dilemma, we investigate the latent representations and observe that efficient reasoning steps naturally cluster into a concentrated region in latent space, while those deviating from this region tend to produce verbose sequences. To leverage this, we keep reasoning focused within this region via a quadratic program which projects deviating hidden states back into the region. Then we propose a novel training-free framework to achieve efficient reasoning that reduces token generation costs without sacrificing performance. Extensive experiments conducted on four models ranging from 1.5B to 14B, and across six benchmarks in math reasoning, coding, and scientific QA, validate the effectiveness of our method, up to a 12.1% improvement in accuracy while reducing generated tokens by 11.8% to 52.8%. Codes are available at \href{https://github.com/hzn18/Opt4Reasoning}{https://github.com/hzn18/Opt4Reasoning}.

Online resource allocation is a fundamental problem in revenue management, sponsored search, and platform operations. Most prior work assumes nonincreasing assignment rewards, capturing diminishing returns. We instead study increasing returns, where assigning more customers to the same product can unlock larger value through scale, visibility, or network effects. We consider capacity-limited products assigned to sequentially arriving customers, where the reward of each product depends on the total number of customers assigned to it. We focus on full compatibility, where every product can be assigned to every customer. The number of customers is unknown to the online algorithm. We show that representative classical approaches, such as online greedy algorithm and LP-based independent rounding, can perform arbitrarily bad in this setting, even when products share a common bonus function. For homogeneous bonus functions, we give a simple polynomial-time algorithm that achieves a tight $1/2$-competitive ratio. For arbitrary heterogeneous bonus functions, we show that no constant competitive ratio independent of the number of products is possible: for $m$ products, the optimal competitive can be as small as $\Theta\left({1}/{\sqrt m}\right)$. This shows that heterogeneous delayed rewards can force any online algorithm to guess the realized arrival counts. We then identify a structured heterogeneous regime that restores a constant guarantee. When each bonus function is nonnegative, nondecreasing, and discrete concave, we design an intermediate target repair algorithm with a deterministic pathwise guarantee of $0.2466$. The algorithm repeatedly computes an offline target for a larger demand level and repairs the current allocation toward it using a prefix-robust assignment order. This coordinates buildup while remaining robust to early stopping and yields a distribution-free guarantee.

We consider the convex problem $(P): {\min}_{y\in \mathbb{R}^n}\, f(Ay)+g(y)$, where $A:\mathbb{R}^n\to \mathbb{R}^m$ is a linear operator, $f$ is convex and $(M,\nu)$-Hölder smooth on $\mathbb{R}^m$, and $g$ is proper, closed convex on $\mathbb{R}^n$, and admits a (generalized) linear minimization oracle (LMO). An active-hull (AH) method calls the LMO in each iteration and outputs a point that lies in the convex hull of all the past LMO returns. We show that under the Euclidean geometry (and in the high-dimensional regime), the optimal LMO complexity for the class of AH methods to solve $(P)$ is $O\big(M^{{2}/({1+\nu})}D^2\varepsilon^{-{2}/({1+\nu})}\big)$, where $D$ denotes the diameter of the feasible region and $\varepsilon$ denotes the objective sub-optimality gap. In particular, the optimal LMO complexity is achieved by a single-loop first-order primal-dual splitting method. Since this method requires knowledge of the problem parameters (e.g., $M$, $D$ and $\nu$), we propose two meta parameter-search algorithms that aim to resolve this issue. These algorithms are parameter-free, but come with the price of an additional log-factor in their LMO complexities.

Constrained nonconvex optimization has seen increasing application in modern machine learning, such as safe LLM alignment/finetuning, transfer learning and rank-constrained continual learning. The commonly used approach is Projected SGD which requires projection at every iteration. However, projection onto a functional constraint can cost substantially more than a stochastic first-order update. We study whether this operation can be deferred until the end of nonconvex stochastic optimization, and only do it once. For weakly convex, possibly nonsmooth objectives with regular convex constraints, we give a penalized proximal method that uses $\widetilde O(\epsilon^{-4})$ stochastic subgradients and one projection onto the constraint set. The returned point is exactly feasible and has a small Moreau-envelope stationarity measure; for smooth objectives, the same algorithm controls the projected-gradient mapping. For smooth nonconvex constraints, we assume a global lower bound on the infeasible constraint slope which permits arbitrary, possibly infeasible initialization. An exact-penalty variant attains the same stochastic-oracle order and returns an exactly feasible point within $O(\epsilon)$ of an $\epsilon$-KKT point. All intermediate updates use only projections onto a simple Euclidean ball. These are oracle-complexity guarantees: the cost of the single terminal projection is separate, and no efficient projection algorithm for a general nonconvex set is assumed to follow from regularity.

In facility location problems, planners can reduce demand uncertainty by collecting data, but these actions are costly. In this paper, we ask how a planner should jointly decide where to open facilities and invest in demand learning, which comes with a cost. The challenge is that learning changes how future random costs should be evaluated, creating a model that is both endogenous and nonlinear in location decisions. We model this problem within a two-stage stochastic framework, where both location and learning decisions have to be made here-and-now. Under some mild assumptions, we obtain a closed-form reformulation, which further enables a theoretical analysis of the sensitivity of learning decisions to model parameters and design of convergent solution algorithms that alternate between updating location and learning decisions. Numerical experiments on benchmark instances show that learning is most valuable when initial demand uncertainty is high, learning is effective, and learning costs are low. Moreover, the proposed solution algorithms find high-quality solutions much faster than exhaustive search and a standard commercial nonlinear solver. These findings show how information-acquisition decisions can be built directly into strategic facility planning. They help decision makers target limited learning resources where better demand information has the greatest value.

We present a new class of near-linear algorithms for efficiently integrating general tensor fields defined on trees with distance dependent kernels, the Structure-Adaptive Tree Field Integrators (STAD-TFIs). STAD-TFIs exploit the tree's underlying structure through decompositions built around path backbones and single vertex separators, and use two-dimensional fast Fourier transforms to compute interactions jointly. By exploiting this structural information, STAD-TFIs achieve more computationally efficient integration than their regular efficient tree field integrators (TFI) counterparts. We provide a detailed theoretical analysis of our proposed approach and complement it with an exhaustive empirical evaluation, ranging from speed tests on synthetic trees, through accelerated Sinkhorn-based relaxations of the Optimal Transport algorithms on real meshes, to Topological Attention Transformers for vision tasks. To the best of our knowledge, we provide some of the first results showing that efficient to compute and accurate relaxations of the geodesic Sinkhorn-based solutions of the Optimal Transport problem can be derived by applying fast TFI methods.

Standard Big-M mixed-integer programming reformulations of chance-constrained programs (CCPs) with finite support often yield weak linear relaxations and scale poorly as the number of scenarios increases. To address this limitation, we derive two families of intersection cuts from problem-specific S-free sets and develop decomposition-based branch-and-cut frameworks. The first one exploits the submodular structure of the scenario requirement function, while the second one uses probability covers. We propose hybrid separation strategies that combine classical mixing inequalities and quantile cuts with the proposed intersection cuts. Computational experiments on integrated production-distribution planning instances show that the proposed hybrid methods reduce computational time compared to pure methods.

Most existing studies on distributionally robust optimization (DRO) with a decision-dependent ambiguity set focus on the computational and theoretical challenges posed by this class of problems. In this paper, we adopt a combined modeling and computational perspective to understand the trade-offs between modeling fidelity, solution quality, and computational effort, particularly in comparison with DRO models using decision-independent ambiguity sets. Motivated by representative applications in joint pricing-stocking newsvendor problems with price-dependent demand and facility location problems with location-dependent demand, we consider a two-stage stochastic mixed-integer program with (non)convex continuous recourse. Assuming a finite sample space, we model the decision-dependent distributional ambiguity with a polyhedral ambiguity set and reformulate the problem as a nonconvex mixed-integer nonlinear program. To efficiently solve the reformulations, we propose decomposition-based cutting-plane algorithms. Our experiments on benchmark instances indicate that a decision-dependent ambiguity set substantially reduces the postdecision disappointment for the newsvendor and out-of-sample cost for the facility location problems, up to 65% and 7% on average, respectively; thereby mitigating the optimizer's curse relative to a decision-independent DRO. However, this improvement is achieved at the expense of increased computational effort, with the absolute runtime ratio falling between 2 and 25 for medium- to large-sized newsvendor instances and between 2 and 12 for facility location instances.

Plug-and-Play (PnP) algorithms are a class of iterative methods for inverse imaging. Within an optimization algorithm, they combine a flexible fidelity term, encoding the forward operator, and a pretrained image denoiser, in order to deal with more severe corruptions such as blurring or downsampling when reconstructing an image. This work studies provably convergent PnP methods for mixed-noise forward processes by using the infimal convolution as a fidelity term, providing a statistical interpretation as a joint maximum a-posteriori estimator over both noises, and preserving the Bayesian MAP interpretation of PnP methods. Independently, we extend the PnP formulation to multiple prior terms using the Davis–Yin three-operator splitting. This extension can be combined with either standard fidelities or the proposed mixed-noise infimal-convolution fidelities. We verify that these generalized PnP methods are convergent under standard Kurdyka–Lojasiewicz conditions. Numerical experiments on Laplace-Gaussian and Poisson–Gaussian noise demonstrate stable convergence of single-prior and multiple-prior PnP methods, and divergence under fidelity mismatch. Furthermore, PnP with infimal convolution fidelities are able to scale to noise up to 38% standard deviation, with multiple priors reaching different stationary points that qualitatively preserve more textural properties.

Neural surrogates can accelerate large-scale optimization by replacing expensive or intractable model components with efficient learned approximations, but solving the resulting embedded problems can remain prohibitively costly. For instance, standard exact encodings of neural networks with ReLU activations allow the problem to be solved by mixed-integer solvers, but add large numbers of binary variables to accommodate the nonlinearity of the activations, which can render the problem computationally prohibitive. To address this, we propose DCEmbed, a heuristic for optimization problems with embedded neural surrogates that leverages the difference-of-convex (DC) representation of the network and avoids adding activation binaries. Exploiting shared structure within the DC representation of a ReLU network, we derive a reduced-size, exact formulation for its convex components that can be embedded in optimization problems using just two linear inequalities and one continuous auxiliary variable per hidden neuron. Using this, the problem is solved via an iterative penalty convex-concave procedure, where only the concave portions of the neural terms are approximated at each stage. The original objective, constraints, and any discrete decisions are retained, allowing standard convex or mixed-integer optimization solvers to optimize the host and surrogate jointly at each iteration. In experiments on quadratic programs, mixed-integer resource allocation, and neural two-stage stochastic programming, our method demonstrates much faster progress toward high-quality feasible solutions than approaches using exact mixed-integer embeddings. In particular, DCEmbed achieves $4\times$ lower normalized primal integral than the best exact baseline on the resource allocation problem, while in two-stage stochastic programming it reaches the global surrogate optimum $\sim 5\times$ faster than Gurobi ML.

Mean-field control barrier functions (MF-CBFs) enforce swarm safety through constraints on the agents' distribution. Previous formulations, including applications to stochastic coverage and shepherding, use first-order differential inequalities. These cannot directly enforce constraints whose barrier functions yield control-free first derivatives. Thus, we develop the theory of higher-order MF-CBFs, which ensures the positivity of the barrier functionals through higher-order differential inequalities, as in their finite-dimensional analogs. More specifically, we introduce the notion of relative degree for mean-field safety functionals and develop an analytic framework for computing their higher-order derivatives. Moreover, for practical examples of cross-correlation and self-correlation functionals, we show that repeated differentiation along mean-field dynamics is structure-preserving and reduces to kernel recursions. As an illustration, we apply higher-order MF-CBFs to double-integrator swarm tracking and avoidance problems, where position-only constraints lead to MF-CBFs of relative degree two.

Optimization layers enable the incorporation of structured constraints and decision problems into learning systems. Training such systems requires differentiating through the embedded optimization problem, which can be challenging for general conic programs. We introduce dOPT, a solver-agnostic framework that, rather than differentiating the full conic formulation, reduces it at a computed primal-dual solution to an equality-constrained quadratic program that preserves the reference solution and its first-order sensitivity. The reduction captures the local first- and second-order conic geometry relevant to differentiation and remains well defined at singular configurations. Computing solution derivatives then requires a single symmetric linear solve, independently of the forward solver. We derive explicit reductions for convex NLPs, QPs, SOCPs, and SDPs. Numerical experiments validate the computed gradients and show favorable backward-pass scalability, with substantial speedups over existing differentiable conic optimization methods as problem size increases.

We study average consensus over slowly time-varying networks and its application to decentralized optimization. We introduce WAVE, a windowed Chebyshev method that limits the accumulated effect of network variation by restarting the recurrence after finite windows. If $\chi$ bounds the network condition number and successive communication operators satisfy $\|L_{k+1}-L_k\|\leq\beta$, WAVE reaches $\epsilon$-consensus in $O((\sqrt{\chi}+\min\lbrace\beta\chi^2,\chi\rbrace)\ln(e/\epsilon))$ communication rounds. This rate matches the fixed-network $\sqrt{\chi}$ dependence when $\beta=O(\chi^{-3/2})$ and smoothly interpolates across the full range of network variation. For piecewise-constant networks where each change is detected when it occurs and consecutive changes are at least $\tau$ rounds apart, WAVE reaches $\epsilon$-consensus in $O((\sqrt{\chi}+\chi/\tau)\ln(e/\epsilon))$ communication rounds. Finally, we use WAVE as the consensus step in an accelerated decentralized optimization method for $\alpha$-smooth convex local objectives with a $\mu$-strongly convex average. For sufficiently slow network variation, every agent obtains an $\epsilon$-solution to the global optimization problem after $\widetilde O(\sqrt{\alpha/\mu}\,\sqrt{\chi})$ communication rounds, matching the fixed-network dependence.

We characterize Bregman Douglas-Rachford splitting (BDRS) for unregularized discrete optimal transport and develop an anytime primal-dual certificate in linear memory. We first establish that BDRS coincides with warm-started Inexact Proximal point method for exact Optimal Transport (IPOT) using a single inner Sinkhorn iteration. By eliminating the primal transport plan from the updates, we derive an equivalent dual formulation that reveals BDRS as annealed Sinkhorn under an implicit inverse-linear temperature schedule, with an additional log-scaling momentum term and a cooler kernel. While this explains the role of the temperature parameter in BDRS as an initial temperature, it also reduces the solver's memory requirement from quadratic to linear. Utilizing this annealing perspective, we introduce overrelaxed BDRS, which combines annealing and overrelaxed scaling within a single recursion. We derive a primal-dual certificate for both methods that can be evaluated in linear memory without transport plan construction, thus providing a computable stopping rule. On the DOTmark benchmark, combining momentum with the cooler kernel produces substantially smaller optimality gaps than annealed Sinkhorn under the same schedule. For pixel-level color transfer between $1024\times1024$ images, BDRS attains a lower repaired transport cost than MDOT-TNT with a 9$\times$ speed up, reaching a relative duality gap of $1.59\%$ in 24 minutes. We further demonstrate a color transfer with $4238\times2365$ images, yielding 10 million pixels per image and approximately one hundred trillion implicit transport entries, reaching a best relative duality gap of $2.41\%$ and $2.80\%$ within 35 hours in each direction on a single NVIDIA L40S GPU.

We develop positivity certificates for polynomials that are affine in auxiliary parameters, with sum-of-squares multipliers depending only on the principal variables. Under a restricted Archimedean order-unit condition, every strictly positive scalar polynomial admits such a certificate. We construct a convergent restricted moment-SOS hierarchy for polynomial optimization problems with affine parameters and establish a flatness criterion for global optimality detection and global minimizer recovery. The scalar theorem extends to polynomial-matrix constraints with any number of affine parameters. For matrix-valued objectives, pointwise positivity is sufficient with one parameter but can fail with two or more parameters; uniform free positivity provides a general sufficient replacement. Applications include robust linear programming, robust Lyapunov design, min-max polynomial optimization, analysis and control of input-affine dynamical systems. Numerical experiments show that the proposed approach substantially reduces semidefinite block sizes and solution times while incurring little or no degradation in bound quality for most test cases.

Information diffusion can be controlled by restricting or removing links (edges) in online social networks, as well as in real-world networks. To identify the most influential links to remove while minimizing diffusion, previous studies have proposed upper bounds for spreading processes in SIR and SIS models, using supermodularity and weighted matrices to identify critical links in contact networks. However, in some cases, existing upper bounds are not sufficiently tight to accurately capture the effect of important edges, as in the SDIR model of [14] (Khanh-Cho-Dung, Proceedings of 40th ICOIN, 2026). We therefore propose a tighter upper bound for controlling diffusion in the SDIR model by directly analyzing the dynamics of the two state vectors D and I in a $2N$-dimensional space. This approach yields an improved spectral-radius convergence condition and outperforms the previous method. Simulations on the synthetic Erdos-Renyi network and the real-world Haslemere dataset using a Greedy edge-deletion algorithm demonstrate its effectiveness for influence minimization on social networks.

Differentiable optimization brings the structural guarantees of mathematical optimization to network pipelines, allowing them to be trained end-to-end. However, its application remains challenging for nonconvex constrained problems, as existing differentiable solvers often suffer from limited modeling expressiveness due to their reliance on specialized problem structures, while also incurring substantial computation time and memory overhead in both the forward and backward passes. To address these challenges, we propose lapanda, a matrix-free differentiable solver for nonconvex optimization with general constraints. It reformulates the problem to a sequence of augmented Lagrangian subproblems, each handled by a first-order inner solver through a proximal averaged quasi-Newton algorithm with adaptive linesearch, thus enabling efficient forward optimization. We establish local well-posedness of the solution map and convergence of the outer iterations, and further derive a sensitivity alignment between the original problem and the final subproblem in the backward pass, demonstrating that the subproblem sensitivity, which can be computed efficiently in a matrix-free manner, provides a principled approximation to the exact optimizer sensitivity. We evaluate lapanda on nonconvex constrained Rosenbrock benchmarks, imitation learning with several representative constrained optimal control problems, and embedded robotic obstacle-avoidance tasks. Compared with state-of-the-art differentiable solvers, lapanda delivers substantial reductions in computation time and memory footprint while maintaining reliable constraint satisfaction and learning performance.

What information is needed to turn a natural-gradient direction into a useful finite update? Under a population Kullback-Leibler (KL) budget, we call a step useful if it is feasible and loses at most a fraction $\varepsilon$ of the best feasible gain along the direction. We construct a four-state exponential family whose laws share their initial gradient, scalar Fisher information and natural gradient, yet two laws have disjoint useful-step sets. With these quantities supplied exactly and the law otherwise known only through draws, the family's worst-case sample complexity is $\Theta(\log(1/\delta)/(p\varepsilon^2))$ for small $\varepsilon$, where $p$ scales rare-state probabilities and $\delta$ is the failure probability. The budget is fixed and the optimal gain stays bounded away from zero, so the step length, not the direction, carries this cost. For succinctly described event-tilt models, returning a useful step is NP-hard even with the exact natural gradient and efficient exact sampling. Recovering the unit natural gradient to constant error is also NP-hard even in a two-parameter logistic family with Fisher condition number at most 3. We also give matching sample bounds for event tilts, sample bounds for damped Fisher solves and a population-KL certificate for affine classifiers. In frozen-feature classifier heads, stopping at a sampled KL boundary succeeds in about half of the trials, and a 10% KL margin raises joint success above 93% at a KL budget of 0.01. Thus, knowing where to move is not enough: how far to move can carry an update's entire cost.

Decision-focused surrogates can recover downstream decisions without identifying the quotient report. We characterize the equality set of the convex Smart Predict-then-Optimize surrogate (SPO+) population risk. Under central symmetry, the centered mean class is the unique Bayes minimizer exactly when every nonzero effective displacement makes the old optimizer leave the shifted optimal face with positive probability. This condition separates face crossing from selected-oracle disagreement and gives quantitative local coercivity. Without symmetry, strict crossing alone need not identify the mean; selection balance with reflected crossing restores quotient-report identification, and conditional versions extend the result to measurable predictors. These are population statements, without finite-sample report-recovery or generic transfer-regret guarantees. Closed-form mechanisms reproduce the analytic identities and rates. Portfolio, complete-matrix KuaiRec, and Energy/Storage studies measure predictive fidelity, shifted regret, and fitted-report geometry. A known data-generating process (DGP) companion retains their application geometries while isolating conditional-mean recovery and crossing, without testing the original observational assumptions.

We present DAOCP, a dual active set solver for linear quadratic optimal control problems with stage-wise equality and inequality constraints. Active set methods are leading Model Predictive Control benchmarks for full-body robotics, but existing solvers operate on dense QPs, while typical problem dimensions favor methods that exploit the optimal control structure. DAOCP combines the warm-starting capabilities of active set algorithms with the better computational scaling of structure exploiting solvers, by relying on a generalization of the relationship between the Riccati recursion and the Cholesky factorization of the condensed Hessian. This result enables dual active set iterations to operate directly on the original optimal control problem through recursive computations, without explicitly forming the condensed quadratic program. On robotics benchmarks, DAOCP is the fastest of four solvers in four of five scenarios, cutting average solve time on a 58-state Atlas model by 9$\times$ relative to state of the art solvers DAQP and HPIPM.

Let $\Delta_u(d,n)$ be the maximum graph diameter of a pointed $d$-dimensional polyhedron with $n$ facets. Using a curved-path encoding of the iterated Kalai–Kleitman recurrence, we prove, uniformly over $n\geq d\geq 4$, \[ \Delta_u(d,n) \leq (n-d)^{\log_2 G_d},\qquad G_d = (4\ln 2+o(1))\frac{d}{(\ln d)^2}, \] where the asymptotic expression for $G_d$ is understood as $d\to\infty$, improving the exponent of the previous best quasi-polynomial bound by an additional logarithmic factor. With the quantitative $d$-step reduction, we also obtain the complementary excess-based bound \[ \Delta_u(d,n) \leq (n-d)^{\frac{1}{2}\log_2(n-d)+O(1)}, \] where the implied constant is absolute. In the regime $n - d = \Theta(d)$, the latter bound is asymptotically stronger and halves the leading coefficient in the exponent. We further examine their behavior as $n$ grows relative to $d$, obtaining sharper exponents when $n = d^{1/\gamma+o(1)}$ for fixed $0<\gamma<1$ and an almost-linear bound in the deep-tail regime $(\ln n)/d\to\infty$. We also show that the leading term of the general bound is sharp within this positive path-counting framework.

Recovering probability measures from their moments has numerous applications, esp. in connection with the method of moments in statistics and optimization. In the setting where measure need not be finitely atomic, but its support is known to be compact and semialgebraic with codimension at least one, the problem is still open. We combine moment-matrix kernel information with the Christoffel–Darboux kernel to provide a discrete approximation of the support. To validate the proposed approach, we test our algorithm on analytically computed moments and pseudo-moments arising from polynomial optimization problems without unique global minimizers. This complements well-known recent work on recovery of measures with algebraic support, where the kernel of a moment matrix can reveal polynomials vanishing on the support, and on recovery of sufficiently regular full-dimensional supports, where estimators constructed by thresholding the Christoffel–Darboux kernel are known to converge asymptotically to the support.

We develop an accelerated stochastic method for convex $(H_0,H_1)$-smooth optimization under heavy-tailed noise. The unbiased oracle has a finite $p$-th noise moment, $1<p\le2$, with constant, gradient-dependent, and gap-dependent terms. Our method combines accelerated updates with clipping, projection, and phase restarts. Each iteration uses a single stochastic-gradient sample, without minibatching. We prove high-probability convergence to any desired accuracy despite clipping bias. The deterministic terms retain square-root dependence on both $H_0$ and $H_1$. In our three-component noise model, strong convexity leaves a polynomial accuracy cost only for the constant component; without it, the dependence is logarithmic even for $p<2$. More generally, for noise moments scaling as $(f(x)-f^\star)^\alpha$, the stochastic cost becomes logarithmic at $\alpha=p$ for convex objectives and $\alpha=p/2$ under strong convexity.

The local linear minimization oracle (Ferris and Zavriev, 1996; arXiv:2605.08850), or Local LMO, solves constrained convex problems without having to compute a projection: it minimizes a linear model over the intersection of the feasible set with a ball around the current iterate. We show that, whenever the ball radius does not exceed the Polyak radius, the Local LMO step is the Euclidean projection of the current iterate onto the intersection of the feasible set with a half-space that separates the iterate from the solution set; this projection is nonetheless computable by a linear oracle alone. We demonstrate that Local LMO belongs to a broader family of projection methods which may be indexed by the depth of the localizing half-space. For objectives with $\vartheta$-Hölder continuous gradient, every method from this family whose half-space lies sufficiently deep drives the best of its first $K$ iterates to optimality at the universal rate $\mathcal{O}(K^{-(1+\vartheta)/2})$, matching the non-accelerated universal gradient method of Nesterov (2015). Run at the Polyak radius, Local LMO attains the same rate when the constrained optima are also unconstrained ($\|\nabla f(x_\star)\| = 0$), and the rate $\mathcal{O}(K^{-1/(2-\vartheta)})$ otherwise.

We determine the exact worst-case function-value suboptimality after $N$ first-order oracle calls on $L$-smooth, $\mu$-strongly convex functions under a nonnegative weighted combination of the initial squared distance, function-value suboptimality, and squared gradient norm. Excluding the case where no strict improvement in function-value suboptimality can be guaranteed, the exact deterministic minimax risk in dimension $d\geq2N+1$ is characterized by the unique solution of a single scalar equation involving an $N$-step recurrence. The proof constructs an explicit hard instance for the lower bound and a constant-memory first-order method, ITEM-w, whose worst-case performance matches this lower bound. On the standard initial conditions, the result recovers the exact convex bound attained by the Optimized Gradient Method of Kim and Fessler, and establishes the minimax optimality of the recently introduced ITEM-f method of Kim, Ryu, and Das Gupta for the initial function-value condition.

The Forward-Forward Algorithm (FFA) replaces backpropagation (BP) with layer-wise local contrastive objectives, eliminating the backward pass and the need to retain intermediate activations, yet suffers a persistent performance gap with BP that worsens with depth. This paper diagnoses two structural failure modes: an optimization floor arising from concurrent local updates; and a geometric collapse of layer representations driven by the local update mechanism. On the optimization side, we prove that the FFA loss satisfies the Polyak–Lojasiewicz inequality at each layer; however, simultaneous layer updates induce inter-layer representation-distribution drift, so each layer optimizes against a moving input distribution and incurs an error floor. On the representational side, the pairwise similarity kernel of layer representations contracts exponentially toward rank one as depth increases, collapsing the diversity of per-layer error signals. This collapse bounds FFA's effective learning capacity, which measures the diversity of gradient information across layers, independently of depth, whereas BP's chain-rule signal preserves per-layer diversity, yielding a capacity that scales with depth.

Generalized smoothness assumptions have attracted growing attention in recent years, motivated in part by machine learning problems in which the gradient of the objective function may not be Lipschitz continuous. Among the most prominent of these is the $(\alpha, L_0, L_1)$-smoothness assumption. Existing methods that attain the best known complexity bounds require knowledge of, or upper bounds on, $\alpha$, $L_0$ and $L_1$, while the few parameter-agnostic methods available do not recover those bounds. In this paper, we show that trust-region methods attain the best known bounds for $(\alpha, L_0, L_1)$-smooth objective functions without prior knowledge of these parameters. Establishing these results requires new analytical tools, beyond those used in classical trust-region complexity analyses. We further show that our working model assumption allows the use of general model Hessian approximations and, in particular, accommodates limited-memory quasi-Newton updates. Finally, we show that our complexity bound is sharp in the nonconvex setting.

We study the tensor nuclear norm minimization with general linear constraints, a fundamental model for low-rank tensor optimization that includes many practical applications in computer vision, recommendation systems, revenue management, transportation research. Although the tensor nuclear norm is a convex surrogate for the tensor rank, optimizing it remains computationally challenging because evaluating the nuclear norm itself is NP-hard. We propose a semidefinite programming framework that exposes a finite structure hidden in this problem, even though it appears to possess infinitely many constraints. In particular, we show that the tensor nuclear norm admits an exact SDP representation supported by a finite number of points on the unit sphere. Based on this representation, we develop two complementary solution approaches. The first replaces the unknown support set by a finite sphere discretization which yields a solution with a provable worst-case approximation guarantee. The second selects spherical points adaptively, returns an exact optimum upon finite termination, and otherwise converges asymptotically to an optimal solution. Numerical experiments on synthetic tensor completion instances and real highway traffic data demonstrate that the adaptive method achieves competitive, and often substantially improved, recovery accuracy relative to existing methods.