Papers

Stochastic inverse variational inequalities (SIVIs) arise in applications in which equilibrium responses are observed under uncertainty, such as inverse road pricing and network equilibrium control. Existing methods typically rely on co-coercivity or strong monotonicity, while general monotone SIVIs remain less understood. We propose a regularized projection algorithm that combines Tikhonov regularization with increasing batch sizes. Under monotonicity and Lipschitz continuity, we prove almost sure boundedness of the iterates and almost sure convergence of their distance to the SIVI solution set. We further establish, to the best of our knowledge, the first explicit nonasymptotic rate of $O(T^{-1/2})$ for the expected squared residual under general monotonicity. This yields $O(\epsilon^{-2})$ iterations and $O(\epsilon^{-4-2\delta})$ stochastic oracle calls, for any $\delta>0$, to obtain an $\epsilon$-solution in expected squared residual. A deterministic variant attains the same iteration complexity using $O(\epsilon^{-2})$ exact operator evaluations. Numerical experiments illustrate the proposed methods on monotone SIVI problems.

We present two algorithms based on the Newton-Raphson method to calculate the floor of y**(1/m) for natural integer numbers y>2 and m >1. One could use such an algorithm to establish whether y is an integer power of an integer in number theory problems, even though binary search methods are traditionally considered simpler to implement.

We continue the study of the natural polytope $\mathcal{D}$ in $\mathbb{R}^{n+d}$ associated with the disjunction of a set of $n+1$ polytopes in $\mathbb{R}^d$, managed by $n$ binary variables. Already $\mathcal{D}$ had been characterized for arbitrary $n\geq 1$ and (i) $d\in\{1,2\}$, and (ii) for a broad generalization of hyper-rectangles. In both cases, the complete characterization employs full optimal big-M lifting. Here, we give a complete description of $\mathcal{D}$ for the case of $n=1$ and arbitrary $d$, when the (two) polytopes are arbitrary generalized cross polytopes. Furthermore, we characterize when our complete description employs only optimal big-M lifting. For $n>1$, we generalize the family of facet-describing inequalities used for $n=1$. Finally, we carry out some computational experiments demonstrating the value of our theoretical results.

The energetic response of inertial particles to turbulent flow motions is important for both a fundamental understanding of the multi-phase dynamics at play, and for applications such as reduced-order models of particle dispersion processes, and their two-way coupled effects onto the flow phase. Numerous studies focus on the energetics of ensembles of particles in homogeneous isotropic turbulence, where the influence of flow anisotropy (such as that provided by boundary conditions, or other external forcing) is not considered a priori. In this work, we investigate the role of flow anisotropy on the Eulerian scale-wise particle phase energetics in a turbulent wall bounded flow for settling inertial Lagrangian particles. By using coupled Eulerian-Lagrangian direct numerical simulations at moderate Reynolds number, we aim to unravel the complex dependency of the scale-wise particle energetics on the turbulence intensity, particle inertia, and particle settling. In particular, we focus on how the developing anisotropy of the underlying turbulent flow (derived from the presence of the wall) is donated to the particle phase, and how particle inertia and settling preserve this large scale anisotropy into the formally isotropic scale range of the flow. We derive an exact (but unclosed) conservation law for the particle phase energetics at arbitrary scale, and use an asymptotic argument to help elucidate our DNS data. We discuss the relative changes to the quasi-streamwise and vertical components of the fluctuating particle field, and finish by discussing the implications of anisotropic non-local effects for more general flows, and the implications for continuum models of inertial settling Lagrangian particles.

We establish a one-dimensional Pólya-Szegő principle for the Riesz $(p,\alpha)$-variation $\mathcal{V}_p^\alpha$, a family of functionals that interpolates between Wiener $p$-variation and the Sobolev seminorm $\|f'\|_{L^p}$. More precisely, for every measurable function $f$, we prove that its non-increasing rearrangement $f^*$ satisfies $$ \mathcal{V}_p^\alpha(f^*)\le\mathcal{V}_p^\alpha(f) $$ for all $1\le p<\infty$ and $0\le\alpha\le 1-1/p$. This result contains and extends the classical variation-diminishing property of rearrangements from Sobolev spaces to a scale of spaces admitting fractional smoothness and, in particular, applies to functions that are nowhere differentiable. Our methods also yield a sharp rearrangement inequality for a modulus of continuity defined via $p$-variation, providing an analogue of a problem posed by Ul'yanov. Finally, we sketch how our inequality can be useful in the study of nonlinear Fredholm integral equations.

We obtain results on cut and choose games for complete Boolean algebras. Zapletal proved that there is a Boolean algebra $\mathbb{B}$ such that $\mathcal{G}_\omega^\textsf{candc}(\mathbb{B})$, the version of the game which ends on the $\omega$'th round, is undetermined. We prove that, assuming the consistency of a proper class of supercompact cardinals, the limit version $\mathcal{G}^{\textsf{candc}}_{< \lambda}(\mathbb{B})$, in which there are $\lambda$-many rounds but no concluding round, is consistently determined for all complete Boolean algebras $\mathbb{B}$ and all successor cardinals $\lambda$. In particular, this answers a question of Zapletal \cite[Question 2]{Zapletal1995}. We also show that undetermined instances of the game $\mathcal{G}^\textsf{candc}_\lambda(\mathbb{B})$ follow from the approachability property, extending results of Dobrinen, and we prove that undetermined instances are compatible with $\textsf{MM}^{++}$.

Physics-informed neural networks are often evaluated by residual losses sampled at finitely many points, which do not by themselves certify pointwise values of a partial differential equation solution. In this work, deterministic pointwise error intervals are developed for mesh-based, piecewise-linear numerical physics-informed neural networks. The proposed error estimation is given for a compatible field, which is the finite-element reconstruction of an admissible prediction on a mesh. The certifying residual is then obtained by applying the finite-dimensional numerical system to this compatible field. For compatible square linear systems, the pointwise error relative to the discrete target has an exact adjoint Green representation, and the computed signed error recovers the finite element solution exactly. Norm-based, inexact, localized, and randomized variants provide computable intervals when the exact correction computation is impractical. The extension from the discrete target to the continuous solution is supplied by comparison estimates. For a one-dimensional coercive reaction-diffusion class, this transfer layer is made fully computable by an explicit residual-based a posteriori estimator with querywise constants. The error bound derivation is extended to nonlinear residual systems with explicit Taylor remainders. Numerical experiments assess compatibility and calibration on manufactured examples, on a large-scale public three-dimensional elasticity benchmark, and on projected neural load families on the same benchmark.

We present a geometric multigrid preconditioner for high-order continuous finite elements that operates entirely on redundant, cell-wise stored vectors: the assembled global vector is never formed on any level of the hierarchy. In this storage paradigm the machinery that classically complicates adaptive multigrid dissolves. Hanging-node constraints are never assembled: we prove that the plain tensor-product transfer operators, applied to the unassembled residual, algebraically reproduce the classical constrained restriction, including the action of the transposed constraint matrix, and the edge operators of local smoothing reduce to a pointwise masking of the residual, with no splitting of the level operator into interior and edge blocks. As a consequence, the single inter-cell primitive of the whole V-cycle can use a topologically oblivious structured kernel even on adaptively refined meshes. We prove that the resulting cell-wise V-cycle is equivalent, iterate by iterate, to the classical local multigrid method, and therefore inherits its convergence theory. Numerical experiments for the Laplace operator confirm grid-independent convergence that is essentially unaffected by local refinement; on a single GPU, using nothing more than a masked point-Jacobi smoother, the solver sustains up to $1.1$\, GDoF/s per V-cycle in double precision and reaches end-to-end solve throughput on par with patch-smoother-based solvers.

This paper presents a cooperative distributed model predictive control (MPC) scheme for nonlinear continuous-time systems. The centralized optimal control problem is solved asynchronously via a fixed number of sensitivity-based distributed programming (SBDP) iterations. The proposed scheme requires only neighbor-to-neighbor communication and no synchronization between agents during optimization. Under nominal MPC stability and bounded information delay, local exponential stability is established for a sufficiently large number of per-agent SBDP iterations. Numerical and hardware-in-the-loop results on both Ethernet and Wi-Fi demonstrate the benefits of an asynchronous execution, reducing execution times by over 60% while maintaining comparable closed loop performance.

We study irreducible polynomials admitting a single real root in any real closed field extension of the base field, called monoreal polynomials. We show some stability properties satisfied by the induced monoreal field extensions, and define the monoreal closure of a field. We make the link with a notion of real radiciality for ring extensions, and an injectivity property at the real spectrum level. We end with a geometric application, showing that injectivity implies surjectivity for the real spectrum mapping, under certain assumptions.

Unmanned underwater vehicles (UUV) integrate ocean forecasts with path planning algorithms in order to identify energy- or time-minimizing paths that enable mission completion. Typically, a well-defined deterministic ocean forecast is assumed to be available for path planning; however, in practice, different ocean forecasts can disagree. In this paper, we extend previous work on deterministic optimal path planning to identify optimal mean-time paths when presented with an ensemble of possible ocean forecasts. In particular, we formulate a system of time-independent Hamilton-Jacobi partial differential equations that incorporates forecast uncertainty and yields the optimal mean reachability travel time and the necessary controls to find the associated optimal path. An efficient numerical solution of this system of PDEs is obtained through an extension of the Fast Sweeping Method; verification and benchmarking results are provided. Additional numerical examples illustrate the impact uncertainty can have on the optimal path; in particular, these results demonstrate that the vehicle's optimal path can deviate significantly from the deterministic optimal paths associated with the individual ensemble members.

We show that the existence of an oddomorphism from a graph $F$ to a graph $G$ does not imply that $G$ is a minor of $F$. This answers a question posed by Roberson (2022) and shows that the CFI graphs cannot be used to prove the Strong Roberson Conjecture. Additionally, we introduce the concept of a split-off minor and show that the existence of an oddomorphism from $F$ to $G$ implies that $G$ is a split-off minor of $F$. Consequently, every class that is closed under taking split-off minors and disjoint unions is homomorphism distinguishing closed. The split-off minor relation is the first minor-like structural relation shown to have this property, marking a meaningful advancement in our understanding of the interaction between structural graph containment and homomorphism indistinguishability relations.

Let $X$ be a complex normed space. A map $P: X \rightarrow X$ is called idempotent if $P^2 = P$. A collection $\mathcal{C} = \{P_1, P_2\}$ of nonzero distinct orthogonal ($P_1P_2 = P_2P_1 = 0$) idempotent maps on $X$ is said to be a family of generalized bi-circular idempotents if there exist distinct unit modulus complex numbers $\lambda_1, \lambda_2$ such that $P_1 + P_2 = I$ (identity operator on $X$) and $\lambda_1P_1 + \lambda_2P_2$ is a surjective isometry on $X$. This generalizes the notion of generalized bi-circular projections on Banach spaces introduced by Fošner, Ilišević and Li \cite{MDC} to nonlinear maps. In this paper, we describe the structure of generalized bi-circular idempotents over the space of analytic functions on the open unit disk with bounded derivatives.

We study how sketching techniques from randomized numerical linear algebra can be incorporated into the dynamical low-rank approximation (DLRA) of large-scale matrix differential equations. A natural approach is to sketch the Galerkin condition that defines the DLRA, which leads to an oblique tangent space projection. We show that this oblique projection approximately reproduces the standard DLRA only under restrictive conditions on the vector field, and that it fails on problems with a large perpendicular residual. As an alternative, we propose an orthogonal sketch DLRA that evolves sketch-orthogonal bases while using standard orthogonal projections for the dynamics. This approach preserves the geometric structure of the classical DLRA and is numerically stable. The computational advantage of randomized Gram–Schmidt over Householder QR lies in fewer global synchronizations on a row-distributed basis, at a comparable flop count; when the basis is well conditioned, randomized Gram–Schmidt can be replaced by randomized Cholesky QR, which additionally shifts the basis update from BLAS-2 to BLAS-3 kernels, making it well-suited to modern accelerators. We derive sketch versions of the projector-splitting and BUG integrators, and demonstrate the approach on the Allen–Cahn, Fokker–Planck, and Vlasov–Poisson equations.

We study the power and limitations of subset selection in statistical estimation through the framework of super-teaching, where a teacher selects a subset of i.i.d. data to optimize a learner's estimator. Unlike prior work focused on specific distributions or fixed subset sizes, we develop a general theory under minimal assumptions. For mean estimation, we prove that super-teaching is possible for any distribution whose density is bounded away from zero in some neighborhood of the mean, allowing subset sizes growing as $k = o(n^{1/3})$ and achieving error on the order of roughly $k!/n^{k}$. This significantly extends existing results on admissible distributions and subset scaling. We also extend the analysis to parameters expressed as smooth functionals of expectations, such as variance and scale parameters in classical parametric families, including settings with heavy tails. Moreover, we show that super-teaching can greatly improve estimation rates for nonlinear estimators like the sample median, achieving rates beyond classical asymptotics. Through examples, including cases where maximum likelihood estimators are inconsistent or fail to be asymptotically normal, we demonstrate that super-teaching can succeed even when standard statistical guarantees break down. Our results establish a unified theory of data selection to enhance statistical efficiency.

Integrated sensing and communications (ISAC), empowered by dynamic metasurface antennas (DMAs), has emerged as a promising paradigm for next-generation wireless networks. However, existing DMA-based designs commonly rely on the frequency-flat response model for DMA elements, which is accurate only in narrowband scenarios and can cause significant phase and magnitude mismatches in wideband and ultra-wideband systems. This paper investigates a DMA-based wideband ISAC system under a frequency-selective Lorentzian response model, which accurately captures the frequency-dependent behavior of DMA elements. We aim to jointly balance the aggregate signal-to-interference-plus-noise ratio (SINR) of communication users and the signal-to-noise ratio (SNR) of the radar target. To this end, we first develop an alternating optimization framework based on projected gradient ascent (PGA), deriving closed-form gradients of the objective function with respect to the digital beamforming vectors, resonance frequencies, and damping factors under the frequency-selective Lorentzian DMA model. We then propose an unfolded PGA architecture that preserves the interpretability of model-based optimization while learning key hyperparameters to accelerate convergence. Simulation results show that the frequency-selective Lorentzian model improves performance by approximately 20% over its frequency-flat approximation. Moreover, deep-unfolded PGA achieves up to 20-fold faster convergence and improves the objective value by up to 7% compared with PGA-based benchmarks.

The Laplacian conjecture of Brouwer asserts that for any graph \(G\) of order n with \(m\) edges, the sum of the \(k\) largest Laplacian eigenvalues satisfies \(s_k(G) \le m + \binom{k+1}{2}\) for $k=1, \ldots, n$. Later, Li and Guo in 2022 further proposed the full Brouwer's Laplacian spectrum conjecture. Recently, Kothari and Tudose in 2026 proved the Brouwer's conjecture. Motivated by their perfect proof and methods, we proved that for a simple graph of order $n$ with $m$ edges and $1\le k\le n-1$, \(s_k(G) = m + \binom{k+1}{2}\) if and only if $G$ is a threshold graph with clique number \(k+1\), which confirms the full Brouwer conjecture proposed by Li and Guo.

We show a sum-product phenomenon in fields equipped with abstract dimension theories, which simultaneously generalizes the dimensions in geometric theories and Hrushovski's coarse pseudo-finite dimensions. More precisely, we show that for type-definable sets of positive non-zero dimension, non-expansion in dimension of both the sumset and product set implies the existence of a definable field in the same dimension. Main ingredients of the proof include dimensional analogues of the Ruzsa triangle inequality and the Plünnecke-Ruzsa inequality.

We obtain the unique asymptotics of $SO(k)\times SO(n-k+1)$-invariant, compact, simply-connected, {factorwisely non-self-similar} $n$-dimensional $\kappa$-solutions of the Ricci flow $(M^n, g(t))$, where $n\geq 4$ and $2\leq k\leq n-2$. More precisely, these $\kappa$-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere $S^n$, having a positive curvature operator metric $g(t)$ and a cylindrical tangent flow at $-\infty$, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric $g(t)$ of every $SO(k)\times SO(n-k+1)$-invariant ancient oval is represented in the form $g(t)=dz\otimes dz + F^2(z,t) g_{S^{k-1}} + G^2(z,t)g_{S^{n-k}}$ (up to flipping $k-1$ and $n-k$). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function $G(z, t)$, and prove that the uniqueness of $G(z, t)$ implies the uniqueness of $F(z, t)$. In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional $\kappa$-solutions of the Ricci flow.

We determine when the bigness of the tangent bundle is preserved under point blow-ups of a ruled surface $S=\mathbb{P}_C(\mathcal{E})$, giving a sharp bound on the number of blow-up points in terms of the first Segre invariant $s_1(\mathcal{E})$.

In this article, we study the weighted homogeneous Sobolev–Morrey spaces on domains in $\mathbb{R}^n$ with higher co-dimensional boundaries. Precisely, we systematically establish a real-variable theory of these spaces, including completeness, embedding theorems, Riesz potential characterizations, continuity, trace and extension theorems, and complex interpolation. Applying the boundedness of the trace and the extension operators, we obtain sharp weighted a priori estimates for solutions to the Dirichlet problem of divergence-form degenerate second-order elliptic equations on such domains in weighted Lebesgue spaces. The absence of a boundary manifold structure of these domains poses some essential difficulties, which are overcome by using some tools, such as the intrinsic properties of distance weights and the geometric structure of domains, different from those available in Lipschitz domains.

We consider the poisson problem $\wt{\mc{L}} u = E$ for the operator $\wt{\mc{L}} = \Delta_{\mb{R}^2} + 2\cosh^{-2}(s)$ on domains of the form $\Lambda : = \{ (s, z): |s| \leq \ell(z) \}$, where the width $\ell(z)$ of the domain $\Lambda$ varies with $z$. We prove the existence of solutions satisfying weighted estimates when the source term satisfies certain natural orthogonality conditions, and when $e^{-\ell}$ is small and slowly varying.

The Richelot $(2,2)$-step is the standard step of explicit genus-2 isogeny computation. We determine the exact stratification of its input space over a finite field $\mathbb{F}_q$ of odd characteristic: ordered factorizations $f=uvw$ of a square-free sextic into monic quadratics fall into three strata by the geometric type of the quotient, governed by the incidence geometry of the discriminant locus. This yields closed formulas for each stratum and, modulo affine coordinate changes, mass formulas of degree four in $q$ with a complete classification of stabilisers. The classification is decided by data the step already computes, at $5\mathrm{M}+6\mathrm{S}$ beyond the brackets, and an output post-check is provably redundant. On the decomposable stratum the square class of one resultant determines the field of definition of the elliptic factors and the shape of the Weil polynomial of the Jacobian; the two cases are counted exactly, and in the nonsplit case the curve $y^2=f$ has $q+1$ rational points. Exhaustive enumeration over small finite fields verifies every proved count.

We investigate the existence, uniqueness, and regularity of the gradient flow of the Ambrosio-Tortorelli functional, viewed as an initial-boundary value problem, in arbitrary dimension. For any initial data, using a time-discrete Euler scheme, we establish the existence of a weak gradient flow satisfying a maximum principle. We also identify a functional space in which uniqueness holds. We further show that such gradient flow is smooth in the interior of the space-time domain. Under additional assumptions on the initial data, the regularity of the boundary of the domain, we prove optimal regularity for the solution, up to the space-time boundary.

In this paper we consider the Stefan problem endowed with a doubly nonlinear phase relaxation. More precisely we assume that the rate of convergence of the phase function depends on the temperature via an increasing continuous function, and it depends on the phase via a maximal antimonotone graph. We prove the existence and uniqueness of the solution of the relaxed problem, and we perform the asymptotic analysis toward the Stefan problem as the relaxation parameter approaches zero.

Kernel Stein Discrepancy (KSD) compares a sample to a fixed target distribution known only through its score, and is widely used for goodness-of-fit testing, sample quality assessment, and approximate inference. We study the estimation of $\operatorname{KSD}(P_0,P)$ from $n$ independent observations and identify the sharp spectral constant governing the minimax risk: it is the Hilbert-Schmidt norm of the Stein covariance operator $C_\star$, giving the minimax scale $\sqrt{\|C_\star\|_{\mathrm{HS}}/n}$. This scale is attained by the positive-part square-root U-statistic, whereas the standard plug-in V-statistic remains at the trace scale $\sqrt{\operatorname{tr}(C_\star)/n}$ and is therefore suboptimal by the fourth root of the effective rank of $C_\star$; for a Gaussian target with a fixed-bandwidth Gaussian kernel this factor is exponential in the dimension.

The Elekes-Szabó theorem establishes that an irreducible algebraic hypersurface $Z(F)$ contains few grid points unless it exhibits a specific group-related structure. Identifying this structure from the polynomial $F$ is a challenging problem in combinatorial geometry. Our first main result (Theorem 2.3) provides a local geometric criterion to detect such group-related hypersurfaces. By applying this criterion, we develop a geometric framework for boundary varieties, which are defined by the vanishing of partial derivatives along $Z(F)$. In Theorem 2.4, we show that for group-related varieties, these boundary varieties must be contained in coordinate slices. This gives a strict geometric constraint on the loci where $Z(F)$ becomes tangent to coordinate directions. As an application, we study configurations formed by $d$ coordinate-grid hyperplane families together with a one-parameter polynomial family of hyperspheres in $\mathbb{R}^d$. If one chooses $n$ members from each of these $d+1$ families and obtains $\Omega(n^{d-\eta})$ common incidence points, then the hypersphere family is forced to have a very restricted form: it is concentric in dimensions $d \geq 3$, and in dimension $2$ it is either concentric or consists of fixed-radius circles whose centres lie on a line parallel to a coordinate axis. We also generalize the pinned distance problem initiated by Elekes and Szabó for three points in the plane to $d+1$ points in $\mathbb{R}^d$. More precisely, in Theorem 2.8 we prove that if $d+1$ families of hyperspheres centred at fixed points determine $\Omega(n^{d-\eta})$ points, each lying on one hypersphere from each family, then the centres must be affinely dependent.

Optimal power flow (OPF) solved by an interior-point method (IPM) requires repeatedly solving Newton linear systems. When variational quantum linear solvers (VQLS) are used, each IPM iteration involves an additional nested inner variational optimization loop, which can significantly slow the overall quantum-assisted IPM convergence. To address this challenge, this paper proposes a dual-level trainable quantum IPM framework for OPF that leverages early solver-generated trajectories rather than relying on single-point prediction. The key observation is that early IPM iterates provide informative primal-dual, slack, and barrier-variable evolution about the path to optimality, while early VQLS parameter updates provide useful information about the later variational search. At the quantum-solver level, a trainable parameter model uses a short prefix of the VQLS parameter trajectory to project the remaining variational search toward a lower-cost region. At the OPF-solver level, a second trainable model uses early primal-dual IPM iterates to project a later central path state, which is restored to an admissible point before IPM refinement continues. Simulation studies show that the proposed approach reduces the number of variational updates by up to $95\%$ while maintaining OPF objective values close to the classical IPM reference. A 2-bus demonstration on real quantum hardware is also included to validate the implementation of the proposed workflow.

This work establishes a simultaneous Smoluchowski-Kramers approximation and Khasminskii averaging principle for a class of second-order stochastic differential equations (SDEs) in nonequilibrium random environments. The system describes the motion of a particle of mass $m>0$ subject to external forces, friction, and noise, all of which depend on a fluctuating environment such as a stochastic heat bath. The environment is modeled by a fast-varying first-order SDE, where a parameter $0<\epsilon \ll 1$ encodes the time-scale separation. Under the scaling $m=\epsilon^2$, the slow process converges in probability to an effective dynamics with averaged drift and noise-induced coefficients. Our analysis utilizes a pathwise integration-by-parts formula and Poisson equations associated with the fast dynamics. Finally, numerical experiments are provided for demonstration.

Dynamic congestion pricing is an important tool for managing congestion and coordinating distributed energy resources in active distribution networks. However, scalable mechanisms that preserve participant autonomy remain computationally challenging because the operator-resource interaction is naturally bilevel. This paper develops a convex-analytic framework in which a distribution system operator computes dynamic congestion-price adders, while decentralized energy hubs schedule flexible demand, storage, local generation, renewable curtailment, and grid import/export. Unlike conventional single-level reformulations that replace lower-level problems by Karush-Kuhn-Tucker (KKT) conditions, complementarity constraints, and big-M linearizations, the proposed model represents follower feasibility and optimality through a Fenchel-Young equality involving the convex conjugate of an extended follower objective. The remaining bilinear price-response term is handled through a penalized difference-of-convex reformulation and sequential convex approximation. The method solves continuous convex subproblems and avoids the constraint-wise complementarity and branch-and-bound scaling of mixed-integer KKT reformulations; its main computational drivers are price-response dimension and conjugate evaluation rather than binary encodings of follower inequalities. On augmented IEEE 13- and 34-node feeders, it reduces congestion by 96.89% and 96.45%, respectively, approaches centralized full-information dispatch, certifies price-response consistency to numerical precision, and yields lower residual congestion than time-limited KKT incumbents within the computational budget.