Papers

This paper proposes a preconditioned ordinary hyperbolic smoothing modified L-BFGS framework for finite minimax problems. To mitigate the deterioration of Euclidean conditioning as the smoothing parameter decreases, a problem-derived hyperbolic majorization preconditioner is incorporated into the initial L-BFGS metric; it combines component-gradient Lipschitz curvature with the curvature induced by the hyperbolic smoothing Jacobian, is uniformly positive definite, and provably majorizes the Hessian of the smoothed objective. To exploit strict negative secant curvature, three correction strategies are introduced—direct sign reflection, a Euclidean nearest-point correction, and a $B_k^{-1}$-metric nearest-point correction—for which explicit formulas are derived and Clarke-stationary accumulation points are obtained for the idealized continuation. For a fixed smoothing parameter, Q-linear convergence of the HMLBFGS objective values is obtained under a Polyak–Lojasiewicz condition, local strong convexity further yields R-linear convergence of the iterates, and the corresponding full-memory modified BFGS methods attain local Q-superlinear convergence under the usual unit-step assumptions. Numerical experiments demonstrate the effectiveness of the proposed preconditioned methods and their numerical advantages over the selected comparison methods.

Three classical problems-linear assignment, graph matching, and optimal transport-share a common structure: they seek an optimal matrix-valued coupling on a matrix polytope. We study regularized coupling maps that encompass two families of regularization: entropy/KL regularization, which exponentiates the input score matrix and projects it onto the relevant matrix polytope via the KL-projection; and the Frobenius/Euclidean regularization, which performs the Euclidean projection of the scaled input matrix onto the same polytope. We trace how these two geometries yield complementary algorithmic tools for graph matching (CSGO, ASM, FRAM) and optimal transport (IP-EOT, PSN). The regularized coupling map serves as a direction generator in graph matching and as a solution representation in optimal transport, with the regularizer determining the geometry and the outer task determining the precision standard.

We study the value of information in predicting the evolving state of a finite Markov chain. At each stage, a decision maker chooses a state and observes only whether the current state of the chain matches her choice; the resulting information is used to make a prediction on the state at the final stage of the problem. We show that, when the chain starts from an invariant distribution, the optimal terminal value is non-decreasing with the number of observations and converges at a uniform exponential rate. We introduce the predictive learning index, which measures whether all attainable informational value is extracted after finitely many observations, and show that both finite and infinite indices may occur. In contrast, for nonstationary initial distributions, the value may strictly decrease with the horizon.

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cos\theta)^3$ where $e$ is the relative eccentricity of the primaries and $\theta$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

Optimal Control Problems (OCPs) have been applied to a variety of applications. Such problems are often transcribed into a Nonlinear Program (NLP) for which solution methods typically solve a Linear Quadratic (LQ) subproblem. To efficiently solve this LQ subproblem, the Riccati recursion exploits the block-diagonal structure that arises from discretizing an OCP using a Multiple Shooting method. However, reformulating general dynamics constraints to fit the required problem structure to use the Riccati recursion introduces additional micro-structure within these block matrices that remains unexploited. This work demonstrates how structure-exploiting solvers such as Fatrop can benefit from exploiting not only the block-diagonal structure but also the micro-structure within those blocks. We propose a structure-exploiting LU decomposition algorithm and integrate it within the Riccati recursion. Numerical results on randomized linear systems and two OCP examples demonstrate a reduction in computation time of the search direction of up to 20%, with the largest speedup observed for relative zero block sizes around 0.25 and large stage-wise equality constraint jacobians.

Risk-averse design optimization with conditional value-at-risk (CVaR) constraints requires accurate tail estimates, but repeated high-fidelity (HF) evaluations are costly. We propose a multifidelity (MF) tail-region correction method that uses dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) as a global surrogate and directs a limited HF budget to responses that most strongly influence CVaR. Covariance of the estimated DD-GPCE coefficients quantifies finite-sample prediction uncertainty and defines a confidence-interval-based tail region. Within this region, a two-stage strategy explores locations with high prediction uncertainty and then exploits locations with large correction-induced shifts. A Tikhonov-regularized residual expansion using the same polynomial basis yields a unified tail-corrected surrogate. For a ten-bar truss, the method produced a design close to the crude-MCS reference with approximately 72 to 274 times fewer HF evaluations than the considered MF importance sampling configurations. For a suction valve, it reduced mass by 1.05% and bending and bulging CVaR by 17.68% and 21.53%, respectively, while satisfying the valve-opening requirement. The results demonstrate accurate, HF-sample-efficient CVaR-constrained design through localized MF correction.

In a real Hilbert space, we study a bilevel optimization problem that consists in minimizing an outer convex function over the zero set of a maximally monotone operator. In the smooth setting, where the outer objective is convex and Fréchet differentiable and the inner operator is single-valued, continuous and monotone, we associate with the problem a first-order dynamical system that can be viewed as a monotone flow applied to a dynamically regularized operator. Under a suitable Attouch-Czarnecki-type condition for the inner level, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the trajectories to optimal solutions of the bilevel problem. In the smooth+nonsmooth setting, we enrich the outer objective with a proper, convex, and lower semicontinuous function, while the inner operator is augmented by the subdifferential of a function with the same properties. We propose a regularized proximal-extragradient algorithm in which both the forward and backward steps are performed with respect to dynamically regularized operators and functions, respectively. Under an analogous Attouch-Czarnecki-type condition formulated for the smooth+nonsmooth case, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the iterates to optimal solutions of the bilevel problem.

We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss. Since the potentials are only unique up to additive constants, we measure the error using the quotient supremum norm, defined as $d_\infty([u],[v]) = \inf_{a\in\mathbb{R}}\|u-v-a\|_\infty$. For a fixed regularization parameter $\varepsilon>0$, we establish a non-asymptotic statistical rate of $n^{-1/2}$. This is achieved by combining the Birkhoff-Hopf contraction theorem with entropy bounds on normalized kernel sections. However, the constant in this bound grows exponentially with $1/\epsilon$. To improve this, we isolate geometric conditions under which the empirical estimator maintains the $n^{-1/2}$ rate but features polynomial dependence on $1/\varepsilon$. The key requirement is a polynomial residual-stability estimate for the population Sinkhorn map. We provide sufficient criteria for this, including a polynomial contraction property and a local inverse estimate. Furthermore, we introduce two rigorously verifiable model classes an $\varepsilon$-weak residual-interaction class obtained after separable centering and another based on connected tight-edge graphs for fixed discrete costs where the polynomial rate is guaranteed without relying on abstract resolvent assumptions. Finally, we establish matching minimax lower bounds demonstrating that the $\varepsilon n^{-1/2}$ rate cannot be uniformly improved in the bounded-interaction regime.

We consider a resource allocation problem with strategic agents that have private stochastic satisfaction functions and local constraints. To achieve a global optimal solution, we propose an incentive mechanism that induces a game among the agents. For the payment function of the mechanism, we construct a family of quadratic functions using the linear matrix inequality (LMI) approach that implements the social welfare maximizing outcome on the unique Nash equilibrium (NE) of the induced game while ensuring budget balance and individual rationality. Moreover, we propose a decentralized variable sample-size proximal best-response (VS-PBR) algorithm with Krasnoselskij iteration where only aggregate information is available to the agents. The algorithm is dynamically stable, as it is proven to converge in the mean-square sense to the NE of the game. The efficiency of the mechanism is then investigated on the Sioux Falls City transportation network, where electric vehicle (EV) users jointly select their destination and route.

This paper analyses the Stochastic Team Orienteering Problem (STOP), a stochastic variant of the Team Orienteering Problem (TOP). In the STOP, travel times are represented by random variables. The objective is to determine a set of routes that maximises the expected collected profit. We model the problem as a two-stage stochastic integer program and propose an exact solution method based on the Integer L-shaped method. We also consider a set of non-linear chance constraints to restrict the search to highly reliable routes. In the first-stage, a set of routes is selected, while in the second-stage the recourse evaluates the expected profit of the selected routes. We present a new set of optimality cuts and strengthen the formulation by considering valid inequalities and constraints liftings. Computational results provide a comprehensive analysis of the algorithm using the standard TOP benchmark dataset from the literature.

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $\Gamma(\sigma)=e^{\kappa\sigma}\omega(\sigma)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $\epsilon>0$, an algorithm terminates with a rational interval of width at most $\epsilon$ containing $C_2^*$. Additional results—sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria—are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $\delta<\delta_{\mathrm{sharp}}(k/r_{\star})$, where $\delta_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $\delta_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}\lambda$ for all $\lambda\gtrsim\|\mathcal{A}^{*}(\xi)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $\delta<\delta_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

Convex-concave saddle-point problems are ubiquitous across diverse domains, including machine learning, image processing, economics, and equilibrium problems. Primal-dual algorithms provide a highly effective and powerful framework for convex-concave saddle-point problems. Convex combination has become a crucial acceleration technique for primal-dual algorithms, and the integration of this technique has recently made these algorithms a highly active research topic. The choice of the convex combination parameter often has a significant impact on both the theoretical analysis and the numerical performance of the corresponding algorithms. However, the requirements on this parameter imposed by theory are sometimes inconsistent with those suggested by numerical experiments. For instance, theoretical analysis often requires the parameter to be small, while numerical experiments tend sometimes to favor larger values. To address this inconsistency and further advance primal-dual algorithms with convex combination, we develop a novel strategy based on two convex combinations, integrate it into a primal-dual framework, and propose a new primal-dual algorithm with linesearch, termed NPDAL-n, for general convex-concave saddle-point problems. The proposed two convex combinations in NPDAL-n ensure that the permissible range of the convex combination parameters is mainly determined by theoretical considerations, with little regard for numerical performance. Through rigorous Lyapunov energy descent analysis, we establish the global convergence and a sublinear ergodic convergence rate of $\mathcal{O}(1/N)$ for NPDAL-n under standard assumptions. When the primal function is strongly convex, we develop an accelerated version of NPDAL-n that achieves an optimal $\mathcal{O}(1/N^2)$ rate.

Piecewise-linear (PWL) optimization problems arise in many mixed-integer programming (MIP) optimization applications, including portfolio optimization, workforce scheduling, and resource allocation. But solving them to global optimality remains computationally expensive because branch-and-bound repeatedly solves LP relaxation subproblems. Existing solvers are largely CPU-centric, leaving the scalability of modern GPUs underutilized. Few prior GPU-accelerated branch-and-bound either targets neural network which is not suitable for general PWL optimization, or accelerates only auxiliary subroutines such as strong branching heuristics within CPU-centric MIP solvers. To bridge this gap, we propose B$^3$-PWL, a GPU-centric batched branch-and-bound framework for piecewise-linear optimization with Special Ordered Set of type 2 (SOS2) constraints. Our method solves batches of LP relaxation subproblems concurrently on the GPU using a first-order primal-dual solver, enabled by a specialized batched block-tiled sparse matrix kernel. To complement bound computation, we further introduce a unified feasibility search module that combines an SOS2 repair primal heuristic with a batched feasibility pump to rapidly obtain feasible incumbents and improve pruning efficiency. On a benchmark of 43 PWL-MIP instances, B$^3$-PWL achieves a 9.25x geometric-mean speedup over NVIDIA cuOpt while reaching high-quality feasible incumbents on every tested instance. On a public valve-point unit-commitment benchmark, it further outperforms NVIDIA cuOpt and the open-source CPU solvers SCIP and HiGHS, demonstrating the potential of first-order LP methods as the central engine of GPU-accelerated branch-and-bound.

We develop a few calculus rules to compute or lower bound the facial distance of a polytope. We illustrate our calculus rules on various popular polytopes. In particular, we provide a provide a lower bound on the facial distance of the Birkhoff polytope.

This paper studies the 2D Navier–Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochastic heat equation. We prove exponential mixing for the associated Navier–Stokes–heat system, and consequently exponential convergence of the velocity law under stationary forcing. The main difficulty is that the white noise is confined to a subdomain and reaches the velocity indirectly through the heat component. To address the degeneracy, the proof combines Malliavin calculus with PDE control theory. The key ingredient is a stabilization scheme that converts localized Navier–Stokes controls into time-regular controls compatible with the heat dynamics.

Many open systems are observed only through aggregate snapshots, while the rule that moves objects between states stays hidden. This paper studies the inverse problem of recovering the transfer operator of a multicluster open system from two adjacent snapshots. The consistent operators fill an affine manifold: the data fix the aggregate flows and leave the internal routing free, so every recovery method commits to a prior that selects one operator. Tikhonov regularisation with a zero reference gives the uniform outflow on each source simplex, which underestimates retention in an inertial system. We propose least action on the operator trajectory, under which the rule changes as little as the data permit. In discrete time this is exact as a probability statement: the action prices a trajectory by its likelihood under a Gaussian reference drawn toward full retention, so least action selects the most probable trajectory consistent with the balance. A causal construction keeps the balance exact by moving the operator only inside the null space of the constraints, and we relate it to optimal transport, to Perron-Frobenius coarse-graining, and to the Schrödinger bridge. Two instantiations carry it: the Russian labour market and a radiation-damped relativistic electron ensemble. The construction holds the balance to machine precision where temporal smoothing corrupts it by twelve percent on the labour series and sixty percent in the physical one. The spectral gap is non-identifiable from aggregate snapshots but available from micro-trajectories, where Ulam's method reproduces the Landau-Lifshitz rate to within six percent. On the short labour series no method beats the naive forecast in point accuracy at conventional significance levels. The value is structural: an interpretable operator, an exactly preserved balance, and an explicit information limit of aggregate observation.

Differentiating through parametric optimization problems is central to bilevel programming and meta-learning, often accomplished using approximate implicit differentiation. The implicit function theorem requires inverting a partial Jacobian of the optimality condition, which fails when there are many solutions. Nonetheless, in such cases it is possible to relax invertibility to a strictly weaker uniform range condition, under which it is shown that the minimal-norm solution mapping admits generalized derivatives by using a limiting Tikhonov regularization argument and conservative set-valued field theory. With additional control on the eigenvalues of the generalized Hessians, a pseudoinverse formula is justified. This is established for a class of smooth convex objectives and extended to nonsmooth composite problems. These assumptions are verified for Least-Squares, Huber regression and LASSO. The resulting extension of nonsmooth implicit differentiation to ill-posed settings is examined experimentally on data poisoning and data hypercleaning problems.

Capacity expansion modeling plays a critical role in optimizing the deployment of new generation, storage, and transmission, typically at national and regional levels. To support long-term planning, these models consider a large set of energy technologies and policies, along with decades of weather and demand data. Realistic capacity expansion models thus become high-dimensional optimization problems, with hundreds of millions of variables and constraints, which are challenging to solve. A common strategy to address this complexity is to omit non-linear, non-convex AC optimal power flow (ACOPF) constraints and instead use linearized power balance equations or transport formulations. While these simplifications improve tractability, they limit our understanding of how power flow and the physical properties of power networks impact investment decisions across generation, storage, and transmission infrastructure. This paper addresses this gap by extending the GenX capacity expansion model to incorporate fixed point theorem-based network constraints. These embed ACOPF-based considerations while maintaining the tractability of the planning model, nearly preserving the dimensionality of the transport formulation and incurring only modest runtime increases. This approach is much cheaper than embedding ACOPF directly, making it appropriate for large-scale capacity planning problems. We compare our approach to the original transport-based GenX model as well as a non-linear, non-convex version that incorporates the full ACOPF constraints, for a case study of the ISO New England grid.

We develop a novel approach to matrix discrepancy based on matrix small-ball estimates. Specifically, we use a determinantal weight (obtained from the log-barrier) to scale the small-ball probability into a partition function of a tilt of the Gaussian measure. We then employ matrix-weighted Poincaré inequalities to compare this partition function to that of a pinched or diagonal part of the matrix, obtaining dimension-free constants. Our technique yields a hereditary small-ball estimate for Gaussian series that should be of independent interest. As the main application, we resolve the Matrix Spencer conjecture: for symmetric $n\times n$ matrices $A_1,\dots,A_n$ with $\|A_i\|\le1$, one can efficiently find a coloring $x\in\{\pm1\}^n$ with $\|\sum_{i=1}^n x_iA_i\|=O(\sqrt n)$.

Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.

We investigate the role of quartic terms in the small-time local controllability of scalar-input systems. First, we prove a new sufficient condition for controllability, which exploits simultaneously more good quartic Lie brackets than previous results. Second, we identify a family of quartic obstructions to controllability, relying on Lie brackets whose coordinates of the second kind are positive-definite functionals of the control. Third, we show that the complementarity of these results can be seen as an answer to Kawski's 1987 open problem concerning the construction of a Hall basis which somehow separates good and bad quartic brackets. We give examples and remarks illustrating some of the intricacies of these notions.

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\bar{\nu}$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis–Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee–Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $\chi^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee–Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $\chi^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

Composite optimization plays a central role in modern machine learning and signal processing, as it offers a natural balance between data fidelity and structural properties. In this paper, we study composite optimization in the setting where both components are nonsmooth and nonconvex. We start with a deterministic Bregman proximal subgradient method that converges under subgradient upper-bound conditions. This approach relaxes the standard requirement on the convexity of the regularization term, thus accommodating a broader range of applications. To extend this to the stochastic regime, we develop a model-based minimization method under a relative Lipschitz condition and establish a convergence rate of $\mathcal{O}(\varepsilon^{-4})$. We also extend the framework with convergence guarantees to the setting where the distance generating function and its gradient are accessible only through a stochastic oracle.

An operator–theoretic formulation is developed for the synthesis of parameter-dependent controllers and observers for the Vlasov–Maxwell system. The linearized dynamics are modeled as a non-autonomous evolution system whose generators depend on measurable plasma quantities. Well-posedness of the associated evolution family is established together with uniform growth bounds. Parameter-dependent Lyapunov operators yield operator differential LMIs ensuring uniform exponential stability and observer convergence. An $H_\infty$ extension provides disturbance attenuation conditions consistent with the intrinsic energy structure of the coupled Vlasov–Maxwell equations. Galerkin projections lead to finite-dimensional LMIs consistent with the operator inequalities, enabling reliable numerical synthesis while preserving the analytical structure of the original model. Numerical results on a reduced Vlasov–Maxwell benchmark confirm the predicted convergence properties.

In the present paper, a study is made of first-order differential properties of certain multifunctions, which are defined as a solution map associated with a family of parameterized “split" feasibility problems. The latter are a class of convex feasibility problems, whose specific structure makes them suitable for well-established applications in several areas of engineering and systems biology. As a result, inner and outer approximations of the graphical (contingent) derivative of the solution map are provided, which are expressed in terms of derivatives of the problem data. As an application, a perspective is also discussed for employing some of these achievements in the formulation of optimality conditions for mathematical programs with “split" feasibility constraints.

In a recent paper, Attouch, Chbani, Fadili and Riahi introduced the inertial gradient algorithm with Hessian-driven daming, called (IGAHD). Under the condition $0\leq\beta<2\sqrt{s}$, where $\beta$ is the Hessian driven damping parameter and $s>0$ is the gradient step size, their Lyapunov analysis shows an accelerated estimate of the objective function as well as a weighted summability of the gradient for the case $\beta>0$. For any fixed value of $\beta>0$, this condition excluded sufficiently small step sizes $s>0$. The present note refines one of the estimates in this Lyapunov analysis by retaining two coefficients that were previously replaced by lower bounds. This leads to the following less restrictive and sufficient condition: $$0<\beta L\sqrt{s}<1+\sqrt{1+sL(1-sL)},$$ where $L>0$ is the constant Lipschitz of the gradient objective function and $0<s\leq1/L$. In particular, the simplest condition $0<\beta <\frac{2}{L\sqrt{s}}$ is sufficient. For fixed values of $\beta>0$ and $L>0$, this new condition is satisfied for sufficiently small $s>0$. Consequently, under this new refined sufficient condition, the conclusions in the original paper stay valid. For convex quadratic functions in finite dimensions, a separate spectral analysis leads to a larger region parameters. It gives a necessary and sufficient condition for Schur stability of the limit modal matrices, as well as the geometrical decay of the objective residuals and the gradients. This modal analysis raises the question of whether a different Lyapunov function would allow us to recover part (or all) of this extended spectral domain for non-quadratic objective functions.

We introduce a new family of non-negative polynomials, constructed via the arithmetic harmonic inequality, called AHI polynomials. We derive explicit algebraic conditions for this family and prove that, for AHI polynomials, the cone of non-negative polynomials coincides with the cone of sum of squares (SOS) polynomials. We then study their convexity, showing that although AHI polynomials are generally nonconvex, certain monomial substructures are SOS convex. We further locate the family precisely among the standard nonnegativities certificates; every AHI polynomial is simultaneously SOS and a sum of non negative circuit polynomials (SONC), and the containment in the intersection of these two cones is strict. By closing this family under multiplication, we obtain a cone Pi AHI that is, by construction, still SOS, yet we prove that it lies outside both the SONC cone and the smaller SDSOS cone. Moreover, membership in this cone admits a closed form certificate that does not require solving any semidefinite programs. Finally, we demonstrate the usefulness of these structures in optimization, numerical experiments indicate that exploiting AHI sparsity yields a computation time over 300 times faster than dense SOS relaxations and enables solving high degree polynomial optimization problems (up to degree 40) that standard methods cannot handle due to computational limits, and a factorized hierarchy for Pi AHI decomposes products into independent small subproblems that generic sparsity techniques do not detect.

The continuous-discrete projection filter offers a rigorous framework to approximate the solution of the nonlinear state estimation problems. However, it suffers from numerical instability during the prediction phase when integration errors force the natural parameters outside their admissible domain. To address this issue, we introduce the sum-of-squares (SOS) relaxation to constrain the evolution of the natural parameters within the admissible domain. By parameterizing the underlying SOS matrix using the log-Cholesky map, we derive a projected ordinary differential equation (ODE) that inherently preserves the necessary positivity constraints without requiring the computationally expensive online optimization checks associated with previous semi-infinite programming approach. We provide a theoretical derivation of this positivity-preserving propagation scheme and present the explicit SOS-relaxed evolution equations for the Gaussian case.