arXiv subjects

Hoang Anh Tran

Publications and source records attributed to Hoang Anh Tran.

10 recordsLinked to original sources

arXiv2026

A Sum-of-Squares Hierarchy with Quadratic Convergence for Quantum Channel Coding

Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.

quant-ph↗

arXiv2026

Quadratic Optimization over Probability Measures with Coupling Constraints

We consider solving an optimization instance in which the objective is quadratic and where the decision variable is a probability measure. Our class of problems are motivated by applications arising from optimal transport (with the Gromov-Wasserstein problem being a prominent example) as well as energy landscape minimization. Because the objective depends quadratically on the decision variable, our class of problems fall outside the standard modeling framework of the Generalized Moment Problems (which requires the objective to be linear). To this end, we propose a hierarchy of convex relaxations based on searching over probability measures over products of the base space. These have a natural interpretation with the moment Sum-of-squares hierarchy-a prominent framework for solving polynomial optimization instances, which we adapt to accommodate probability measures. A key conceptual contribution is to introduce a notion of positive-semidefiniteness that extends the usual notion over matrices. Under the assumption that the decision variables satisfy certain marginal constraints (as in the Kantorovich formulation of the optimal transport problem), we establish convergence of our hierarchy towards the globally optimal solution. Under the additional assumption that the objective is a polynomial, we propose a moment-SOS type hierarchy of finite dimensional semidefinite programs whose optimal solution converges to that of the original quadratic optimization over measures. We demonstrate our framework with numerical experiments. More generally, optimization over measures where the objective and/or constraint depends on the decision in a polynomial way is a fundamental problem. It is hoped that our work provides a road-map as to how the ideas of the SOS-ordinarily developed for polynomial optimization-may be applied to a broader class of non-linear problems involving measures.

math.OC↗

arXiv2026

Sum-of-Squares Hierarchy for the Gromov Wasserstein Problem

The Gromov-Wasserstein (GW) problem is a variant of the classical optimal transport problem that allows one to compute meaningful transportation plans between incomparable spaces. At an intuitive level, it seeks plans that minimize the discrepancy between metric evaluations of pairs of points. The GW problem is typically cast as an instance of a non-convex quadratic program that is, unfortunately, intractable to solve. In this paper, we describe tractable semidefinite relaxations of the GW problem based on the Sum-of-Squares (SOS) hierarchy. We describe how the Putinar-type and the Schmüdgen-type moment hierarchies can be simplified using marginal constraints, and we prove convergence rates for these hierarchies towards computing global optimal solutions to the GW problem. The proposed SOS hierarchies naturally induce a distance measure analogous to the distortion metrics, and we show that these are genuine distances in that they satisfy the triangle inequality. In particular, the proposed SOS hierarchies provide computationally tractable proxies of the GW distance and the associated distortion distances (over metric measure spaces) that are otherwise intractable to compute.

math.OC↗

arXiv2025

Convergence rates of S.O.S hierarchies for polynomial semidefinite programs

We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set $\mathcal{X}$ defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on $\mathcal{X}$, based on Jackson's theorem and a variant of the Łojasiewicz inequality.

math.OC↗

arXiv2025

On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences

The moment-SOS hierarchy is a widely applicable framework to address polynomial optimization problems over basic semi-algebraic sets based on positivity certificates of polynomial. Recent works show that the convergence rate of this hierarchy over certain simple sets, namely, the unit ball, hypercube, and standard simplex, is of the order $O(1/r^2)$, where r denotes the level of the moment-SOS hierarchy. This paper aims to provide a comprehensive understanding of the convergence rate of the moment-SOS hierarchy by estimating the Hausdorff distance between the set of truncated pseudo-moment sequences and the set of truncated moment sequences specified by Tchakaloff's theorem. Our results provide a connection between the convergence rate of the moment-SOS hierarchy and the Lojasiewicz exponent L of the domain under the compactness assumption, where we establish the convergence rate of $O(1/r^L)$. Consequently, we obtain the convergence rate of $O(1/r)$ for polytopes and sets satisfying the constraint qualification condition, $O(1/\sqrt{r})$ for domains that either satisfy the Polyak-Lojasiewicz condition or are defined by locally strongly convex polynomials. We also obtain the convergence rate of $O(1/r^2)$ for general polynomials over a sphere.

math.OC↗

arXiv2025

AUTOBargeSim: MATLAB(R) toolbox for the design and analysis of the guidance and control system for autonomous inland vessels

This paper introduces AUTOBargeSim, a simulation toolbox for autonomous inland vessel guidance and control system design. AUTOBargeSim is developed using MATLAB and provides an easy-to-use introduction to various aspects of autonomous inland navigation, including mapping, modelling, control design, and collision avoidance, through examples and extensively documented code. Applying modular design principles in the simulator structure allows it to be easily modified according to the user's requirements. Furthermore, a GUI interface facilitates a simple and quick execution. Key performance indices for evaluating the performance of the controller and collision avoidance method in confined space are also provided. The current version of AUTOBargeSim attempts to improve reproducibility in the design and simulation of marine systems while serving as a foundation for simulating and evaluating vessel behaviour considering operational, system, and environmental constraints.

eess.SY↗

arXiv2025

Moment Sum-of-Squares Hierarchy for Gromov Wasserstein: Continuous Extensions and Sample Complexity

The Gromov-Wasserstein (GW) problem is an extension of the classical optimal transport problem to settings where the source and target distributions reside in incomparable spaces, and for which a cost function that attributes the price of moving resources is not available. The sum-of-squares (SOS) hierarchy is a principled method for deriving tractable semidefinite relaxations to generic polynomial optimization problems. In this work, we apply ideas from the moment-SOS hierarchy to solve the GW problem. More precisely, we identify extensions of the moment-SOS hierarchy, previously introduced for the discretized GW problem, such that they remain valid for general probability distributions. This process requires a suitable generalization of positive semidefiniteness over finite-dimensional vector spaces to the space of probability distributions. We prove the following properties concerning these continuous extensions: First, these relaxations form a genuine hierarchy in that the optimal value converges to the GW distance. Second, each of these relaxations induces a pseudo-metric over the collection of metric measure spaces. Crucially, unlike the GW problem, these induced instances are tractable to compute -- the discrete analogs are expressible as semidefinite programs and hence are tractable to solve. Separately from these properties, we also establish a statistical consistency result arising from sampling the source and target distributions. Our work suggests fascinating applications of the SOS hierarchy to optimization problems over probability distributions in settings where the objective and constraint depend on these distributions in a polynomial way.

math.OC↗

arXiv2025

Asynchronous distributed collision avoidance with intention consensus for inland autonomous ships

This paper focuses on the problem of collaborative collision avoidance for autonomous inland ships. Two solutions are provided to solve the problem in a distributed manner. We first present a distributed model predictive control (MPC) algorithm that allows ships to directly negotiate their intention to avoid collision in a synchronous communication framework. Moreover, we introduce a new approach to shape the ship's behavior to follow the waterway traffic regulations. The conditional convergence toward a stationary solution of this algorithm is guaranteed by the theory of the Alternating Direction Method of Multipliers (ADMM). To overcome the problem of asynchronous communication between ships, we adopt a new asynchronous nonlinear ADMM and present an asynchronous distributed MPC algorithm based on it. Several simulations and field experiments show that the proposed algorithms can prevent ship collisions even in complex scenarios.

eess.SY↗

arXiv2025

Distributed MPC for autonomous ships on inland waterways with collaborative collision avoidance

This paper presents a distributed solution for the problem of collaborative collision avoidance for autonomous inland waterway ships. A two-layer collision avoidance framework that considers inland waterway traffic regulations is proposed to increase navigational safety for autonomous ships. Our approach allows for modifying traffic rules without changing the collision avoidance algorithm, and is based on a novel formulation of model predictive control (MPC) for collision avoidance of ships. This MPC formulation is designed for inland waterway traffic and can handle complex scenarios. The alternating direction method of multipliers is used as a scheme for exchanging and negotiating intentions among ships. Simulation results show that the proposed algorithm can comply with traffic rules. Furthermore, the proposed algorithm can safely deviate from traffic rules when necessary to increase efficiency in complex scenarios.

eess.SY↗

arXiv2021

Applications of resultant of two $p$-adic power series

Given a prime $p$, and $v_p(a)$ stand for the $p$-adic valuation of the element $a$ in a finite extension $K$ of $\mathbf{Q}_p$, or more generally the field $\mathbf{C}_p$ which is the complete field of the algebraic closure $\mathbf{Q}_p$ with respect to the $p$-adic absolute value, denoted by $\lvert \cdot \rvert_p$. Let $F$ and $G$ be two ($p$-adic) power series with no common roots. We aim to estimate the maximal value $S$ and the minimal value $s$ of the function $ϕ(x)=\min(v_p(F(x)), v_p(G(x)))$ over various domains, namely open and closed unit discs of $K$ or $\mathbf{C}_p$. To do this, we use partial resultants of two power series over certain domains defined by varied versions of the Weierstrass preparation theorem. Furthermore, the resultant of power series provides an efficient tool while studying the irreducibility and calculating the maximal value of $ϕ$.

math.NT↗