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Tien Son Pham

Publications and source records attributed to Tien Son Pham.

At least 19 recordsLinked to original sources
arXiv2026

DPS: Dual-Mode Precision LLM Serving with Semi-Unified Memory

Existing LLM serving systems virtualize and optimize KV-cache memory, but treat model-weight memory as fixed throughout execution. Recent work on multi-precision model representations challenges this design by allowing a single stored model to support both full-accuracy and lower-precision execution, making the effective weight footprint runtime-dependent. This creates an opportunity under bursty workloads, where temporary spikes in KV-cache demand often determine throughput and SLO compliance. We present DPS, a dual-precision LLM serving system that turns weight memory into an elastic resource: under normal load, DPS serves the full-accuracy model; under KV pressure, it switches to a nested, lower-precision variant and repurposes unused weight memory for KV cache blocks. DPS is built on Semi-Unified Memory (SUM), which partitions the weight region into a persistent lower-precision sub-region and a shared region that alternates between residual weight tensors and KV-cache blocks, preserving compatibility with paged KV-cache management. We implement DPS on top of vLLM and evaluate it across both dense and MoE models and various production workload traces. Our results show that \sysname improves sustained throughput by $2.1$--$3.3\times$ and effective pass@1 by up to $+41$\,pp over Static FP16, while preserving FP16-class accuracy.

cs.DC↗
arXiv2026

Perturbation of monic matrix polynomials

In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(λ)=A_d(u)λ^d+A_{d-1}(u)λ^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$. Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(λ)$ define set-valued maps that are Hölder continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(λ)$ depend analytically on $u$. We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges.

math.RA↗
arXiv2023

Existence theorems for optimal solutions in semi-algebraic optimization

Consider the problem of minimizing a lower semi-continuous semi-algebraic function $f \colon \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ on an unbounded closed semi-algebraic set $S \subset \mathbb{R}^n.$ Employing adequate tools of semi-algebraic geometry, we first establish some properties of the tangency variety of the restriction of $f$ on $S.$ Then we derive verifiable necessary and sufficient conditions for the existence of optimal solutions of the problem as well as the boundedness from below and coercivity of the restriction of $f$ on $S.$ We also present a computable formula for the optimal value of the problem.

math.OC↗
arXiv2023

Subdifferentials at infinity and applications in optimization

In this work, the notions of normal cones at infinity to unbounded sets and limiting and singular subdifferentials at infinity for extended real value functions are introduced. Various calculus rules for these notions objects are established. A complete characterization of the Lipschitz continuity at infinity for lower semi-continuous functions is given. The obtained results are aimed ultimately at applications to diverse problems of optimization, such as optimality conditions, coercive properties, weak sharp minima and stability results.

math.OC↗
arXiv2022

Characterizations of directional openness for set-valued mappings

We provide necessary and sufficient conditions for a set-valued mapping between finite dimensional spaces to be directionally open by relating this property with directional regularity, Hölder continuity of the inverse mapping, coderivatives and variations. These generalize and refine some previously known results.

math.OC↗
arXiv2022

Limits of real bivariate rational functions

Given two nonzero polynomials $f, g \in\mathbb R[x,y]$ and a point $(a, b) \in \mathbb{R}^2,$ we give some necessary and sufficient conditions for the existence of the limit $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}.$ We also show that, if the denominator $g$ has an isolated zero at the given point $(a, b),$ then the set of possible limits of $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}$ is a closed interval in $\overline{\mathbb{R}}$ and can be explicitly determined. As an application, we propose an effective algorithm to verify the existence of the limit and compute the limit (if it exists). Our approach is geometric and is based on Puiseux expansions.

math.CA↗
arXiv2021

On definable open continuous mappings

For a definable continuous mapping $f$ from a definable connected open subset $Ω$ of $\mathbb R^n$ into $\mathbb R^n,$ we show that the following statements are equivalent: (i) The mapping $f$ is open. (ii) The fibers of $f$ are finite and the Jacobian of $f$ does not change sign on the set of points at which $f$ is differentiable. (iii) The fibers of ${f}$ are finite and the set of points at which $f$ is not a local homeomorphism has dimension at most $n - 2.$ As an application, we prove that Whyburn's conjecture is true for definable mappings: A definable open continuous mapping of one closed ball into another which maps boundary homeomorphically onto boundary is necessarily a homeomorphism.

math.AG↗
arXiv2021

Some classical analysis results for continuous definable mappings

In this paper, we show that some fundamental results for smooth mappings (e.g., the Brouwer degree formula, the implicit function and inverse function theorems, the mean value theorem, Sard's theorem, Hadamard's global invertibility criteria, Pourciau's surjectivity and openness results) have natural extensions for continuous mappings that are definable in o-minimal structures. The arguments rely on nice properties of definable mappings and sets.

math.FA↗
arXiv2021

Some variational properties of tangent directions at infinity of real algebraic sets

In this paper, we relate the set of asymptotic critical values of a polynomial function $f$ with the set of discontinuity of two functions, the multivalued function which associate to each value $t$ the set of tangent directions at infinity of the fiber $f^{-1}(t)$ and the composition of the $(n-2)$-dimensional volume function with the first one. This gives necessary conditions of equisingularity at infinity for the family of the fibers of a real polynomial function.

math.AG↗
arXiv2021

Nichtnegativstellensätze for definable functions in o-minimal structures

This paper addresses to Nichtnegativstellensätze for definable functions in o-minimal structures on $(\mathbb{R}, +, \cdot).$ Namely, let $f, g_1, \ldots, g_l \colon \mathbb{R}^n \to \mathbb{R}$ be definable $C^p$-functions ($p \ge 2$) and assume that $f$ is non-negative on $S := \{x \in \mathbb{R}^n \ | \ g_1(x) \ge 0, \ldots, g_l(x) \ge 0 \}.$ Under some natural hypotheses on zeros of $f$ in $S,$ we show that $f$ is expressible in the form $f = ϕ_0 + \sum_{i = 1}^l ϕ_i g_i,$ where each $ϕ_i$ is a sum of squares of definable $C^{p - 2}$-functions. As a consequence, we derive global optimality conditions which generalize the Karush--Kuhn--Tucker optimality conditions for nonlinear optimization.

math.AG↗
arXiv2021

The mountain pass theorem in terms of tangencies

This paper addresses the Mountain Pass Theorem for locally Lipschitz functions on finite-dimensional vector spaces in terms of tangencies. Namely, let $f \colon \mathbb R^n \to \mathbb R$ be a locally Lipschitz function with a mountain pass geometry. Let $$c := \inf_{γ\in \mathcal A}\max_{t\in[0,1]}f(γ(t)),$$ where $\mathcal{A}$ is the set of all continuous paths joining $x^*$ to $y^*.$ We show that either $c$ is a critical value of $f$ or $c$ is a tangency value at infinity of $f.$ This reduces to the Mountain Pass Theorem of Ambrosetti and Rabinowitz in the case where the function $f$ is definable (such as, semi-algebraic) in an o-minimal structure.

math.AP↗
arXiv2021

Stability of closedness of closed convex sets under linear mappings

We study the problem of when the continuous linear image of a fixed closed convex set $X \subset\mathbb{R}^n$ is closed. Specifically, we improve the main results in the papers \cite{Borwein2009, Borwein2010} by showing that for all, except for at most a $σ$-porous set, of the linear mappings $T$ from $\mathbb{R}^n$ into $\mathbb{R}^m,$ not only $T(X)$ is closed, but there is also an open neighborhood of $T$ whose members also preserve the closedness of $X.$

math.OC↗
arXiv2021

Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings

Given a closed semi-algebraic set $X \subset \mathbb{R}^n$ and a continuous semi-algebraic mapping $G \colon X \to \mathbb{R}^m,$ it will be shown that there exists an open dense semi-algebraic subset $\mathscr{U}$ of $L(\mathbb{R}^n, \mathbb{R}^m),$ the space of all linear mappings from $\mathbb{R}^n$ to $\mathbb{R}^m,$ such that for all $F \in \mathscr{U},$ the image $(F + G)(X)$ is a closed (semi-algebraic) set in $\mathbb{R}^m.$ To do this, we study the tangent cone at infinity $C_\infty X$ and the set $E_\infty X \subset C_\infty X$ of (unit) exceptional directions at infinity of $X.$ Specifically we show that the set $E_\infty X$ is nowhere dense in $C_\infty X \cap \mathbb{S}^{n - 1}.$

math.AG↗
arXiv2020

Limits of tangent spaces to definable sets

We study the set of tangent limits at a given point to a set definable in any o-minimal structure by characterizing the set of exceptional rays in the tangent cone to the set at that point and investigating the set of tangent limits along these rays. Several criteria for determining exceptional rays will be given. The main results of the paper generalize, to the o-minimal setting and to arbitrary dimension, the main results of O'Shea--Wilson which deals with algebraic surfaces in $\mathbb R^3$.

math.AG↗
arXiv2018

Volume estimates of sublevel sets of real polynomials

We give upper bounds for volume of sublevel sets of real polynomials. Our method is to combine a version of global Lojasiewicz inequality with some well known estimate on volume of tubes around real algebraic sets. Some applications to oscillatory integrals and integration indices of real polynomial are also given.

math.CV↗
arXiv2016

The bifurcation set of a real polynomial function of two variables and Newton polygons of singularities at infinity

In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called "cleaving" and "vanishing" in the same setting. Finally, we give an upper bound of the number of elements in the bifurcation set in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively.

math.GT↗