arXiv subjects

Shenyuan Ma

Publications and source records attributed to Shenyuan Ma.

5 recordsLinked to original sources

arXiv2026

When Can One Obtain Certificates of Optimality Using Positivstellensaetze?

We study certificates of positivity and optimality for learning problems whose objectives and constraints need not be polynomial. We isolate an axiomatic core of Fischer's constructive strict and weak Positivstellens\"{a}tze and prove the resulting theorems for abstract function algebras over ordered fields. The framework separates two roles that can otherwise be conflated: objective and constraint functions may be built from broad classes of continuous or definable operations, while the auxiliary primitives used to construct a certificate satisfy explicit scalar and closure axioms. We give instances over continuous and definable function algebras, including ordered fields not closed under square roots, derive lower-bound and global-optimality certificates, and analyze both expanded term length and shared computation-graph complexity.

cs.AI↗

arXiv2026

On Moment-Based Recovery of Measures with Atomic and Continuous Parts

Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one dimension. A typical strategy when recovering a measure from moments is to verify the flat-extension property, which certifies that the underlying measure is finitely atomic and ultimately leads to recovery. For many classes of measures, however, the flat extension never occurs and thus if one aims to recover the measure corresponding to the moments, assumptions need to be made. We formulate a new kind of recovery problem, where one assumes that the measure has compact support and a fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights, which apply to algorithms widely used for the recovery of atomic measures from moments. Furthermore, the statements proven lead to novel algorithms, which we benchmark, further confirming the theoretical findings.

math.OC↗

arXiv2026

Hierarchy of bounds in free orthotropic material optimization: From convex relaxations to Hashin-Shtrikman via sequential global programming

We study free orthotropic material optimization for two-dimensional plane-stress compliance minimization with two well-ordered isotropic phases, motivated by the gap between tensors admissible in classical free-material optimization and tensors realizable by composites. To reduce this gap, we construct a hierarchy of realizability-aware admissible sets induced by zeroth-order, Voigt, and Hashin--Shtrikman (HS) energy bounds, moving from convex relaxations to a tighter nonconvex model. In the convex zeroth-order and Voigt settings, the Voigt set is strictly tighter for intermediate volume fractions and coincides with the zeroth-order set at pure-phase endpoints, and the Voigt model reduces to an isotropic variable-thickness-sheet formulation. In the single-loadcase continuum zeroth-order problem, at least one optimal solution can be chosen orthotropic. For HS constraints, we rewrite the bound as a Voigt term minus a nonnegative correction, clarifying strict tightening for interior volume fractions and local nonconvexity. We further prove that the convex hull of the HS feasible set equals the Voigt set and derive reduced formulations via active-constraint analysis and explicit elementwise volume characterization, including reductions specialized to orthotropic effective tensors. In the single-loadcase continuum setting, the HS relaxation is tight with the Allaire--Kohn relaxed problem, attained in the relaxation sense by sequential laminates, whereas in generic multi-loadcase settings it provides a lower bound on optimal compliance over general microstructures. The resulting nonconvex orthotropic HS problem is solved by sequential global programming, and numerical results confirm the predicted compliance hierarchy and show close agreement with finite-rank laminate references.

math.OC↗

arXiv2025

On Taylor-like Estimates for Chebyshev Series Approximations of $e^x$

Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and expansions derived based on families of $L^{2}-$orthogonal polynomials on bounded intervals. Traditionally, however, the properties of these series are studied in an isolated manner and most results on the $L^{2}-$based approximations are restricted to the interval of approximation. In many applications, theoretical guarantees require global estimates. Since these often follow trivially from Taylor's remainder theorem, Taylor's series is often favored in these applications. Its suboptimal numerical performance on compact intervals, however, ultimately results in such numerical algorithms being suboptimal. We show that "Taylor-like" bounds valid outside of the interval of approximation may be derived for the important case of $e^{x}$ and Chebyshev polynomials on $[-1,1]$ and provide links to substantial numerical applications.

math.GM↗

arXiv2025

Truss topology design under harmonic loads: Peak power minimization with semidefinite programming

Designing lightweight yet stiff structures that can withstand vibrations is a crucial task in structural optimization. Here, we present a novel framework for truss topology optimization under undamped harmonic oscillations. Our approach minimizes the peak power of the structure under harmonic loads, overcoming the limitations of single-frequency and in-phase assumptions found in previous methods. For this, we leverage the concept of semidefinite representable (SDr) functions, demonstrating that while compliance readily conforms to an SDr representation, peak power requires a derivation based on the non-negativity of trigonometric functions. Finally, we introduce convex relaxations for the minimization problem and provide promising computational results.

math.OC↗