About

Jiawang Nie is a professor in the Department of Mathematics at UC San Diego. His research is in the broad area of Numerical Analysis and Scientific Computing, with an emphasis on Polynomial and Semidefinite Optimization, Moment and Tensor Computation, Convex and Real Algebraic Geometry. He has received numerous honors including the Young Researcher Prize, NSF CAREER Award, Tucker Prize Finalist, Hellman Fellowship, SIAG/LA Best Paper Prize, the Kalman Visiting Fellowship, and the Feng Kang Prize in 2021. His educational background includes a Ph.D. in Applied Mathematics from the University of California, Berkeley, obtained in 2006.

Research topics

• Computer Science
• Mathematics
• Pure mathematics
• Physics
• Mathematical analysis
• Quantum mechanics
• Geometry
• Mathematical optimization
• Combinatorics
• Applied mathematics

Selected publications

• Robust Completion for Rank-1 Tensors with Noises

  Journal of Scientific Computing · 2026-05-12

  articleOpen access1st author

  Abstract This paper studies the rank-1 tensor completion problem for cubic tensors when there are noises for observed tensor entries. First, we propose a robust biquadratic optimization model for obtaining rank-1 completing tensors. When the observed tensor is sufficiently close to be rank-1, we show that this biquadratic optimization produces an accurate rank-1 tensor completion. Second, we give an efficient convex relaxation for solving the biquadratic optimization. When the optimizer matrix is separable, we show how to get optimizers for the biquadratic optimization and how to compute the rank-1 completing tensor. When that matrix is not separable, we apply its spectral decomposition to obtain an approximate rank-1 completing tensor. The software is applied to solve the resulting large size semidefinite programs. Numerical experiments are given to explore the efficiency of this biquadratic optimization model and the proposed convex relaxation.

  PublisherOA PDFDOI

• Lagrange multiplier expressions for matrix polynomial optimization and tight relaxations

  ArXiv.org · 2025-06-14

  preprintOpen access

  This paper studies matrix constrained polynomial optimization. We investigate how to get explicit expressions for Lagrange multiplier matrices from the first order optimality conditions. The existence of these expressions can be shown under the nondegeneracy condition. Using Lagrange multiplier matrix expressions, we propose a strengthened Moment-SOS hierarchy for solving matrix polynomial optimization. Under some general assumptions, we show that this strengthened hierarchy is tight, or equivalently, it has finite convergence. We also study how to detect tightness and how to extract optimizers. Numerical experiments are provided to show the efficiency of the strengthened hierarchy.

  PublisherOA PDFDOI

• A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem

  ArXiv.org · 2025-10-31

  preprintOpen access

  This paper studies how to compute global minimizers of the cubic-quartic regularization (CQR) problem \[ \min_{s \in \mathbb{R}^n} \quad f_0+g^Ts+\frac{1}{2}s^THs+\fracβ{6} \| s \|^3+\fracσ{4} \| s \|^4, \] where $f_0$ is a constant, $g$ is an $n$-dimensional vector, $H$ is a $n$-by-$n$ symmetric matrix, and $\| s \|$ denotes the Euclidean norm of $s$. The parameter $σ\ge 0$ while $β$ can have any sign. The CQR problem arises as a critical subproblem for getting efficient regularization methods for solving unconstrained nonlinear optimization. Its properties are recently well studied by Cartis and Zhu [cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods, Math. Program, 2025]. However, a practical method for computing global minimizers of the CQR problem still remains elusive. To this end, we propose a semidefinite programming (SDP) relaxation method for solving the CQR problem globally. First, we show that our SDP relaxation is tight if and only if $\| s^* \| ( β+ 3 σ\| s^* \|) \ge 0$ holds for a global minimizer $s^*$. In particular, if either $β\ge 0$ or $H$ has a nonpositive eigenvalue, then the SDP relaxation is shown to be tight. Second, we show that all nonzero global minimizers have the same length for the tight case. Third, we give an algorithm to detect tightness and to obtain the set of all global minimizers. Numerical experiments demonstrate that our SDP relaxation method is both effective and computationally efficient, providing the first practical method for globally solving the CQR problem.

  PublisherOA PDFDOI

• Finite convergence of the Moment-SOS hierarchy for polynomial matrix optimization

  Mathematical Programming · 2025-02-15 · 2 citations

  articleSenior author
  PublisherDOI

• Optimization over the weakly Pareto set and multi-task learning

  ArXiv.org · 2025-03-31

  preprintOpen access

  We study the optimization problem over the weakly Pareto set of a convex multiobjective optimization problem given by polynomial functions. Using Lagrange multiplier expressions and the weight vector, we give three types of representations for the weakly Pareto set. Using these representations, we reformulate the optimization problem over the weakly Pareto set as a polynomial optimization problem. We then apply the Moment--SOS hierarchy to solve it and analyze its convergence properties under certain conditions. Numerical experiments are provided to demonstrate the effectiveness of our methods. Applications in multi-task learning are also presented.

  PublisherOA PDFDOI

• Preface: Recent advances in polynomial optimization theory and methods

  Numerical Algebra Control and Optimization · 2025-10-23

  articleOpen access
  PublisherOA PDFDOI

• A characterization for tightness of the sparse Moment-SOS hierarchy

  Mathematical Programming · 2025-05-02 · 1 citations

  articleOpen access1st authorCorresponding

  Abstract This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.

  PublisherOA PDFDOI

• Polynomial optimization relaxations for generalized semi-infinite programs

  Mathematical Programming Computation · 2025-04-18 · 1 citations

  articleOpen accessSenior author

  Abstract This paper studies generalized semi-infinite programs (GSIPs) given by polynomials. We propose a hierarchy of polynomial optimization relaxations to solve them. They are based on Lagrange multiplier expressions and polynomial extensions. Moment-SOS relaxations are applied to solve the polynomial optimization. The convergence of this hierarchy is shown under certain conditions. In particular, the classical semi-infinite programs can be solved as a special case of GSIPs. We also study GSIPs that have convex infinity constraints and show that they can be solved exactly by a single polynomial optimization relaxation. The computational efficiency is demonstrated by extensive numerical results.

  PublisherOA PDFDOI

• Moment-SOS relaxations for moment and tensor recovery problems

  Numerical Algebra Control and Optimization · 2025-10-20

  articleOpen access

  This paper studies moment and tensor recovery problems whose decomposing vectors are contained in some given semialgebraic sets. We propose Moment-SOS relaxations with generic objectives for recovering moments and tensors, whose decomposition lengths are expected to be low. The tensor recovery problem is treated as a structured instance of the moment recovery problem. This kind of problems have broad applications in various tensor decomposition questions. Numerical experiments are provided to demonstrate the efficiency of this approach.

  PublisherOA PDFDOI

• A Global Approach for Generalized Semi-Infinite Programs with Polyhedral Parameter Sets

  Journal of Optimization Theory and Applications · 2025-08-13

  article
  PublisherDOI

Recent grants

• Semidefinite Programming Methods for Moment and Optimization Problems

  NSF · $210k · 2014–2018

• Computational Methods for Symmetric Tensor Problems

  NSF · $150k · 2016–2020

• CAREER: Linear Matrix Inequality Representations in Optimization

  NSF · $500k · 2009–2015

• Lagrange Multiplier Expression Methods for Optimization

  NSF · $350k · 2021–2025

Frequent coauthors

• Zi Yang

  University at Albany, State University of New York

  19 shared

• Suhan Zhong

  Texas A&M University

  17 shared

• Xindong Tang

  Hong Kong Baptist University

  16 shared

• J. William Helton

  University of California, San Diego

  15 shared

• Ya-xiang Yuan

  Chinese Academy of Sciences

  13 shared

• Xinzhen Zhang

  Hunan Agricultural University

  12 shared

• James Demmel

  12 shared

• Bernd Sturmfels

  7 shared

Awards & honors

• Young Researcher Prize
• NSF CAREER Award
• Tucker Prize
• Finalist Hellman Fellowship
• SIAG/LA Best Paper Prize