arXiv subjects

Jiawang Nie

Publications and source records attributed to Jiawang Nie.

At least 19 recordsLinked to original sources

arXiv2026

A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem

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 an $n$-by-$n$ symmetric matrix, and $\| s \|$ denotes the Euclidean norm of $s$. The parameter $σ$ is nonnegative 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 {\it [cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods, Math. Program, 2025]}. We propose a structured semidefinite programming (SDP) relaxation method for solving the CQR problem globally. The SDP relaxation has only three symmetric positive semidefinite matrix variables of sizes $(n+1)$-by-$(n+1)$, $3$-by-$3$ and $2$-by-$2$ respectively. We show that our SDP relaxation is tight if and only if $\| s^* \| ( β+ 3 σ\| s^* \|) \ge 0$ holds for a global minimizer $s^*$. When $s^* \ne 0$, this aligns with the sufficient global optimality condition $β+ 3 σ\| s^* \| \ge 0$ given by Cartis and Zhu. 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 Euclidean norm 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. This paper gives a polynomial time algorithm for solving the CQR problem globally, under the sufficient global optimality condition.

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arXiv2026

Generalized Nash equilibrium problems with quasi-linear constraints

We study generalized Nash equilibrium problems (GNEPs) such that objectives are polynomial functions, and each player's constraints are linear in their own strategy. For such GNEPs, the KKT sets can be represented as unions of simpler sets by Carathéodory's theorem. We give a convenient representation for KKT sets using partial Lagrange multiplier expressions. This produces a set of branch polynomial optimization problems, which can be efficiently solved by Moment-SOS relaxations. By doing this, we can compute all generalized Nash equilibria or detect their nonexistence. This method may not be very scalable to large scale GNEPs. Numerical experiments are provided to demonstrate the computational efficiency.

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arXiv2026

Algebraic Degrees of Generalized Nash Equilibrium Problems

This paper studies the algebraic degree of generalized Nash equilibrium problems (GNEPs) given by polynomials. Their generalized Nash equilibria (GNEs), as well as their KKT or Fritz-John points, are algebraic functions in the coefficients of defining polynomials. We study the degrees of these algebraic functions, which also count the numbers of complex KKT or Fritz-John points. Under some genericity assumptions, we show that a GNEP has only finitely many complex Fritz-John points and every Fritz-John point is a KKT point. We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases.

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arXiv2026

Robust Completion for Rank-1 Tensors with Noises

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 SDPNAL+ 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.

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arXiv2026

Rank One Completion for Higher Order Tensors

We study the rank one completion problem for tensors of arbitrary orders. The notion of rank one determinable tensors is introduced. We explore its properties and propose a recursive algorithm for computing rank one tensor completion. This algorithm only requires solving linear systems and computing singular vectors. In the absence of noise, it produces a unique rank one completion under some assumptions. In the presence of noise, we show that the computed rank one tensor completion is close to the exact one when the noise is sufficiently small. Numerical experiments demonstrate the efficiency and accuracy of the proposed method.

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arXiv2026

Sparse Copositive Polynomial Optimization

This paper studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions are shown for these relaxations to be tight. In particular, we prove they are tight under the cop-SOS convexity assumption. Compared to the traditional dense ones, the sparse Moment-SOS relaxations are more computationally efficient. Numerical experiments are given to show the efficiency.

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arXiv2026

Partial Lagrange Multiplier Expressions and Disjunctive Decompositions for Bilevel Optimization

This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker (KKT) conditions with a sparse type of Lagrange multipliers. This kind of Lagrange multipliers can be conveniently represented by polynomials, for which we call partial Lagrange multiplier expressions (PLMEs). By doing this, each branch problem of the disjunctive program can be solved efficiently by polynomial optimization techniques. Solving each branch problem either returns infeasibility or gives a candidate local or global optimizer for the original bilevel optimization. We give necessary and sufficient conditions for these candidates to be global optimizers, and sufficient conditions for the local optimality. Numerical experiments are also presented to show the efficiency of the method.

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arXiv2026

Lagrange multiplier expressions for matrix polynomial optimization and tight relaxations

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.

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arXiv2026

Log-Polynomial Optimization

We study an optimization problem in which the objective is given as a sum of logarithmic-polynomial functions. This formulation is motivated by statistical estimation principles such as maximum likelihood estimation, and by loss functions including cross-entropy and Kullback-Leibler divergence. We propose a hierarchy of moment relaxations based on the truncated $K$-moment problems to solve log-polynomial optimization. We provide sufficient conditions for the hierarchy to be tight and introduce a numerical method to extract the global optimizers when the tightness is achieved. In addition, we modify relaxations with optimality conditions to better fit log-polynomial optimization with convenient Lagrange multipliers expressions. Various applications and numerical experiments are presented to show the efficiency of our method.

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arXiv2025

Learning Polynomial Activation Functions for Deep Neural Networks

Activation functions are crucial for deep neural networks. This novel work frames the problem of training neural network with learnable polynomial activation functions as a polynomial optimization problem, which is solvable by the Moment-SOS hierarchy. This work represents a fundamental departure from the conventional paradigm of training deep neural networks, which relies on local optimization methods like backpropagation and gradient descent. Numerical experiments are presented to demonstrate the accuracy and robustness of optimum parameter recovery in presence of noises.

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arXiv2025

Sparse Polynomial Optimization with Matrix Constraints

This paper studies the hierarchy of sparse matrix Moment-SOS relaxations for solving sparse polynomial optimization problems with matrix constraints. First, we prove a sufficient and necessary condition for the sparse hierarchy to be tight. Second, we discuss how to detect the tightness and extract minimizers. Third, for the convex case, we show that the hierarchy of the sparse matrix Moment-SOS relaxations is tight, under some general assumptions. In particular, we show that the sparse matrix Moment-SOS relaxation is tight for every order when the problem is SOS-convex. Numerical experiments are provided to show the efficiency of the sparse relaxations.

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arXiv2025

A global approach for generalized semi-infinte programs with polyhedral parameter sets

This paper studies generalized semi-infinite programs (GSIPs) defined with polyhedral parameter sets. Assume these GSIPs are given by polynomials. We propose a new approach to solve them as a disjunctive program. This approach is based on the Karush-Kuhn-Tucker (KKT) conditions of the robust constraint and a technique called partial Lagrange multiplier expressions. We summarize a semidefinite algorithm and study its convergence properties. Numerical experiments are given to show the efficiency of our method. In addition, we checked its performance in gemstone cutting and robust control applications.

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arXiv2025

Distributionally Robust Optimization with Polynomial Robust Constraints

This paper studies distributionally robust optimization (DRO) with polynomial robust constraints. We give a Moment-SOS relaxation approach to solve the DRO. This reduces to solving linear conic optimization with semidefinite constraints. When the DRO problem is SOS-convex, we show that it is equivalent to the linear conic relaxation and it can be solved by the Moment-SOS algorithm. For nonconvex cases, we also give concrete conditions such that the DRO can be solved globally. Numerical experiments are given to show the efficiency of the method.

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arXiv2025

A Characterization for Tightness of the Sparse Moment-SOS Hierarchy

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.

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arXiv2025

Polynomial Optimization Relaxations for Generalized Semi-Infinite Programs

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 (SIPs) 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.

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arXiv2025

Optimization over the weakly Pareto set and multi-task learning

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.

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arXiv2025

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

This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the finite convergence of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second order sufficient condition hold at every minimizer, under the Archimedean property. A useful criterion for detecting the finite convergence is the flat truncation. Our second result is to show that every minimizer of the moment relaxation must have a flat truncation when the relaxation order is big enough, under the above mentioned optimality conditions. These results give connections between nonlinear semidefinite optimization theory and Moment-SOS methods for solving polynomial matrix optimization.

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arXiv2024

Positivstellensätze and Moment problems with Universal Quantifiers

This paper studies Positivstellensätze and moment problems for sets $K$ that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$. Then $K$ is described as the set of all points $x$ such that each $g_j(x, y) \ge 0$ for all $y \in Q$. Fix a finite nonnegative Borel measure $ν$ with $supp(ν) = Q$, and assume it satisfies the multivariate Carleman condition. The first main result of the paper is a Positivstellensatz with universal quantifiers: if a polynomial $f(x)$ is positive on $K$, then it belongs to the quadratic module $QM(g,ν)$ associated to $(g,ν)$, under the archimedeanness assumption on $QM(g,ν)$. Here, $QM(g,ν)$ denotes the quadratic module of polynomials in $x$ that can be represented as \[τ_0(x) + \int τ_1(x,y)g_1(x, y)\, dν(y) + \cdots + \int τ_s(x,y) g_s(x, y)\, dν(y), \] where each $τ_j$ is a sum of squares polynomial. Second, necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in $K$ are given. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set $K$. Finally, applications of these results for solving semi-infinite optimization problems are presented.

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