Jie Wang

Jie Wang
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Jie verified their affiliation via an institutional email.
Verified
Jie verified their affiliation via an institutional email.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Doctor of Mathematics
structured polynomial optimization, large-scale semidefinite programming, quantum information

About

73
Publications
3,991
Reads
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592
Citations
Introduction
polynomial optimization, real algebraic geometry, symbolic computation and their applications.
Additional affiliations
July 2021 - present
July 2019 - June 2021
Laboratory for Analysis and Architecture of Systems
Position
  • PostDoc Position
July 2017 - June 2019
Peking University
Position
  • PostDoc Position
Education
September 2012 - June 2017
Chinese Academy of Sciences
Field of study
  • Mathematics
September 2008 - July 2012

Publications

Publications (73)
Preprint
Full-text available
This paper studies Positivstellens\"atze for a polynomial matrix subject to polynomial matrix inequality constraints with universal quantifiers. We first present a Scherer-Hol-type Positivstellensatz under the Archimedean condition. When the objective is a scalar polynomial, we further provide a sparse Scherer-Hol-type Positivstellensatz in the pre...
Preprint
Full-text available
A polynomial matrix inequality is a statement that a symmetric polynomial matrix is positive semidefinite over a given constraint set. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of polynomial matrix inequalities. This work explores the use of sparsity methods in red...
Article
Full-text available
This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to solve. In addition, we prove that global optimality is achieved when the ranks of the moment matrix and certain s...
Chapter
Full-text available
Barrier certificates, serving as differential invariants that witness system safety, play a crucial role in the verification of cyber-physical systems (CPS). Prevailing computational methods for synthesizing barrier certificates are based on semidefinite programming (SDP) by exploiting Putinar Positivstellensatz . Consequently, these approaches are...
Article
We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is also defined by a PMI. These can be viewed as matrix generalizations of semi-infinite polynomial programs because they involve actually infinitely many PMI constraints in general. Under certain sum-of-...
Article
Full-text available
A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states a...
Preprint
Interpolation-based techniques become popular in recent years, as they can improve the scalability of existing verification techniques due to their inherent modularity and local reasoning capabilities. Synthesizing Craig interpolants is the cornerstone of these techniques. In this paper, we investigate nonlinear Craig interpolant synthesis for two...
Preprint
Full-text available
This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum of squares (SOS) relaxations that is dual to the generalized moment problem approach due to Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: 1) it allows us to cond...
Preprint
Full-text available
We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre's hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensi...
Preprint
Full-text available
This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the run...
Article
In this paper, we develop a dynamical system counterpart to the term sparsity sum-of-squares (TSSOS) algorithm proposed for static polynomial optimization. This allows for computational savings and improved scalability while preserving convergence guarantees when sum-of-squares methods are applied to problems from dynamical systems, including the p...
Article
This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin’s solution to Hilbert’s 17th problem is proved showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Somewhat surprisingly, it is also est...
Preprint
Full-text available
A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalisation quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: energy is minimised over a subset of all possible states a...
Preprint
Full-text available
This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to solve. In addition, we prove that global optimality is achieved when the ranks of the moment matrix and certain s...
Preprint
Full-text available
This note proposes a novel reformulation of complex semidefinite programs (SDPs) as real SDPs by using Lagrange duality. As an application, we present an economical reformulation of complex SDP relaxations of complex polynomial optimization problems as real SDPs and derive some further reductions by exploiting structure of the complex SDP relaxatio...
Preprint
Full-text available
We study a class of polynomial optimization problems with a robust polynomial matrix inequality constraint for which the uncertainty set is defined also by a polynomial matrix inequality (including robust polynomial semidefinite programs as a special case). Under certain SOS-convexity assumptions, we construct a hierarchy of moment-SOS relaxations...
Preprint
Full-text available
We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions. This approach incorporates the augmented Lagrangian method and the Burer-Monteiro factorization, and features the adaptive strategies for updating the factorization size and the penalty parameter. We prove that the present algorithm can...
Preprint
Full-text available
This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin's solution to Hilbert's 17th problem is proved showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Somewhat surprisingly, it is also est...
Article
We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy—a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor’s relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarch...
Article
This work proposes a new moment-SOS hierarchy, called CS-TSSOS , for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [40] while the latte...
Preprint
The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of finite-dimensional convex problems. At each step of the associated hierarchy, one needs to solve a fixed size semidefinite prog...
Article
We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result, such moment relaxations can be solved efficiently by first-order methods that exploit CTP, e.g., the conditional gradie...
Article
The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constr...
Preprint
Full-text available
We study optimal second-order cone representations for weighted geometric means, which turns out to be closely related to minimum mediated sets. Several lower bounds and upper bounds on the size of optimal second-order cone representations are proved. In the case of bivariate weighted geometric means (equivalently, one dimensional mediated sets), w...
Article
Full-text available
In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierar...
Article
Control systems can show robustness to many events, like disturbances and model inaccuracies. It is natural to speculate that they are also robust to sporadic deadline misses when implemented as digital tasks on an embedded platform. This letter proposes a comprehensive stability analysis for control systems subject to deadline misses, leveraging a...
Preprint
Full-text available
In this paper we present term sparsity sum-of-squares (TSSOS) methods applied to several problems from dynamical systems, such as region of attraction, maximum positively invariant sets and global attractors. We combine the TSSOS algorithm of Wang, Magron and Lasserre [SIAM J. Optim., 31(1):30-58, 2021] with existing infinite dimensional linear pro...
Article
Full-text available
We provide a new hierarchy of semidefinite programming relaxations, called NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the exploitation of term sparsity hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits correlat...
Preprint
Full-text available
In this paper, we report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB with up to 24464 buses via the CS-TSSOS hierarchy - a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor's relaxation. Our numerical experiment...
Article
Full-text available
In this paper, we examine the potential of optimization-based computer-assisted proof methods to be applied much more widely than commonly recognized by engineers and computer scientists. More specifically, we contend that there are vast opportunities to derive valuable mathematical results and properties that may be narrow in scope, such as in hig...
Preprint
Full-text available
In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierar...
Preprint
Full-text available
The Julia library TSSOS aims at helping polynomial optimizers to solve large-scale problems with sparse input data. The underlying algorithmic framework is based on exploiting correlative and term sparsity to obtain a new moment-SOS hierarchy involving potentially much smaller positive semidefinite matrices. TSSOS can be applied to numerous problem...
Preprint
Full-text available
Control systems can show robustness to many events, like disturbances and model inaccuracies. It is natural to speculate that they are also robust to alterations of the control signal pattern, due to sporadic late completions (called deadline misses) when implemented as a digital task on an embedded platform. Recent research analysed stability when...
Preprint
Full-text available
We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result such moment relaxations can be solved efficiently by first-order methods that exploit CTP, e.g., the conditional gradien...
Preprint
Full-text available
The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constr...
Preprint
Full-text available
We provide a new hierarchy of semidefinite programming relaxations, called NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the exploitation of term sparsity hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits correlat...
Preprint
Full-text available
This paper focuses on the computation of joint spectral radii (JSR), when the involved matrices are sparse. We provide a sparse variant of the procedure proposed by Parrilo and Jadbabaie, to compute upper bounds of the JSR by means of sum-of-squares (SOS) relaxations. Our resulting iterative algorithm, called SparseJSR, is based on the term sparsit...
Preprint
Full-text available
This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. while the latter was...
Article
Full-text available
We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically...
Preprint
Full-text available
This work is a follow-up and a complement to arXiv:1912.08899 [math.OC] for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose is a new sparse moment-SOS framework based on term-sparsity and chordal extension. By exploiting term-sparsity of the input polynomials we obtain a two-level hierarchy of semidefini...
Preprint
Full-text available
The second-order cone is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using second-order cones, given that they have a strong expressive ability. In this paper, we prov...
Preprint
Full-text available
This paper is concerned with polynomial optimization problems. We show how to exploit term (or monomial) sparsity of the input polynomials to obtain a new converging hierarchy of semidefinite programming relaxations. The novelty (and distinguishing feature) of such relaxations is to involve block-diagonal matrices obtained in an iterative procedure...
Article
Full-text available
In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials admitting at least one positive real zero in terms of their Newton polytopes and combinatorial structure. Moreover,...
Conference Paper
A new sparse SOS decomposition algorithm is proposed based on a new sparsity pattern, called cross sparsity patterns. The new sparsity pattern focuses on the sparsity of terms and thus is different from the well-known correlative sparsity pattern which focuses on the sparsity of variables though the sparse SOS decomposition algorithms based on thes...
Preprint
Full-text available
The second-order cone is a class of simple convex cones and the optimization problem over second-order cones can be solved more efficiently than semidefinite programming. Given that second-order cones have a strong expressive ability, it is interesting to investigate which convex cones admit a representation using second-order cones. In this paper,...
Preprint
Full-text available
We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically...
Preprint
Full-text available
We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically...
Preprint
Full-text available
In this paper we examine the potential of computer-assisted proof methods to be applied much more broadly than commonly recognized. More specifically, we contend that there are vast opportunities to derive useful mathematical results and properties that are extremely narrow in scope, and of practical relevance only to highly-specialized engineering...
Preprint
Full-text available
We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically...
Preprint
Full-text available
In this paper, we consider a new pattern of sparsity for SOS Programming named by cross sparsity patterns. We use matrix decompositions for a class of PSD matrices with chordal sparsity patterns to construct sets of supports for a sparse SOS decomposition. The method is applied to the certificate of the nonnegativity of sparse polynomials and uncon...
Preprint
Full-text available
In this paper, we prove several theorems on systems of polynomial equations with at least one positive real zero, which can be viewed as a generalization of Birch's theorem. Moreover, we give a class of polynomials attaining their minimums, which is useful in polynomial optimization.
Preprint
Full-text available
In this paper, we prove several theorems on systems of polynomial equations with at least (exactly) one positive real zero, which can be viewed as some kind of multivariate Descartes' rule. Moreover, we give a class of polynomials attaining minimums, which is useful in polynomial optimization.
Preprint
In this paper, we prove that every SONC polynomial decomposes into a sum of nonnegative circuit polynomials with the same support. By virtue of this fact, we can decide if a polynomial is SONC through relative entropy programming more efficiently.
Preprint
Full-text available
A new sparse SOS decomposition algorithm is proposed based on a new sparsity pattern, called cross sparsity patterns. The new sparsity pattern focuses on the sparsity of terms and thus is different from the well-known correlative sparsity pattern which focuses on the sparsity of variables though the sparse SOS decomposition algorithms based on thes...
Preprint
Full-text available
Recently, S. Iliman and T. Wolff introduced the concept of sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity of polynomials. For a polynomial with a simplex Newton polytope satisfying certain conditions, it is nonnegative if and only if it is a sum of nonnegative circuit polynomials. In this paper, we generalize t...
Preprint
Full-text available
In this note, we describe a class of systems of linear equations which have nonnegative solutions or positive solutions.
Article
Full-text available
In this paper, we prove a finite basis theorem for radical well-mixed difference ideals generated by binomials. As a consequence, every strictly ascending chain of radical well-mixed difference ideals generated by binomials in a difference polynomial ring is finite, which solves an open problem of difference algebra raised by E. Hrushovski in the b...
Article
Full-text available
In this paper, we prove a finite basis theorem for radical well-mixed difference ideals generated by binomials. As a consequence, every strictly ascending chain of radical well-mixed difference ideals generated by binomials in a difference polynomial ring is finite, which answers a question raised by E. Hrushovski in the binomial case.
Article
Full-text available
In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebra geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine P [...
Preprint
In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebra geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine P[x...
Article
This paper is devoted to studying difference indices of quasi-prime difference algebraic systems. We define the quasi dimension polynomial of a quasi-prime difference algebraic system. Based on this, we give the definition of the difference index of a quasi-prime difference algebraic system through a family of pseudo-Jacobian matrices. Some propert...
Article
Full-text available
This paper is devoted to studying difference indices of quasi-regular difference algebraic systems. We give the definition of difference indices through a family of pseudo-Jacobian matrices. Some properties of difference indices are proved. In particular, a Jacobi-type upper bound for the sum of the order and the difference index is given. As appli...
Article
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-...
Preprint
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-...
Article
In this paper, basic properties of monomial difference ideals are studied. We prove the finitely generated property of well-mixed difference ideals generated by monomial- s. Furthermore, a finite prime decomposition of radical well-mixed monomial difference ideals is given. As a consequence, we prove that every strictly ascending chain of radical w...

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