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1. Secretory expression of recombinant small laccase genes in Gram-positive bacteria

Author

Silja Välimets, Patricia Pedetti, Ludovika Jessica Virginia, Mai Ngoc Hoang, Michael Sauer, and Clemens Peterbauer

Subjects
Streptomyces lividans, Bacillus subtilis, Heterologous expression, Bacterial protein secretion, Microbiology, QR1-502
Abstract

Abstract Background Laccases are multicopper enzymes that oxidize a wide range of aromatic and non-aromatic compounds in the presence of oxygen. The majority of industrially relevant laccases are derived from fungi and are produced in eukaryotic expression systems such as Pichia pastoris and Saccharomyces cerevisiae. Bacterial laccases for research purposes are mostly produced intracellularly in Escherichia coli, but secretory expression systems are needed for future applications. Bacterial laccases from Streptomyces spp. are of interest for potential industrial applications because of their lignin degrading activities. Results In this study, we expressed small laccases genes from Streptomyces coelicolor, Streptomyces viridosporus and Amycolatopsis 75iv2 with their native signal sequences in Gram-positive Bacillus subtilis and Streptomyces lividans host organisms. The extracellular activities of ScLac, SvLac and AmLac expressed in S. lividans reached 1950 ± 99 U/l, 812 ± 57 U/l and 12 ± 1 U/l in the presence of copper supplementation. The secretion of the small laccases was irrespective of the copper supplementation; however, activities upon reconstitution with copper after expression were significantly lower, indicating the importance of copper during laccase production. The production of small laccases in B. subtilis resulted in extracellular activity that was significantly lower than in S. lividans. Unexpectedly, AmLac and ScLac were secreted without their native signal sequences in B. subtilis, indicating that B. subtilis secretes some heterologous proteins via an unknown pathway. Conclusions Small laccases from S. coelicolor, S. viridosporus and Amycolatopsis 75iv2 were secreted in both Gram-positive expression hosts B. subtilis and S. lividans, but the extracellular activities were significantly higher in the latter.

Published
2023
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2. Network analysis of the sick-pig commodity chain in Northern Vietnam: Risk of disease dissemination

Author

Mai Ngoc Hoang, Phuong Thi Nguyen, Hanh Quang Han, Tien Dinh Nguyen, Ton Dinh Vu, and Marisa Peyre

Subjects
Swine, food supply chain, risk assessment, animal diseases, fraud, Vietnam, Animal culture, SF1-1100
Abstract

A study was carried out between January and July 2016 to investigate the sickpig trade network which is believed to have been operating in Northern Vietnam for decades. From early 2015, many warnings about the existence of this illegal network have been published in local newspapers, causing great concern among consumers and authorities. The aim of this study was to investigate the commodity chain to identify the trade network, evaluate the risk of disease dissemination, and propose preventive solutions. The results showed that the sick-pig commodity chain included farmers, middlemen, slaughterhouses and consumers in the three studied districts (Van Lam, Van Giang and My Hao, in Hung Yen Province). The qualitative risk assessment determined that middlemen and slaughterhouses were the main actors in this uncontrolled chain. Appropriate administrative regulations, regardless of the importance of these actors, would be crucial to limit negative effects of sick-pig products. Furthermore, proper intervention would remove threats to other farms and long-term effects on human health.

Published
2020
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7. Double-Labeling Method for Visualization and Quantification of Membrane-Associated Proteins in Lactococcus lactis

Author

Peterbauer, Mai Ngoc Hoang and Clemens

Subjects
L. lactis, biarsenical labeling, nickel NTA, visualization, quantification, membrane proteins, post-translational pathway
Abstract

Lactococcus lactis displaying recombinant proteins on its surface can be used as a potential drug delivery vector in prophylactic medication and therapeutic treatments for many diseases. These applications enable live-cell mucosal and oral administration, providing painless, needle-free solutions and triggering robust immune response at the site of pathogen entry. Immunization requires quantitative control of antigens and, ideally, a complete understanding of the bacterial processing mechanism applied to the target proteins. In this study, we propose a double-labeling method based on a conjugated dye specific for a recombinantly introduced polyhistidine tag (to visualize surface-exposed proteins) and a membrane-permeable dye specific for a tetra-cysteine tag (to visualize cytoplasmic proteins), combined with a method to block the labeling of surface-exposed tetra-cysteine tags, to clearly obtain location-specific signals of the two dyes. This allows simultaneous detection and quantification of targeted proteins on the cell surface and in the cytoplasm. Using this method, we were able to detect full-length peptide chains for the model proteins HtrA and BmpA in L. lactis, which are associated with the cell membrane by two different attachment modes, and thus confirm that membrane-associated proteins in L. lactis are secreted using the Sec-dependent post-translational pathway. We were able to quantitatively follow cytoplasmic protein production and accumulation and subsequent export and surface attachment, which provides a convenient tool for monitoring these processes for cell surface display applications.

Published
2023
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8. A Nichtnegativstellensatz on singular varieties under the denseness of regular loci

Author

Mai, Ngoc Hoang Anh

Subjects
Mathematics - Algebraic Geometry, Optimization and Control (math.OC), FOS: Mathematics, Algebraic Geometry (math.AG), Mathematics - Optimization and Control
Abstract

Let $V$ be a real algebraic variety with singularities and $f$ be a real polynomial non-negative on $V$. Assume that the regular locus of $V$ is dense in $V$ by the usual topology. Using Hironaka's resolution of singularities and Demmel--Nie--Powers' Nichtnegativstellensatz, we obtain a sum of squares-based representation that characterizes the non-negativity of $f$ on $V$. This representation allows us to build up exact semidefinite relaxations for polynomial optimization problems whose optimal solutions are possibly singularities of the constraint sets., 21 pages, 1 table, fixed typos

Published
2022

9. Optimisation polynomiale effective via des certificats de positivité avec ou sans dénominateurs

Author

Mai, Ngoc Hoang Anh, Equipe Polynomial OPtimization (LAAS-POP), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Université Paul Sabatier - Toulouse III, Victor Magron, Jean-Bernard Lasserre, Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, UT3 : Université Toulouse 3 Paul Sabatier, Victor MAGRON, and Jean-Bernard LASSERRE (co-encadrant)

Subjects
Putinar-Vasilescu's Positivstellensatz, Hiérarchies moment/sommes-de-carrés, Taux de convergence, Polynomial optimization, Positivstellensatz de Polya, Putinar's Positivstellensatz, Programmation semidéfinie, constant trace property, Propriété de la trace constante, Positivstellensatz de Putinar, Polya's Positivstellensatz, Hiérarchie Moment-SOS, Reznick's Positivstellensatz, correlative and term sparsity, convergence rate, Positivstellensatz de Reznick, Programmation semi-définie, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Positivstellensatz de Putinar-Vasilescu, Optimisation polynomiale, Sparsité corrélative et termique, Semidefinite programming, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Moment-SOS hierarchy, Moment/sums-of-squares hierarchy
Abstract

National audience; Positivity certificates or Positivstellens"atze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set dollarSdollar can be written as a linear weighted combination of the polynomials describing dollarSdollar, under a certain condition on dollarSdollar slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on dollarSdollar, and therefore this alternative description provides a certificate of positivity on dollarSdollar. Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite programming to solve polynomial optimization problems (POPs) that arise in various real-life applications, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the emph{Moment-SOS hierarchy}. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators.; Les certificats de positivité ou Positivstellens"atze fournissent des représentations de polynômes positifs sur des ensembles semialgébriques de basiques, c'est-à-dire des ensembles définis par un nombre fini d'inégalités polynomiales. Le célèbre Positivstellensatz de Putinar stipule que tout polynôme positif sur un ensemble semialgébrique basique fermé dollarSdollar peut être écrit comme une combinaison pondérée linéaire des polynômes décrivant dollarSdollar, sous une certaine condition sur dollarSdollar légèrement plus forte que la compacité. Lorsqu'il est écrit comme ceci, il devient évident que le polynôme est positif sur dollarSdollar, et donc cette description alternative fournit un certificat de positivité sur dollarSdollar. De plus, comme les poids polynomiaux impliqués dans le Positivstellensatz de Putinar sont des sommes de carrés (SOS), de tels certificats de positivité permettent de concevoir des relaxations convexes basées sur la programmation semidéfinie pour résoudre des problèmes d'optimisation polynomiale (POP) qui surviennent dans diverses applications réelles, par exemple dans la gestion des réseaux d'énergie et l'apprentissage automatique pour n'en citer que quelques unes. Développée à l'origine par Lasserre, la hiérarchie des relaxations semidéfinies basée sur le Positivstellensatz de Putinar est appelée la emph{hiérarchie Moment-SOS}. Dans cette thèse, nous proposons des méthodes d'optimisation polynomiale basées sur des certificats de positivité impliquant des poids SOS spécifiques, sans ou avec dénominateurs.

Published
2022

10. On the exactness for polynomial optimization strengthened with Fritz John conditions

Author

Mai, Ngoc Hoang Anh

Subjects
Mathematics - Algebraic Geometry, Optimization and Control (math.OC), ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, Algebraic Geometry (math.AG), Mathematics - Optimization and Control
Abstract

We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions. Based on each representation, we obtain semidefinite programs which return a sequence of values that finitely converges to the optimal value of a given polynomial optimization problem under generic assumption. Consequently, we can compute exactly the minimal value of any polynomial over a basic convex semi-algebraic set which is defined by the inequalities of concave polynomials., It has been included in arXiv:2205.04254

Published
2022

11. A Sparse Version of Reznick's Positivstellensatz.

Author

Mai, Ngoc Hoang Anh, Magron, Victor, and Lasserre, Jean

Subjects
SEMIALGEBRAIC sets, CYBER physical systems, SUM of squares, ARTIFICIAL intelligence, POLYNOMIALS
Abstract

If f is a positive definite form, Reznick's Positivstellensatz states that there exists k ∈ N such that ‖ x ‖ 2 2 k f is a sum of squares of polynomials. Assuming that f can be written as a sum of forms ∑ l = 1 p f l , where each fl depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick's Positivstellensatz. Namely, there exists k ∈ N such that f = ∑ l = 1 p σ l / H l k , where σl is a sum of squares of polynomials, Hl is a uniform polynomial denominator, and both polynomials σ l , H l involve the same variables as fl for each l = 1 , ... , p. In other words, the sparsity pattern of f is also reflected in this sparse version of Reznick's certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly noncompact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate from Putinar and Vasilescu. Funding: V. Magron was supported by the Fondation Mathématique Jacques Hadamard Programme Gaspard Monge for Optimization (Exact Polynomial Optimization with Innovative Certifed Schemes project). This work has benefited from the Tremplin European Research Council Starting [Grant ANR-18-ERC2-0004-01] (Tremplin-Certified Optimization for Cyber-Physical Systems project), the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Actions [Grant 813211] (Polynomial Optimization, Efficiency through Moments and Algebra) as well as from the Artificial Intelligence Interdisciplinary Institute Artificial and Natural Intelligence Toulouse Institute funding, through the French "Investing for the Future Programme d'Investissements d'Avenir3" program [Grant n°ANR-19-PI3A-0004]. [ABSTRACT FROM AUTHOR]

Published
2023
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12. Tractable semidefinite bounds of positive maximal singular values

Author

Magron, Victor, Mai, Ngoc Hoang Anh, Ebihara, Yoshio, Waki, Hayato, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Equipe Polynomial OPtimization (LAAS-POP), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Graduate School of Information Science and Electrical Engineering, Kyushu University, Institute of Mathematics for Industry, Kyushu University, This research has been supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme., ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-18-ERC2-0004,COPS,Optimisation garantie pour la vérification des systèmes cyber-physiques(2018), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019), Magron, Victor, Artificial and Natural Intelligence Toulouse Institute - - ANITI2019 - ANR-19-P3IA-0004 - P3IA - VALID, TREMPLIN-ERC - Optimisation garantie pour la vérification des systèmes cyber-physiques - - COPS2018 - ANR-18-ERC2-0004 - TERC - VALID, Centre International de Mathématiques et d'Informatique (de Toulouse) - - CIMI2011 - ANR-11-LABX-0040 - LABX - VALID, and Polynomial Optimization, Efficiency through Moments and Algebra - POEMA - - H20202019-01-01 - 2022-12-31 - 813211 - VALID

Subjects
Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract

We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative orthant. We provide a hierarchy of tractable semidefinite relaxations to approximate the value of the latter polynomial optimization problem as closely as desired. This hierarchy is based on an extension of P\'olya's representation theorem. Doing so, positive polynomials can be decomposed as weighted sums of squares of $s$-nomials, where $s$ can be a priori fixed ($s=1$ corresponds to monomials, $s=2$ corresponds to binomials, etc.). This in turn allows us to control the size of the resulting semidefinite relaxations., Comment: 4 pages, 1 table, submitted to MTNS as extended abstract

Published
2022

13. Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant

Author

Mai, Ngoc Hoang Anh, Magron, Victor, Lasserre, Jean-Bernard, Toh, Kim-Chuan, Magron, Victor, Artificial and Natural Intelligence Toulouse Institute - - ANITI2019 - ANR-19-P3IA-0004 - P3IA - VALID, TREMPLIN-ERC - Optimisation garantie pour la vérification des systèmes cyber-physiques - - COPS2018 - ANR-18-ERC2-0004 - TERC - VALID, Centre International de Mathématiques et d'Informatique (de Toulouse) - - CIMI2011 - ANR-11-LABX-0040 - LABX - VALID, Polynomial Optimization, Efficiency through Moments and Algebra - POEMA - - H20202019-01-01 - 2022-12-31 - 813211 - VALID, Equipe Polynomial OPtimization (LAAS-POP), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), National University of Singapore (NUS), This research is part of the programme DesCartes and is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme., ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-18-ERC2-0004,COPS,Optimisation garantie pour la vérification des systèmes cyber-physiques(2018), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), and H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019)

Subjects
FOS: Computer and information sciences, Computer Science - Machine Learning, Mathematics - Algebraic Geometry, Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Algebraic Geometry (math.AG), Machine Learning (cs.LG)
Abstract

We consider polynomial optimization problems (POP) on a semialgebraic set contained in the nonnegative orthant (every POP on a compact set can be put in this format by a simple translation of the origin). Such a POP can be converted to an equivalent POP by squaring each variable. Using even symmetry and the concept of factor width, we propose a hierarchy of semidefinite relaxations based on the extension of P\'olya's Positivstellensatz by Dickinson-Povh. As its distinguishing and crucial feature, the maximal matrix size of each resulting semidefinite relaxation can be chosen arbitrarily and in addition, we prove that the sequence of values returned by the new hierarchy converges to the optimal value of the original POP at the rate $O(\varepsilon^{-c})$ if the semialgebraic set has nonempty interior. When applied to (i) robustness certification of multi-layer neural networks and (ii) computation of positive maximal singular values, our method based on P\'olya's Positivstellensatz provides better bounds and runs several hundred times faster than the standard Moment-SOS hierarchy., Comment: 39 pages, 15 tables

Published
2022

14. Exploiting Constant Trace Property in Large-scale Polynomial Optimization.

Author

MAI, NGOC HOANG ANH, LASSERRE, J. B., MAGRON, VICTOR, and WANG, JIE

Subjects
*POLYNOMIALS, *SCALABILITY
Abstract

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 gradient-based augmented Lagrangian method. We also extend this CTP-exploiting framework to large-scale POPs with different sparsity structures. The efficiency and scalability of our framework are illustrated on some moment relaxations for various randomly generated POPs, especially second-order moment relaxations for quadratically constrained quadratic programs. [ABSTRACT FROM AUTHOR]

Published
2022
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15. CS-TSSOS: Correlative and Term Sparsity for Large-Scale Polynomial Optimization.

Author

Wang, Jie, Magron, Victor, Lasserre, J. B., and Mai, Ngoc Hoang Anh

Subjects
POLYNOMIALS, ELECTRICAL load, SEMIDEFINITE programming
Abstract

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 latter was initially proposed by Wang et al. [42] and later exploited in the TSSOS hierarchy [46, 47]. In doing so we obtain CS-TSSOS—a two-level hierarchy of semidefinite programming relaxations with (i) the crucial property to involve blocks of SDP matrices and (ii) the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints. [ABSTRACT FROM AUTHOR]

Published
2022
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16. A hierarchy of spectral relaxations for polynomial optimization

Author

Mai, Ngoc Hoang Anh, Magron, Victor, Lasserre, Jean-Bernard, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-18-ERC2-0004,COPS,Optimisation garantie pour la vérification des systèmes cyber-physiques(2018), European Project: 813211,H2020,POEMA(2019), Equipe Polynomial OPtimization (LAAS-POP), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019), Magron, Victor, Artificial and Natural Intelligence Toulouse Institute - - ANITI2019 - ANR-19-P3IA-0004 - P3IA - VALID, TREMPLIN-ERC - Optimisation garantie pour la vérification des systèmes cyber-physiques - - COPS2018 - ANR-18-ERC2-0004 - TERC - VALID, and Polynomial Optimization, Efficiency through Moments and Algebra - POEMA - - H20202019-01-01 - 2022-12-31 - 813211 - VALID

Subjects
Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Software, Theoretical Computer Science
Abstract

We show that (i) any constrained polynomial optimization problem (POP) has an equivalent formulation on a variety contained in an Euclidean sphere and (ii) the resulting semidefinite relaxations in the moment-SOS hierarchy have the constant trace property (CTP) for the involved matrices. We then exploit the CTP to avoid solving the semidefinite relaxations via interior-point methods and rather use ad-hoc spectral methods that minimize the largest eigenvalue of a matrix pencil. Convergence to the optimal value of the semidefinite relaxation is guaranteed. As a result we obtain a hierarchy of nonsmooth "spectral relaxations" of the initial POP. Efficiency and robustness of this spectral hierarchy is tested against several equality constrained POPs on a sphere as well as on a sample of randomly generated quadratically constrained quadratic problems (QCQPs)., 38 pages, 6 figures, 11 tables

Published
2020

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