[ { "id": "authors:739bj-bet81", "collection": "authors", "collection_id": "739bj-bet81", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121004-133456681", "type": "article", "title": "Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting", "author": [ { "family_name": "Saunderson", "given_name": "J.", "orcid": "0000-0002-5456-0180", "clpid": "Saunderson-James" }, { "family_name": "Chandrasekaran", "given_name": "V.", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "P. A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "A. S.", "clpid": "Willsky-A-S" } ], "abstract": "In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix X formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose X into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points v1, v2, \u2026 , vn \u2208 R^k (where n > k) determine whether there is a centered ellipsoid passing exactly through all of the points. \n\n\nWe show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace U that ensures any positive semidefinite matrix L with column space U can be recovered from D+L for any diagonal matrix D using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.", "doi": "10.1137/120872516", "issn": "0895-4798", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal on Matrix Analysis and Applications", "publication_date": "2012-12-19", "series_number": "4", "volume": "33", "issue": "4", "pages": "1395-1416" }, { "id": "authors:w276e-jcy72", "collection": "authors", "collection_id": "w276e-jcy72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121004-152325296", "type": "article", "title": "The Convex Geometry of Linear Inverse Problems", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Recht", "given_name": "Benjamin", "clpid": "Recht-B" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered includes those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases from many technical fields such as sparse vectors (signal processing, statistics) and low-rank matrices (control, statistics), as well as several others including sums of a few permutation matrices (ranked elections, multiobject tracking), low-rank tensors (computer vision, neuroscience), orthogonal matrices (machine learning), and atomic measures (system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.", "doi": "10.1007/s10208-012-9135-7", "issn": "1615-3375", "publisher": "Springer", "publication": "Foundations of Computational Mathematics", "publication_date": "2012-12", "series_number": "6", "volume": "12", "issue": "6", "pages": "805-849" }, { "id": "authors:w826x-t7e72", "collection": "authors", "collection_id": "w826x-t7e72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130110-100237034", "type": "article", "title": "Convex Graph Invariants", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "The structural properties of graphs are usually characterized in terms of invariants, which\nare functions of graphs that do not depend on the labeling of the nodes. In this paper\nwe study convex graph invariants, which are graph invariants that are convex functions\nof the adjacency matrix of a graph. Some examples include functions of a graph such as\nthe maximum degree, the MAXCUT value (and its semidefinite relaxation), and spectral\ninvariants such as the sum of the k largest eigenvalues. Such functions can be used to\nconstruct convex sets that impose various structural constraints on graphs and thus provide\na unified framework for solving a number of interesting graph problems via convex\noptimization. We give a representation of all convex graph invariants in terms of certain\nelementary invariants, and we describe methods to compute or approximate convex graph\ninvariants tractably. We discuss the interesting subclass of spectral invariants, and also\ncompare convex and nonconvex invariants. Finally, we use convex graph invariants to provide\nefficient convex programming solutions to graph problems such as the deconvolution of\nthe composition of two graphs into the individual components, hypothesis testing between\ngraph families, and the generation of graphs with certain desired structural properties.", "doi": "10.1137/100816900", "issn": "0036-1445", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Review", "publication_date": "2012-08-07", "series_number": "3", "volume": "54", "issue": "3", "pages": "513-541" }, { "id": "authors:gqqee-ncb30", "collection": "authors", "collection_id": "gqqee-ncb30", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130205-141004858", "type": "article", "title": "Rejoinder: Latent variable graphical model selection via convex optimization", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "We thank all the discussants for their careful reading of our paper, and for their insightful critiques. We would also like to thank the editors for organizing this discussion. Our paper contributes to the area of high-dimensional statistics which has received much attention over the past several years across the statistics, machine learning and signal processing communities. In this rejoinder we clarify and comment on some of the points raised in the discussions. Finally, we also remark on some interesting challenges that lie ahead in latent variable modeling.", "doi": "10.1214/12-AOS1020", "issn": "0090-5364", "publisher": "Institute of Mathematical Statistics", "publication": "Annals of Statistics", "publication_date": "2012-08", "series_number": "4", "volume": "40", "issue": "4", "pages": "2005-2013" }, { "id": "authors:a8hmk-wjd18", "collection": "authors", "collection_id": "a8hmk-wjd18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130207-085454891", "type": "article", "title": "Latent Variable Graphical Model Selection via Convex Optimization", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "Suppose we have samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent and observed variables. Is it possible to discover the number of hidden components, and to learn a statistical model over the entire collection of variables? We address this question in the setting in which the latent and observed variables are jointly Gaussian, with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model. As a first step we give natural conditions under which such latent-variable Gaussian graphical models are identifiable given marginal statistics of only the observed variables. Essentially these conditions require that the conditional graphical model among the observed variables is sparse, while the effect of the latent variables is \"spread out\" over most of the observed variables. Next we propose a tractable convex program based on regularized maximum-likelihood for model selection in this latent-variable setting; the regularizer uses both the \u2113_1 norm and the nuclear norm. Our modeling framework can be viewed as a combination of dimensionality reduction (to identify latent variables) and graphical modeling (to capture remaining statistical structure not attributable to the latent variables), and it consistently estimates both the number of hidden components and the conditional graphical model structure among the observed variables. These results are applicable in the high-dimensional setting in which the number of latent/observed variables grows with the number of samples of the observed variables. The geometric properties of the algebraic varieties of sparse matrices and of low-rank matrices play an important role in our analysis.", "doi": "10.1214/11-AOS949", "issn": "0090-5364", "publisher": "Institute of Mathematical Statistics", "publication": "Annals of Statistics", "publication_date": "2012-08", "series_number": "4", "volume": "40", "issue": "4", "pages": "1935-1967" }, { "id": "authors:0t84c-nmc65", "collection": "authors", "collection_id": "0t84c-nmc65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121008-095909823", "type": "article", "title": "Rank-Sparsity Incoherence for Matrix Decomposition", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Sanghavi", "given_name": "Sujay", "orcid": "0000-0003-0754-9154", "clpid": "Sanghavi-S" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components by minimizing a linear combination of the \u2113_1 norm and the nuclear norm of the components. We develop a notion of rank-sparsity incoherence, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and we use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.", "doi": "10.1137/090761793", "issn": "1052-6234", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal of Optimization", "publication_date": "2011-06-30", "series_number": "2", "volume": "21", "issue": "2", "pages": "572-596" }, { "id": "authors:8ea39-50s38", "collection": "authors", "collection_id": "8ea39-50s38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121008-082643123", "type": "book_section", "title": "Sparse and low-rank matrix decompositions", "book_title": "Forty-Seventh annual Allerton Conference on Communication, Control, and Computing", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Sanghavi", "given_name": "Sujay", "orcid": "0000-0003-0754-9154", "clpid": "Sanghavi-S" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "We consider the following fundamental problem: given a matrix that is the sum of an unknown sparse matrix and an unknown low-rank matrix, is it possible to exactly recover the two components? Such a capability enables a considerable number of applications, but the goal is both ill-posed and NP-hard in general. In this paper we develop (a) a new uncertainty principle for matrices, and (b) a simple method for exact decomposition based on convex optimization. Our uncertainty principle is a quantification of the notion that a matrix cannot be sparse while having diffuse row/column spaces. It characterizes when the decomposition problem is ill-posed, and forms the basis for our decomposition method and its analysis. We provide deterministic conditions - on the sparse and low-rank components - under which our method guarantees exact recovery.", "doi": "10.1109/ALLERTON.2009.5394889", "isbn": "978-1-4244-5870-7", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2009-09", "pages": "962-967" }, { "id": "authors:fej1n-59s40", "collection": "authors", "collection_id": "fej1n-59s40", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170411-162244274", "type": "book_section", "title": "Compressed sensing and robust recovery of low rank matrices", "book_title": "42nd Asilomar Conference on Signals, Systems and Computers", "author": [ { "family_name": "Fazel", "given_name": "M.", "clpid": "Fazel-M" }, { "family_name": "Cand\u00e8s", "given_name": "E.", "orcid": "0000-0001-9234-924X", "clpid": "Cand\u00e8s-E-J" }, { "family_name": "Recht", "given_name": "B.", "clpid": "Recht-B" }, { "family_name": "Parrilo", "given_name": "P.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "In this paper, we focus on compressed sensing and recovery schemes for low-rank matrices, asking under what conditions a low-rank matrix can be sensed and recovered from incomplete, inaccurate, and noisy observations. We consider three schemes, one based on a certain Restricted Isometry Property and two based on directly sensing the row and column space of the matrix. We study their properties in terms of exact recovery in the ideal case, and robustness issues for approximately low-rank matrices and for noisy measurements.", "doi": "10.1109/ACSSC.2008.5074571", "isbn": "978-1-4244-2940-0", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2008-10", "pages": "1043-1047" }, { "id": "authors:zq2bs-k7j84", "collection": "authors", "collection_id": "zq2bs-k7j84", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110602-133842599", "type": "article", "title": "Symmetry Analysis of Reversible Markov Chains", "author": [ { "family_name": "Boyd", "given_name": "Stephen", "clpid": "Boyd-S" }, { "family_name": "Diaconis", "given_name": "Persi", "clpid": "Diaconis-P" }, { "family_name": "Parrilo", "given_name": "Pablo", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Xiao", "given_name": "Lin", "clpid": "Xiao-Lin" } ], "abstract": "We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a self-adjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a max-degree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.", "doi": "10.1080/15427951.2005.10129100", "issn": "1944-9488", "publisher": "Taylor & Francis", "publication": "Internet Mathematics", "publication_date": "2008", "series_number": "1", "volume": "2", "issue": "1", "pages": "31-71" }, { "id": "authors:tgzvm-cq850", "collection": "authors", "collection_id": "tgzvm-cq850", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150331-150443251", "type": "book_section", "title": "Finding quantum algorithms via convex optimization", "book_title": "2007 European Control Conference (ECC)", "author": [ { "family_name": "Childs", "given_name": "Andrew M.", "clpid": "Childs-A-M" }, { "family_name": "Landahl", "given_name": "Andrew J.", "clpid": "Landahl-A-J" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "In this paper we describe how to use convex optimization to design quantum algorithms for certain computational tasks. In particular, we consider the ordered search problem, where it is desired to find a specific item in an ordered\nlist of N items. While the best classical algorithm for this\nproblem uses log_2 N queries to the list, a quantum computer\ncan solve this problem much faster. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find quantum algorithms using 4 log_(605) N \u2248 0.433 log_2 N queries, which improves upon the previously best known exact algorithm.", "isbn": "978-3-9524173-8-6", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2007-07", "pages": "854-859" }, { "id": "authors:10hdq-jb858", "collection": "authors", "collection_id": "10hdq-jb858", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20101013-121426898", "type": "book_section", "title": "Solving Commutative Relaxations of Word Problems", "book_title": "46th IEEE Conference on Decision and Control", "author": [ { "family_name": "Tarraf", "given_name": "Danielle C.", "clpid": "Tarraf-D-C" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the\ncommutative word problem, referred to as the \"Zero-to-All\nreachability\" problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for Zero-to-All reachability. We conclude with a set of illustrative examples.", "doi": "10.1109/CDC.2007.4435013", "isbn": "978-1-4244-1497-0", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2007", "pages": "5575-5580" }, { "id": "authors:601jw-f4f86", "collection": "authors", "collection_id": "601jw-f4f86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DOHpra05", "type": "article", "title": "Detecting multipartite entanglement", "author": [ { "family_name": "Doherty", "given_name": "Andrew C.", "clpid": "Doherty-A-C" }, { "family_name": "Spedalieri", "given_name": "Federico M.", "clpid": "Spedalieri-F-M" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "We discuss the problem of determining whether the state of several quantum mechanical subsystems is entangled. As in previous work on two subsystems we introduce a procedure for checking separability that is based on finding state extensions with appropriate properties and may be implemented as a semidefinite program. The main result of this work is to show that there is a series of tests of this kind such that if a multiparty state is entangled this will eventually be detected by one of the tests. The procedure also provides a means of constructing entanglement witnesses that could in principle be measured in order to demonstrate that the state is entangled.", "doi": "10.1103/PhysRevA.71.032333", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "2005-03-01", "series_number": "3", "volume": "71", "issue": "3", "pages": "Art. No. 032333" }, { "id": "authors:5psd3-2y003", "collection": "authors", "collection_id": "5psd3-2y003", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200330-073159726", "type": "book_section", "title": "SOSTOOLS and Its Control Applications", "book_title": "Positive Polynomials in Control", "author": [ { "family_name": "Prajna", "given_name": "Stephen", "clpid": "Prajna-S" }, { "family_name": "Papachristodoulou", "given_name": "Antonis", "clpid": "Papachristodoulou-A" }, { "family_name": "Seiler", "given_name": "Peter", "clpid": "Seiler-P" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "contributor": [ { "family_name": "Henrion", "given_name": "Didier", "clpid": "Henrion-D" }, { "family_name": "Garulli", "given_name": "Andrea", "clpid": "Garulli-A" } ], "abstract": "In this chapter we present SOSTOOLS, a third-party MATLAB toolbox for formulating and solving sum of squares optimization problems. Sum of squares optimization forms a basis for formulating convex relaxations to computationally hard problems such as some that appear in systems and control. Currently, sum of squares programs are solved by casting them as semidefinite programs, which can in turn be solved using interior-point based numerical methods. SOSTOOLS helps this translation in such a way that the underlying computations are abstracted from the user. Here we give a brief description of the toolbox, its features and capabilities (with emphasis on the recently added ones), as well as show how it can be applied to solving problems of interest in systems and control.", "doi": "10.1007/10997703_14", "isbn": "978-3-540-23948-2", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "2005", "pages": "273-292" }, { "id": "authors:mz2xh-b6104", "collection": "authors", "collection_id": "mz2xh-b6104", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111012-135901478", "type": "book_section", "title": "SOSTOOLS: control applications and new developments", "book_title": "2004 IEEE International Symposium on Computer-Aided Control System Design", "author": [ { "family_name": "Prajna", "given_name": "Stephen", "clpid": "Prajna-S" }, { "family_name": "Papachristodoulou", "given_name": "Antonis", "clpid": "Papachristodoulou-A" }, { "family_name": "Seiler", "given_name": "Peter", "clpid": "Seiler-P" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "In this paper we describe recent developments and control applications of SOSTOOLS, a free third-party MATLAB toolbox for formulating and solving sum of squares programs.", "doi": "10.1109/CACSD.2004.1393895", "isbn": "0-7803-8636-1", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2004-09", "pages": "315-320" }, { "id": "authors:skhrj-3tg49", "collection": "authors", "collection_id": "skhrj-3tg49", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110824-111502553", "type": "book_section", "title": "New developments in sum of squares optimization and SOSTOOLS", "book_title": "Proceedings of the 2004 American Control Conference", "author": [ { "family_name": "Prajna", "given_name": "Stephen", "clpid": "Prajna-S" }, { "family_name": "Papachristodoulou", "given_name": "Antonis", "clpid": "Papachristodoulou-A" }, { "family_name": "Seiler", "given_name": "Peter", "clpid": "Seiler-P" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "We describe the latest additions to SOSTOOLS, a freely available MATLAB toolbox for formulating and solving sum of squares programs. Among the many improvements, there are native polynomial objects, structure-exploiting techniques for sparse and structured polynomials, new customized functions, and support for alternative SDP solvers. We sketch some of the theory behind the new improvements, and illustrate the new commands using control-oriented examples.", "isbn": "0-7803-8335-4", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2004-06", "pages": "5606-5611" }, { "id": "authors:5vny8-wby74", "collection": "authors", "collection_id": "5vny8-wby74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PRAieeetac04", "type": "article", "title": "Nonlinear control synthesis by convex optimization", "author": [ { "family_name": "Prajna", "given_name": "Stephen", "clpid": "Prajna-S" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Rantzer", "given_name": "Anders", "clpid": "Rantzer-A" } ], "abstract": "A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used.", "doi": "10.1109/TAC.2003.823000", "issn": "0018-9286", "publisher": "IEEE", "publication": "IEEE Transactions on Automatic Control", "publication_date": "2004-02-01", "series_number": "2", "volume": "49", "issue": "2", "pages": "310-314" }, { "id": "authors:hvjax-bwp85", "collection": "authors", "collection_id": "hvjax-bwp85", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DOHpra04", "type": "article", "title": "Complete family of separability criteria", "author": [ { "family_name": "Doherty", "given_name": "Andrew C.", "clpid": "Doherty-A-C" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Spedalieri", "given_name": "Federico M.", "clpid": "Spedalieri-F-M" } ], "abstract": "We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state rho to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.", "doi": "10.1103/PhysRevA.69.022308", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "2004-02-01", "series_number": "2", "volume": "69", "issue": "2", "pages": "Art. No. 022308" }, { "id": "authors:v84v3-3sg67", "collection": "authors", "collection_id": "v84v3-3sg67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200324-151107160", "type": "article", "title": "Semidefinite programming relaxations for semialgebraic problems", "author": [ { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.", "doi": "10.1007/s10107-003-0387-5", "issn": "0025-5610", "publisher": "Springer", "publication": "Mathematical Programming", "publication_date": "2003-05", "series_number": "2", "volume": "96", "issue": "2", "pages": "293-320" }, { "id": "authors:ctpm8-qpe04", "collection": "authors", "collection_id": "ctpm8-qpe04", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111109-140347153", "type": "book_section", "title": "Introducing SOSTOOLS: A General Purpose Sum of Squares Programming Solver", "book_title": "Proceedings of the 41st IEEE Conference on Decision and Control", "author": [ { "family_name": "Prajna", "given_name": "Stephen", "clpid": "Prajna-S" }, { "family_name": "Papachristodoulou", "given_name": "Antonis", "clpid": "Papachristodoulou-A" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "SOSTOOLS is a MATLAB toolbox for constructing and solving sum of squares programs. It can be used in combination with semidefinite programming software, such as SeDuMi, to solve many continuous and combinatorial optimization problems, as well as various control-related problems. The paper provides an overview on sum of squares programming, describes the primary features of SOSTOOLS, and shows how SOSTOOLS is used to solve sum of squares programs. Some applications from different areas are presented to show the wide applicability of sum of squares programming in general and SOSTOOLS in particular.", "doi": "10.1109/CDC.2002.1184594", "isbn": "0-7803-7516-5", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2002-12", "pages": "741-746" }, { "id": "authors:xxk2w-hj530", "collection": "authors", "collection_id": "xxk2w-hj530", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:VARcdc01", "type": "book_section", "title": "Fast algorithms for solving H\u221e-norm minimization problems", "book_title": "Proceedings of the 40th IEEE Conference on Decision and Control, December 4-7, 2001, Hyatt Regency Grand Cypress, Orlando, Florida, USA", "author": [ { "family_name": "Vargas", "given_name": "Andras", "clpid": "Vargas-A" }, { "family_name": "Parrilo", "given_name": "Pablo", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "We propose an efficient computational approach to minimize the H \u221e-norm of a transfer-function matrix depending affinely on a set of free parameters. The minimization problem, formulated as a semi-infinite convex programming problem, is solved via a relaxation approach over a finite set of frequency values. In this way, a significant speed up is achieved by avoiding the solution of high order LMIs resulting by equivalently formulating the minimization problem as a high dimensional semidefinite programming problem. Numerical results illustrate the superiority of proposed approach over LMIs based techniques in solving zero order H\u221e-norm approximation problems.", "doi": "10.1109/.2001.980109", "isbn": "0-7803-7061-9", "publisher": "IEEE", "place_of_publication": "Los Alamitos, CA", "publication_date": "2002-08-07", "pages": "261-266" }, { "id": "authors:kcfkx-9fq18", "collection": "authors", "collection_id": "kcfkx-9fq18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DOHprl02", "type": "article", "title": "Distinguishing Separable and Entangled States", "author": [ { "family_name": "Doherty", "given_name": "A. C.", "clpid": "Doherty-A-C" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Spedalieri", "given_name": "Federico M.", "clpid": "Spedalieri-F-M" } ], "abstract": "We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.", "doi": "10.1103/PhysRevLett.88.187904", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2002-05-06", "series_number": "18", "volume": "88", "issue": "18", "pages": "Art. No. 187904" }, { "id": "authors:djh5n-6xg64", "collection": "authors", "collection_id": "djh5n-6xg64", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PARacc99", "type": "book_section", "title": "Model reduction for analysis of cascading failures in power systems", "book_title": "American Control Conference, 1999. San Diego, CA, 2-4 June 1999", "author": [ { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Lall", "given_name": "Sanjay", "clpid": "Lall-S" }, { "family_name": "Paganini", "given_name": "Fernando", "clpid": "Paganini-F" }, { "family_name": "Verghese", "given_name": "George C.", "clpid": "Verghese-G-C" }, { "family_name": "Lesieutre", "given_name": "Bernard C.", "clpid": "Lesieutre-B-C" }, { "family_name": "Marsden", "given_name": "Jerrold E.", "clpid": "Marsden-J-E" } ], "abstract": "In this paper, we apply a principal-orthogonal decomposition based method to the model reduction of a hybrid, nonlinear model of a power network. The results demonstrate that the sequence of fault events can be evaluated and predicted without necessarily simulating the whole system.", "doi": "10.1109/ACC.1999.786351", "isbn": "0-7803-4990-3", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "1999-06", "pages": "4208-4212" } ]