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567
Semidefinite Programming Relaxations for Semialgebraic Problems
, 2001
"... 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 mai ..."
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Cited by 361 (22 self)
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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.
GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi
 ACM Trans. Math. Soft
, 2002
"... GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality relaxations of the (generally nonconvex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality or integer constraints. It generates a series of ..."
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Cited by 140 (22 self)
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GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality relaxations of the (generally nonconvex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality or integer constraints. It generates a series of lower bounds monotonically converging to the global optimum. Global optimality is detected and isolated optimal solutions are extracted automatically. Numerical experiments show that for most of the small and mediumscale problems described in the literature, the global optimum is reached at low computational cost. 1
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity
 SIAM Journal on Optimization
, 2006
"... Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of ..."
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Cited by 119 (29 self)
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Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations. Key words.
GloptiPoly 3: Moments, Optimization and Semidefinite Programming
 Soft. PROGRAMMING FOR NPLAYER GAMES 21
, 2007
"... We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming. 1 What is GloptiPoly? Gloptipoly 3 is intended to solve, or at least approximate, the Generalized Problem of Moments (GPM), an infinite ..."
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Cited by 96 (36 self)
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We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming. 1 What is GloptiPoly? Gloptipoly 3 is intended to solve, or at least approximate, the Generalized Problem of Moments (GPM), an infinitedimensional optimization problem which can be viewed as an extension of the classical problem of moments [8]. From a theoretical viewpoint, the GPM has developments and impact in various areas of mathematics such as algebra, Fourier analysis, functional analysis, operator theory, probability and statistics, to cite a few. In addition, and despite a rather simple and short formulation, the GPM has a large number of important applications in various fields such as optimization, probability, finance, control, signal processing, chemistry, cristallography, tomography, etc. For an account of various methodologies as well as some of potential applications, the interested reader is referred to [1, 2] and the nice collection of papers [5]. The present version of GloptiPoly 3 can handle moment problems with polynomial data. Many important applications in e.g. optimization, probability, financial economics and 1
Detecting global optimality and extracting solutions in GloptiPoly
 Chapter in D. Henrion, A. Garulli (Editors). Positive polynomials in control. Lecture Notes in Control and Information Sciences
, 2005
"... GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of po ..."
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Cited by 80 (12 self)
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GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations. 1
Introducing SOSTOOLS: A General Purpose Sum of Squares Programming Solver
 Proceedings of the IEEE Conference on Decision and Control (CDC), Las Vegas, NV
, 2002
"... 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 controlrelated problems. This paper provides ..."
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Cited by 74 (16 self)
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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 controlrelated problems. This 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. 1
Convergent SDPRelaxations in Polynomial Optimization with Sparsity
 SIAM Journal on Optimization
"... Abstract. We consider a polynomial programming problem P on a compact semialgebraic set K ⊂ R n, described by m polynomial inequalities gj(X) ≥ 0, and with criterion f ∈ R[X]. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDPrelaxati ..."
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Cited by 56 (16 self)
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Abstract. We consider a polynomial programming problem P on a compact semialgebraic set K ⊂ R n, described by m polynomial inequalities gj(X) ≥ 0, and with criterion f ∈ R[X]. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDPrelaxation of order r has the following two features: (a) The number of variables is O(κ 2r) where κ = max[κ1, κ2] witth κ1 (resp. κ2) being the maximum number of variables appearing the monomials of f (resp. appearing in a single constraint gj(X) ≥ 0). (b) The largest size of the LMI’s (Linear Matrix Inequalities) is O(κ r). This is to compare with the respective number of variables O(n 2r) and LMI size O(n r) in the original SDPrelaxations defined in [11]. Therefore, great computational savings are expected in case of sparsity in the data {gj, f}, i.e. when κ is small, a frequent case in practical applications of interest. The novelty with respect to [9] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. In such cases, and as a byproduct, we also obtain a new representation result for polynomials positive on a basic closed semialgebraic set, a sparse version of Putinar’s Positivstellensatz [16]. 1.
New upper bounds for kissing numbers from semidefinite programming
 Journal of the American Mathematical Society
, 2006
"... In geometry, the kissing number problem asks for the maximum number τn of unit spheres that can simultaneously touch the unit sphere in ndimensional Euclidean space without pairwise overlapping. The value of τn is only known for n =1, 2, 3, 4, 8, 24. While its determination for n =1, 2 is trivial, ..."
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Cited by 55 (16 self)
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In geometry, the kissing number problem asks for the maximum number τn of unit spheres that can simultaneously touch the unit sphere in ndimensional Euclidean space without pairwise overlapping. The value of τn is only known for n =1, 2, 3, 4, 8, 24. While its determination for n =1, 2 is trivial, it is not the case
Globally optimal estimates for geometric reconstruction problems
 In ICCV
, 2005
"... We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or nonoptimality or a combination of both we pursue the goal of achieving global solutions of the statistically optimal c ..."
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Cited by 53 (15 self)
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We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or nonoptimality or a combination of both we pursue the goal of achieving global solutions of the statistically optimal costfunction. Our approach is based on a hierarchy of convex relaxations to solve nonconvex optimization problems with polynomials. These convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation. The technique is applied to a number of classical vision problems: triangulation, camera pose, homography estimation and last, but not least, epipolar geometry estimation. Experimental validation on both synthetic and real data is provided. In practice, only a few relaxations are needed for attaining the global optimum. 1