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27
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 36 (6 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Nonlinear Systems: Approximating Reach Sets
, 2004
"... We describe techniques to generate useful reachability information for nonlinear dynamical systems. These techniques can be automated for polynomial systems using algorithms from computational algebraic geometry. The generated information can be incorporated into other approaches for doing reachab ..."
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Cited by 27 (6 self)
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We describe techniques to generate useful reachability information for nonlinear dynamical systems. These techniques can be automated for polynomial systems using algorithms from computational algebraic geometry. The generated information can be incorporated into other approaches for doing reachability computation. It can also be used when abstracting hybrid systems that contain modes with nonlinear dynamics. These techniques are most naturally embedded in the hybrid qualitative abstraction approach proposed by the authors previously. They also show that the formal qualitative abstraction approach is well suited for dealing with nonlinear systems.
Abstractions for Hybrid Systems
 Computer Science Laboratory, SRI International, Menlo Park, CA
, 2004
"... Abstract. We present a procedure for constructing sound finitestate discrete abstractions of hybrid systems. This procedure uses ideas from predicate abstraction to abstract the discrete dynamics and qualitative reasoning to abstract the continuous dynamics of the hybrid system. It relies on the ab ..."
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Cited by 17 (2 self)
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Abstract. We present a procedure for constructing sound finitestate discrete abstractions of hybrid systems. This procedure uses ideas from predicate abstraction to abstract the discrete dynamics and qualitative reasoning to abstract the continuous dynamics of the hybrid system. It relies on the ability to decide satisfiability of quantifierfree formulas in some theory rich enough to encode the hybrid system. We characterize the sets of predicates that can be used to create high quality abstractions and we present new approaches to discover such useful sets of predicates. Under certain assumptions, the abstraction procedure can be applied compositionally to abstract a hybrid system described as a composition of two hybrid automata. We show that the constructed abstractions are always sound, but are relatively complete only under certain assumptions.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 13 (3 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
The topology of the Voronoi diagram of three lines
 PROCEEDINGS OF SYMPOSIUM ON COMPUTATIONAL GEOMETRY, ACM PRESS, SOUTHKOREA
, 2007
"... We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four ..."
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Cited by 13 (3 self)
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We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four unbounded branches of either a nonsingular quartic or of a nonsingular cubic and a line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. We introduce a proof technique, which relies heavily upon modern tools of computer algebra, and is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semialgebraic tests for separating the two connected components of each twodimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semialgebraic tests.
Integer polynomial optimization in fixed dimension
 MATHEMATICS OF OPERATIONS RESEARCH
, 2006
"... We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope ..."
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Cited by 9 (4 self)
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We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are nonnegative over the polytope, these sequences of bounds lead to a fully polynomialtime approximation scheme for the optimization problem.
Positive polynomials and projections of spectrahedra
, 2010
"... This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain th ..."
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Cited by 6 (0 self)
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This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of [17] on nonexposed faces. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.
A convex polynomial that is not sosconvex
 Mathematical Programming
"... A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvex ..."
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Cited by 6 (2 self)
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A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it has been recently speculated whether sosconvexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sosconvex. Interestingly, our example is found with software using sum of squares programming techniques and the duality theory of semidefinite optimization. As a byproduct of our numerical procedure, we obtain a simple method for searching over a restricted family of nonnegative polynomials that are not sums of squares. 1
Fastest mixing Markov chain on graphs with symmetries
 SIAM J. Optim
"... We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the secondlargest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant ..."
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Cited by 5 (1 self)
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We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the secondlargest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of largescale instances that are otherwise computationally infeasible. We obtain analytic or semianalytic results for particular classes of graphs, such as edgetransitive and distancetransitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and blockdiagonalization, respectively. We also establish the connection between these two approaches. Key words. Markov chains, eigenvalue optimization, semidefinite programming, graph automorphism, group representation. 1
Exploiting Equalities in Polynomial Programming ∗
, 2006
"... We propose a novel solution approach for polynomial programming problems with equality constraints. By means of a generic transformation, we show that solution schemes for the (typically simpler) problem without equalities can be used to address the problem with equalities. In particular, we propose ..."
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Cited by 3 (2 self)
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We propose a novel solution approach for polynomial programming problems with equality constraints. By means of a generic transformation, we show that solution schemes for the (typically simpler) problem without equalities can be used to address the problem with equalities. In particular, we propose new solution schemes for mixed binary programs, pure 0–1 quadratic programs, and the stable set problem.