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191
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 222 (18 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.
Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
, 2000
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A Geometric Buchberger Algorithm for Integer Programming
 Mathematics of Operations Research
, 1995
"... Let IP denote the family of integer programs of the form Min cx : Ax = b, x ∈ N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of vectors in Z^n such that for each nonoptimal solution α to a program in this family, there is at lea ..."
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Cited by 56 (10 self)
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Let IP denote the family of integer programs of the form Min cx : Ax = b, x ∈ N^n obtained by varying the right hand side vector b but keeping A and c fixed. A test set for IP is a set of vectors in Z^n such that for each nonoptimal solution α to a program in this family, there is at least one element g in this set such that α  g has an improved cost value as compared to α. We describe a unique minimal test set for this family called the reduced Gröbner basis of IP. An algorithm for its construction is presented which we call a Geometric Buchberger Algorithm for integer programming. We show how an integer program may be solved using this test set and examine some geometric properties of elements in the set. The reduced Grobner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A ∈ Z^(n2)×n of full row rank.
Supersingular abelian varieties in cryptology
 Advances in Cryptology  CRYPTO 2002
"... Abstract. For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications. This ..."
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Cited by 45 (7 self)
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Abstract. For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications. This paper determines exactly which values can occur as the security parameters of supersingular abelian varieties (in terms of the dimension of the abelian variety and the size of the finite field), and gives constructions of supersingular abelian varieties that are optimal for use in cryptography. 1
Nonlinear Loop Invariant Generation using Gröbner Bases
, 2004
"... We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generati ..."
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Cited by 41 (4 self)
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We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generation has been focussed on the construction of linear invariants for linear programs. Consequently, there has been little progress toward nonlinear invariant generation. In this paper, we demonstrate a technique that encodes the conditions for a given template assertion being an invariant into a set of constraints, such that all the solutions to these constraints correspond to nonlinear (algebraic) loop invariants of the program. We discuss some tradeoffs between the completeness of the technique and the tractability of the constraintsolving problem generated. The application of the technique is demonstrated on a few examples.
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 37 (7 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.
Solving parametric polynomial systems
 Journal of Symbolic Computation
, 2007
"... We present a new algorithm for solving basic parametric constructible or semialgebraic ..."
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Cited by 36 (2 self)
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We present a new algorithm for solving basic parametric constructible or semialgebraic
Constructing Invariants for Hybrid Systems
 in Hybrid Systems: Computation and Control, LNCS 2993
, 2004
"... Abstract. An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideal ..."
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Cited by 36 (7 self)
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Abstract. An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideals over polynomial rings. We extend our previous work on the generation of algebraic invariants for discrete transition systems in order to generate algebraic invariants for hybrid systems. In doing so, we present a new technique to handle consecution across continuous differential equations. The techniques we present allow a tradeoff between the complexity of the invariant generation process and the strength of the resulting invariants. 1
Introduction to numerical algebraic geometry
 In Solving Polynomial Equations, Series: Algorithms and Computation in Mathematics
, 2005
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