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32
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
Efficient algorithm for computing the EulerPoincaré characteristic of semialgebraic sets defined by few quadratic inequalities
 Computational Complexity
, 2006
"... Abstract. We present an algorithm which takes as input a closed semialgebraic set, S ⊂ R k, defined by P1 ≤ 0,...,Pℓ ≤ 0,Pi ∈ R[X1,...,Xk],deg(Pi) ≤ 2, and computes the EulerPoincaré characteristic of S. The complexity of the algorithm is k O(ℓ). Keywords. Semialgebraic sets, EulerPoincaré chara ..."
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Cited by 13 (8 self)
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Abstract. We present an algorithm which takes as input a closed semialgebraic set, S ⊂ R k, defined by P1 ≤ 0,...,Pℓ ≤ 0,Pi ∈ R[X1,...,Xk],deg(Pi) ≤ 2, and computes the EulerPoincaré characteristic of S. The complexity of the algorithm is k O(ℓ). Keywords. Semialgebraic sets, EulerPoincaré characteristic
Polynomial Instances Of The Positive Semidefinite And Euclidean Distance Matrix Completion Problems
 SIAM J. Matrix Anal. Appl
, 1998
"... Given an undirected graph G = (V; E) with node set V = [1; n], a subset S ` V and a rational vector a 2 Q S[E , the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n \Theta n positive semidefinite matrix X = (x ij ) satisfying: x ii = a ..."
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Cited by 13 (5 self)
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Given an undirected graph G = (V; E) with node set V = [1; n], a subset S ` V and a rational vector a 2 Q S[E , the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n \Theta n positive semidefinite matrix X = (x ij ) satisfying: x ii = a i (i 2 S) and x ij = a ij (ij 2 E). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fillin; the minimum fillin of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph of K 4 .
Polynomialtime computing over quadratic maps. I. Sampling in real algebraic sets
 Comput. Complexity
"... Abstract. Given a quadratic map Q: Kn → Kk defined over a computable subring D of a real closed field K, and p ∈ D[Y1,...,Yk] of degree d we consider the zero set Z = Z(p(Q(X)), Kn) ⊆ Kn of p(Q(X1,...,Xn)) ∈ D[X1,...,Xn]. We present a procedure that computes, in (dn) O(k) arithmetic operations in ..."
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Abstract. Given a quadratic map Q: Kn → Kk defined over a computable subring D of a real closed field K, and p ∈ D[Y1,...,Yk] of degree d we consider the zero set Z = Z(p(Q(X)), Kn) ⊆ Kn of p(Q(X1,...,Xn)) ∈ D[X1,...,Xn]. We present a procedure that computes, in (dn) O(k) arithmetic operations in D, a set S of (real univariate representations of) sampling points in Kn that intersects nontrivially each connected component of Z. As soon as k = o(n), this is faster than the standard methods that all have exponential dependence on n in the complexity. In particular, our procedure is polynomialtime for constant k. In contrast, the best previously known procedure is only capable of deciding in nO(k2) operations the nonemptiness (rather than i Y 2 i constructing sampling points) of the set Z in the case of p(Y) = ∑ and homogeneous Q. A byproduct of our procedure is a bound (dn) O(k) on the number of connected components of Z. The procedure consists of exact symbolic computations in D and outputs vectors of algebraic numbers. It involves extending K by infinitesimals and subsequent limit computation by a novel procedure that utilizes knowledge of an explicit isomorphism between real algebraic sets.
A sharper estimate on the Betti numbers of sets defined by quadratic inequalities
 Discrete and Computational Geometry, to appear. SAUGATA BASU, DMITRII V. PASECHNIK, AND MARIEFRANÇOISE
"... In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semialgebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0,..., Ps ≥ 0, where Pi ∈ R[X1,..., Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 1 + (k − s) + ..."
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In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semialgebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0,..., Ps ≥ 0, where Pi ∈ R[X1,..., Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 1 + (k − s) +
On projections of semialgebraic sets defined by few quadratic inequalities
 in Discrete and Computational Geometry, available at [arXiv:math.AG/0602398]. SEMIALGEBRAIC GEOMETRY AND TOPOLOGY 73
"... Abstract. Let S ⊂ R k+m be a compact semialgebraic set defined by P1 ≥ 0,..., Pℓ ≥ 0, where Pi ∈ R[X1,..., Xk, Y1,..., Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ ℓ. Let π denote the standard projection from R k+m onto R m. We prove that for any q> 0, the sum of the first q Betti numbers of π(S) is bounded b ..."
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Cited by 8 (6 self)
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Abstract. Let S ⊂ R k+m be a compact semialgebraic set defined by P1 ≥ 0,..., Pℓ ≥ 0, where Pi ∈ R[X1,..., Xk, Y1,..., Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ ℓ. Let π denote the standard projection from R k+m onto R m. We prove that for any q> 0, the sum of the first q Betti numbers of π(S) is bounded by (k + m) O(qℓ). We also present an algorithm for computing the the first q Betti numbers of π(S), whose complexity is (k + m) 2O(qℓ). For fixed q and ℓ, both the bounds are polynomial in k + m. 1.
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.