Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set

Greuet, Aurélien; Din, Mohab Safey El

doi:10.1137/130931308

Cited by 46 publications

(52 citation statements)

References 67 publications

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“…This last property is shared with the algorithm in Porkolab and Khachiyan (1997), which, however, cannot be used in practice, since it crucially relies on quantifier elimination techniques. The algorithm in Greuet and Safey El Din (2014) is also exact, but cannot manage semialgebraic constraints and has regularity assumptions on the input, which are not satisfied in our case. The related problem of computing witness points on determinantal algebraic sets has been addressed and solved in Henrion et al (2015b,d).…”

Section: Previous Workmentioning

confidence: 98%

Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic

Naldi

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix, or the dimension of the affine section, is fixed, the algorithm is polynomial time. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We implement it in Maple and discuss practical experiments.

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Section: Previous Workmentioning

confidence: 98%

Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic

Naldi

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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“…We design a parametric variant of [18] that solves this problem. Under the assumption that V is smooth and when the parameters are instantiated, the algorithm in [18] allows to obtain a polynomial of degree singly exponential in the number of decision variables X whose set of roots contains the global optimum of the instantiated polynomial optimization problem.…”

Section: Main Contributionsmentioning

confidence: 99%

“…We use the algebraic nature of the algorithm in [18] to design a parametric variant that returns a list of triples…”

Section: Main Contributionsmentioning

confidence: 99%

“…In the last decade, several approaches have been developed to design dedicated algebraic techniques for polynomial optimization (see [31,17,16,18,2,34] and references therein). In the non-parametric case, they allow to compute polynomials defining the optimum of a polynomial optimization problem whose degrees are singly exponential in the number of decision variables.…”

Section: Introductionmentioning

confidence: 99%

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Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

Guo

Din

Wang

et al. 2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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International audienceWe consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic set. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that when the algebraic set is irreducible and smooth with a compact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of the algebraic set.First, we generalize this approach to non-compact situations. We prove that the graph of the opposite of the optimal value function is still contained in the affine cone over a dual variety similar to the one considered in compact case. In consequence, we present an algorithm for solving the considered parametric optimization problem for generic parameters' values. For some special parameters' values, the representing polynomials of the dual variety can be identically zero, which give no information on the optimal value. We design a dedicated algorithm that identifies those regions of the parameters' space and computes for each of these regions a new polynomial defining the optimal value over the considered region

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“…For some special parameters' values, the representing polynomials of the dual variety V * can be identically zero, which give no information on the optimal value. We designed a parametric variant of [3] that identifies those regions of the parameters' space and computed for each of these regions a new polynomial defining the optimal value over the considered region.…”

Section: Introductionmentioning

confidence: 99%

Optimization Problems over Noncompact Semialgebraic Sets

Zhi

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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In this talk, we will introduce some recent progress in dealing with optimization problems over noncompact semialgebraic sets. We will start with the problem of optimizing a parametric linear function over a noncompact real algebraic variety. Then we will introduce how to compute the semidefinite representation or approximation of the convex hull of a noncompact semialgebraic set. Finally, we will show how to characterize the lifts of noncompact convex sets by the cone factorizations of properly defined slack operators.Optimizing a parametric linear function over a real algebraic variety We consider the problem of optimizing a parametric linear function over a real algebraic variety. . , Xn] and c = (c1, . . . , cn) denotes unspecified parameters. The optimal value c * 0 can be regarded as a function of the parameters c, i.e. the optimal value function.Our goal is to compute a polynomial Φ ∈ Q[c0, c1, . . . , cn] that defines a hypersurface in the parameters' space which contains the graph of this function.Assuming that V is irreducible, smooth and compact in R n , Rostalski and Sturmfels showed that the optimal value function is represented by the defining equation of the irreducible hypersurface V * dual to the projective closure of V in R n [11, Theorem 5.23]. Let C = cl (co (V ∩ R n )) be the closure of the convex hull of V ∩ R n , in [5, Theorem 1.5], we proved that this conclusion is still true for a noncompact real algebraic variety V ∩ R n when it is irreducible, smooth and the recession cone 0 + C of C is pointed (contains no lines). Although the conclusion is correct, the proof of [5, Theorem 1.5] works only if the convex hull of V ∩ R n is closed. The new proof can be found in [4, Corollary 3.2]. Moreover, in [4], we showed (−c * 0 : γ1 : · · · : γn) ∈ V * whenever the optimal value c * 0 is bounded at (γ1, . . . , γn) and V is smooth. When V is not smooth but its real trace is compact, we constructed recursively a finite number of dual varieties such that (−c * 0 : γ1 : · · · : γn) lies in the union of these dual varieties. For some special parameters' values, the representing polynomials of the dual variety V * can be identically zero, which give no information on the optimal value. We designed a parametric variant of [3] that identifies those regions of the parameters' space and computed for each of these regions a new polynomial defining the optimal value over the considered region.

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Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set

Cited by 46 publications

References 67 publications

Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic

Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

Optimization Problems over Noncompact Semialgebraic Sets

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