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23
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 18 (1 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.
An exact duality theory for semidefinite programming based on sums of squares
, 2012
"... Farkas’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certifi ..."
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Cited by 13 (2 self)
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Farkas’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) ≽ 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.
Grothendiecktype inequalities in combinatorial optimization
 COMM. PURE APPL. MATH
, 2011
"... We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. ..."
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Cited by 9 (3 self)
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We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials
 SIAM Journal on Matrix Analysis and Applications
, 2009
"... A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex se ..."
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Cited by 8 (3 self)
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A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form SD(f) = {x ∈ D: f(x) ≥ 0}. Here D = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} is a convex domain defined by some “nice ” concave polynomials gi(x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over D, we prove that SD(f) has some explicit semidefinite representations under certain conditions called preordering concavity or qmodule concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria: f(u) + ∇f(u) T (x − u) − f(x) ≥ 0, ∀ x,u ∈ D. When f(x) is a polynomial or rational function having singularities on the boundary of SD(f), a perspective transformation is introduced to find some explicit semidefinite representations for SD(f) under certain conditions. In the particular case n = 2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that SD(f) always admits an explicitly constructible semidefinite representation. Key words: convex set, linear matrix inequality, perspective transformation, polynomial, Positivstellensatz, preordering convex/concave, qmodule convex/concave, rational function, singularity, semidefinite programming, sum of squares
Semidefinite representation for convex hulls of real algebraic curves
, 2012
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Infeasibility certificates for linear matrix inequalities
"... Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebr ..."
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Cited by 3 (1 self)
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Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality L(x) ≽ 0 is infeasible if and only if −1 lies in the quadratic module associated to L. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certificate, we use a more involved algebraic certificate motivated by the real radical and Prestel’s theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination. A linear matrix inequality (LMI) is a condition of the form n∑ L(x) = A0 + xiAi ≽ 0 (x ∈ R n)
Uniqueness results for minimal enclosing ellipsoids
 Comput. Aided Geom. Design
"... Abstract We prove uniqueness of the minimal enclosing ellipsoid with respect to strictly eigenvalue convex size functions. Special examples include the classic case of minimal volume ellipsoids (Löwner ellipsoids), minimal surface area ellipsoids or, more generally, ellipsoids that are minimal with ..."
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Abstract We prove uniqueness of the minimal enclosing ellipsoid with respect to strictly eigenvalue convex size functions. Special examples include the classic case of minimal volume ellipsoids (Löwner ellipsoids), minimal surface area ellipsoids or, more generally, ellipsoids that are minimal with respect to quermass integrals.
Mechanism Design for Fair Division
, 2012
"... We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and t ..."
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Cited by 2 (0 self)
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We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and the competitive equilibrium, is known to be not implementable in a truthful fashion, which has been its main drawback. To alleviate this issue, we propose a new mechanism, which we call the Partial Allocation mechanism, that discards a carefully chosen fraction of the allocated resources in order to incentivize the agents to be truthful in reporting their valuations. For a multidimensional domain with an arbitrary number of agents and items, and for the very large class of homogeneous valuation functions, we prove that our mechanism provides every agent with at least a 1/e ≈ 0.368 fraction of her Proportionally Fair valuation. To the best of our knowledge, this is the first result that gives a constant factor approximation to every agent for the Proportionally Fair solution. To complement this result, we show that no truthful mechanism can guarantee more than 0.5 approximation, even for the restricted class of additive linear valuations. We also uncover a connection between the Partial Allocation mechanism and VCGbased mechanism design, which introduces a way to implement interesting truthful mechanisms in settings where monetary payments are not an option. We also ask whether better approximation ratios are possible in more restricted settings. In particular, motivated by the massive privatization auction in the Czech republic in the early 90s we provide another mechanism for additive linear valuations that works really well when all the items are highly demanded.
Polynomialsized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones
, 2014
"... We give explicit polynomialsized (in n and k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k in n variables. These convex cones form a family of nonpolyhedral outer approximations of the nonnegative orthant that preserve l ..."
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Cited by 2 (1 self)
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We give explicit polynomialsized (in n and k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k in n variables. These convex cones form a family of nonpolyhedral outer approximations of the nonnegative orthant that preserve lowdimensional faces while successively discarding highdimensional faces. More generally we construct explicit semidefinite representations (polynomialsized in k,m, and n) of the hyperbolicity cones associated with kth directional derivatives of polynomials of the form p(x) = det( ∑n i=1Aixi) where the Ai are m×m symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones. 1