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34
Complete search in continuous global optimization and constraint satisfaction
 ACTA NUMERICA 13
, 2004
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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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Cited by 63 (1 self)
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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
Secondorder cone programming relaxation of sensor network localization
 SIAM J. Optimization
, 2007
"... Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone pro ..."
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Cited by 36 (2 self)
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Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone programming (SOCP) relaxation of this problem, motivated by its simpler structure and its potential to be solved faster than SDP. We show that the SOCP relaxation, though weaker than the SDP relaxation, has nice properties that make it useful as a problem preprocessor. In particular, sensors that are uniquely positioned among interior solutions of the SOCP relaxation are accurate up to the square root of the distance error. Thus, these sensors, which are easily identified, are accurately positioned. In our numerical simulation, the interior solution found can accurately position up to 80–90 % of the sensors. We also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interiorpoint method. Key words. sensor network localization, semidefinite program, secondorder cone program, approximation algorithm, error bound
Generalized Lagrangian Duals and Sums of Squares Relaxations of Sparse Polynomial Optimization Problems
, 2004
"... Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomials) relaxations for a POP (polynomial optimization problem) are introduced. Sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the opt ..."
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Cited by 35 (20 self)
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Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomials) relaxations for a POP (polynomial optimization problem) are introduced. Sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the sequence converge to the optimal value of the POP using a method from the penalty function approach. The sequence of SOS relaxations is transformed into a sequence of SDP (semidefinite program) relaxations of the POP, which correspond to duals of modification and generalization of SDP relaxations given by Lasserre for the POP.
Optimal power flow over tree networks
 PROCEEDINGS OF THE FORTHNINTH ANNUAL ALLERTON CONFERENCE
, 2011
"... The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the conve ..."
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Cited by 28 (12 self)
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The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow oversatisfaction of load. The proof, cast as a complex semidefinite program, makes use of the fact that if the underlying graph of an n n Hermitian positive semidefinite matrix is a tree, then the matrix has rank at least n  1.
Quadratically constrained quadratic programs on acyclic graphs with application to power flow
, 2013
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A General Framework for Convex Relaxation of Polynomial Optimization Problems over Cones
 Journal of Operations Research Society of Japan
, 2002
"... The class of POPs (polynomial optimization problems) over cones covers a wide range of optimization problems such as 01 integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones ..."
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Cited by 20 (10 self)
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The class of POPs (polynomial optimization problems) over cones covers a wide range of optimization problems such as 01 integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones in terms of linear optimization problems over cones. It provides a unified treatment of many existing convex relaxation methods based on the liftandproject linear programming procedure, the reformulationlinearization technique and the semidefinite programming relaxation for a variety of problems. It also extends the theory of convex relaxation methods, and thereby brings flexibility and richness in practical use of the theory.
Convex Relaxation of Optimal Power Flow  Part I: Formulations and Equivalence
, 2014
"... This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. ..."
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Cited by 13 (0 self)
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This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.
Second Order Cone Programming Relaxation of Positive Semidefinite Constraints
, 2001
"... The positive semidefinite constraint for the variable matrix in semidefinite programming (SDP) relaxation is further relaxed by a finite number of second order cone constraints in second order cone programming (SOCP) relaxations. A few types of SOCP relaxations are obtained from different ways of ex ..."
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Cited by 10 (0 self)
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The positive semidefinite constraint for the variable matrix in semidefinite programming (SDP) relaxation is further relaxed by a finite number of second order cone constraints in second order cone programming (SOCP) relaxations. A few types of SOCP relaxations are obtained from different ways of expressing the positive semidefinite constraint of the SDP relaxation. We present how such SOCP relaxations can be derived, and show the relationship between the resulting SOCP relaxations.
Convex Relaxation of Optimal Power Flow Part II: Exactness
, 2014
"... This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. ..."
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Cited by 10 (0 self)
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This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.