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41
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
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A Near Maximum Likelihood Decoding Algorithm for MIMO Systems Based on SemiDefinite Programming
, 2005
"... In MultiInput MultiOutput (MIMO) systems, MaximumLikelihood (ML) decoding is equivalent to finding the closest lattice point in an Ndimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasimaximum likelihood algorithm based on SemiDefinite Pr ..."
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Cited by 28 (4 self)
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In MultiInput MultiOutput (MIMO) systems, MaximumLikelihood (ML) decoding is equivalent to finding the closest lattice point in an Ndimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasimaximum likelihood algorithm based on SemiDefinite Programming (SDP). We introduce several SDP relaxation models for MIMO systems, with increasing complexity. We use interiorpoint methods for solving the models and obtain a nearML performance with polynomial computational complexity. Lattice basis reduction is applied to further reduce the computational complexity of solving these models. The proposed relaxation models are also used for soft output decoding in MIMO systems.
Implementation of a primaldual method for SDP on a shared memory parallel architecture
 Computational Optimization and Applications
, 2006
"... Primal–dual interior point methods and the HKM method in particular have been implemented in a number of software packages for semidefinite programming. These methods have performed well in practice on small to medium sized SDP’s. However, primal–dual codes have had some trouble in solving larger ..."
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Cited by 23 (0 self)
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Primal–dual interior point methods and the HKM method in particular have been implemented in a number of software packages for semidefinite programming. These methods have performed well in practice on small to medium sized SDP’s. However, primal–dual codes have had some trouble in solving larger problems because of the storage requirements and required computational effort. In this paper we describe a parallel implementation of the primaldual method on a shared memory system. Computational results are presented, including the solution of some large scale problems with over 50,000 constraints.
Correlative sparsity in primaldual interiorpoint methods for LP, SDP and SOCP
, 2006
"... Exploiting sparsity has been a key issue in solving largescale optimization problems. The most timeconsuming part of primaldual interiorpoint methods for linear programs, secondorder cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by ..."
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Cited by 22 (16 self)
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Exploiting sparsity has been a key issue in solving largescale optimization problems. The most timeconsuming part of primaldual interiorpoint methods for linear programs, secondorder cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation that is determined by the sparsity of an optimization problem (linear program, semidefinite program or secondorder program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fillin.
Polynomial Filtering for Fast Convergence in Distributed Consensus
, 2008
"... In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature address the consensus averaging problem by distributed linear iterative algorithms, with asymptotic convergenc ..."
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Cited by 21 (1 self)
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In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature address the consensus averaging problem by distributed linear iterative algorithms, with asymptotic convergence of the consensus solution. The convergence rate of such distributed algorithms typically depends on the network topology and the weights given to the edges between neighboring sensors, as described by the network matrix. In this paper, we propose to accelerate the convergence rate for given network matrices by the use of polynomial filtering algorithms. The main idea of the proposed methodology is to apply a polynomial filter on the network matrix that will shape its spectrum in order to increase the convergence rate. Such an algorithm is equivalent to periodic updates in each of the sensors by aggregating a few of its previous estimates. We formulate the computation of the coefficients of the optimal polynomial as a semidefinite program that can be efficiently and globally solved for both static and dynamic network topologies. We finally provide simulation results that demonstrate the effectiveness of the proposed solutions in accelerating the convergence of distributed consensus averaging problems.
Semidefinite characterization and computation of real radical ideals
 Foundations of Computational Mathematics
"... For an ideal I ⊆ R[x] given by a set of generators, a new semidefinite characterization of its real radical I(VR(I)) is presented, provided it is zerodimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute ..."
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Cited by 12 (8 self)
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For an ideal I ⊆ R[x] given by a set of generators, a new semidefinite characterization of its real radical I(VR(I)) is presented, provided it is zerodimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety VR(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components. AMS: 14P05 13P10 12E12 12D10 90C22 1
On the behavior of the homogeneous selfdual model for conic convex optimization
 MIT Operations Research
, 2004
"... There is a natural norm associated with a starting point of the homogeneous selfdual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of εoptimal solutions, and (ii ..."
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Cited by 7 (1 self)
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There is a natural norm associated with a starting point of the homogeneous selfdual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of εoptimal solutions, and (ii) the maximum distance of εoptimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stoppingrule theory for HSDbased interiorpoint methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the εoptimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous selfdual model that might improve the resulting solution time in practice.
Warm starting the homogeneous and selfdual interior point method for linear and conic quadratic problems.
 Mathematical Programming Computation,
, 2013
"... Abstract We present two strategies for warmstarting primaldual interior point methods for the homogeneous selfdual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and th ..."
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Cited by 7 (1 self)
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Abstract We present two strategies for warmstarting primaldual interior point methods for the homogeneous selfdual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and their negligible computational cost. This is a major advantage when compared to previously suggested strategies that require a pool of iterates from the solution process of the initial problem. Consequently our strategies are better suited for users who use optimization algorithms as blackbox routines which usually only output the final solution. Our two strategies differ in that one assumes knowledge only of the final primal solution while the other assumes the availability of both primal and dual solutions. We analyze the strategies and deduce conditions under which they result in improved theoretical worstcase complexity. We present extensive computational results showing work reductions when warmstarting compared to coldstarting in the range 30%75% depending on the problem class and magnitude of the problem perturbation. The computational experiments thus substantiate that the warmstarting strategies are useful in practice.
A secondorder cone cutting surface method: complexity and application
, 2005
"... We present an analytic center cutting surface algorithm that uses mixed linear and multiple secondorder cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can ..."
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Cited by 6 (5 self)
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We present an analytic center cutting surface algorithm that uses mixed linear and multiple secondorder cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding p secondorder cone