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14
DETERMINANT MAXIMIZATION WITH LINEAR MATRIX INEQUALITY CONSTRAINTS
"... The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the s ..."
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Cited by 174 (18 self)
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The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the semidefinite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interiorpoint method, with a simplified analysis of the worstcase complexity and numerical results that indicate that the method is very efficient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interiorpoint method will generally be slower; the advantage is that it handles a much wider variety of problems.
Solving LargeScale Sparse Semidefinite Programs for Combinatorial Optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We present a dualscaling interiorpoint algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational re ..."
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Cited by 113 (11 self)
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We present a dualscaling interiorpoint algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interiorpoint algorithms for approximating the maximum cut semidefinite programs with dimension upto 3000.
Mixed Linear and Semidefinite Programming for Combinatorial and Quadratic Optimization
, 1999
"... We use the semidefinite relaxation to approximate combinatorial and quadratic optimization problems subject to linear, quadratic, as well as boolean constraints. We present a dual potential reduction algorithm and show how to exploit the sparse structure of various problems. Coupled with randomized ..."
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Cited by 19 (4 self)
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We use the semidefinite relaxation to approximate combinatorial and quadratic optimization problems subject to linear, quadratic, as well as boolean constraints. We present a dual potential reduction algorithm and show how to exploit the sparse structure of various problems. Coupled with randomized and heuristic methods, we report computational results for approximating graphpartition and quadratic problems with dimensions 800 to 10,000. This finding, to the best of our knowledge, is the first computational evidence of the effectiveness of these approximation algorithms for solving largescale problems.
Search Directions And Convergence Analysis Of Some Infeasible PathFollowing Methods For The Monotone SemiDefinite LCP
, 1996
"... We consider a family of primal/primaldual/dual search directions for the monotone LCP over the space of n \Theta n symmetric blockdiagonal matrices. We consider two infeasible predictorcorrector pathfollowing methods using these search directions, with the predictor and corrector steps used eith ..."
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Cited by 17 (2 self)
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We consider a family of primal/primaldual/dual search directions for the monotone LCP over the space of n \Theta n symmetric blockdiagonal matrices. We consider two infeasible predictorcorrector pathfollowing methods using these search directions, with the predictor and corrector steps used either in series (similar to the MizunoToddYe method) or in parallel (similar to Mizuno et al./McShane's method). The methods attain global linear convergence with a convergence ratio which, depending on the quality of the starting iterate, ranges from 1 \Gamma O( p n) \Gamma1 to 1 \Gamma O(n) \Gamma1 . Our analysis is fairly simple and parallels that for the LP and LCP cases. 1 Introduction Since the original work of Nesterov and Nemirovskii [26], followed by that of Alizadeh [1] and Jarre [11], there has been very active research on interiorpoint methods for the semidefinite linear programming problem (SDLP) and the semidefinite linear complementarity problem (SDLCP). In particular,...
Using Continuous Nonlinear Relaxations to Solve Constrained MaximumEntropy Sampling Problems
 Mathematical Programming, Series A
, 1998
"... We consider a new nonlinear relaxation for the Constrained MaximumEntropy Sampling Problem  the problem of choosing the s × s principal submatrix with maximal determinant from a given n × n positive definite matrix, subject to linear constraints. We implement a branchandboun ..."
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Cited by 12 (8 self)
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We consider a new nonlinear relaxation for the Constrained MaximumEntropy Sampling Problem  the problem of choosing the s &times; s principal submatrix with maximal determinant from a given n &times; n positive definite matrix, subject to linear constraints. We implement a branchandbound algorithm for the problem, using the new relaxation. The performance on test problems is far superior to a previous implementation using an eigenvaluebased relaxation. A parallel implementation of the algorithm exhibits approximately linear speedup for up to 8 processors, and has successfully solved problem instances that were heretofore intractable.
Continuous Relaxations for Constrained MaximumEntropy Sampling
 In Integer Programming and Combinatorial Optimization
, 1996
"... . We consider a new nonlinear relaxation for the Constrained Maximum Entropy Sampling Problem  the problem of choosing the s \Theta s principal submatrix with maximal determinant from a given n \Theta n positive definite matrix, subject to linear constraints. We implement a branchandbound algo ..."
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Cited by 11 (8 self)
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. We consider a new nonlinear relaxation for the Constrained Maximum Entropy Sampling Problem  the problem of choosing the s \Theta s principal submatrix with maximal determinant from a given n \Theta n positive definite matrix, subject to linear constraints. We implement a branchandbound algorithm for the problem, using the new relaxation. The performance on test problems is far superior to a previous implementation using an eigenvaluebased relaxation. 1 Introduction Let n be a positive integer. For N := f1; : : : ; ng, let YN := fY j j j 2 Ng be a set of n random variables, with jointdensity function g N (\Delta). Let s be an integer satisfying 0 ! s n. For S ae N , j S j = s, let YS := fY j j j 2 Sg, and denote the marginal jointdensity function of YS by gS (\Delta). The entropy of S is defined by h(S) := \GammaE[ln gS (YS )]: Let m be a nonnegative integer, and let M := f1; 2; : : : mg. The Constrained MaximumEntropy Sampling Problem is then the problem of choosing a s...
APPROXIMATING MAXIMUM STABLE SET AND MINIMUM GRAPH COLORING PROBLEMS WITH THE POSITIVE SEMIDEFINITE RELAXATION
"... We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in m ..."
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Cited by 9 (1 self)
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We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.
MaximumEntropy Remote Sampling
, 1998
"... We consider the "remotesampling" problem of choosing a subset S, with jSj = s, from a set N of observable random variables so as to obtain as much information as possible about a set T of target random variables which are not directly observable. Our criterion is that of minimizing the e ..."
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Cited by 8 (1 self)
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We consider the "remotesampling" problem of choosing a subset S, with jSj = s, from a set N of observable random variables so as to obtain as much information as possible about a set T of target random variables which are not directly observable. Our criterion is that of minimizing the entropy of T conditioned on S. We confine our attention to the case in which the random variables have a joint Gaussian distribution. We demonstrate that the problem is NPcomplete. We provide two methods for calculating lower bounds on the entropy: (i) a spectral method, and (ii) a continuous nonlinear relaxation. We employ these bounds in a branchandbound scheme to solve problem instances to optimality. 1 Introduction Let S be a nonempty finite set of s elements, and let Y S := fY j : j 2 Sg be a set of s random variables, with jointdensity function OE S (\Delta). The entropy of S is defined by h(S) := \GammaE [ln(OE S (Y S ))] : In the case where Y S has a joint Gaussian distribution with...
Algorithms and Software for LMI Problems in Control
 IEEE Control Systems Magazine
, 1997
"... this article is to provide an overview of the state of the art of ..."
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Cited by 5 (0 self)
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this article is to provide an overview of the state of the art of
Method of Approximate Centers for SemiDefinite Programming
, 1996
"... The success of interior point algorithms for largescale linear programming has prompted researchers to extend these algorithms to the semidefinite programming (SDP) case. In this paper, the method of approximate centers of Roos and Vial [13] is extended to SDP. Key words: Semidefinite programmi ..."
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Cited by 3 (3 self)
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The success of interior point algorithms for largescale linear programming has prompted researchers to extend these algorithms to the semidefinite programming (SDP) case. In this paper, the method of approximate centers of Roos and Vial [13] is extended to SDP. Key words: Semidefinite programming, pathfollowing algorithm, approximate centers Running title: Method of approximate centers for semidefinite programming. iii 1 Introduction In recent years, a great revival of interest in semidefinite programming (SDP) has taken place. The reason is basically twofold: Important applications in control theory [4], structural optimization [3], and combinatorial optimization [1], to name but a few, have been formulated as SDP problems. (A review of applications may be found in [15]). Secondly, it has become clear that most interior point algorithms for linear programming (LP) may be extended to semidefinite programming. The polynomial complexity of these methods and their success...