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Interiorpoint Methods
, 2000
"... The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
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Cited by 612 (15 self)
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The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by selfconcordant barrier functions.
Monotonicity of primaldual interiorpoint algorithms for semidefinite programming problems
, 1998
"... We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly imp ..."
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Cited by 216 (35 self)
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We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.
Semidefinite optimization
 Acta Numerica
, 2001
"... Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the ..."
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Cited by 152 (2 self)
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Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.
Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework
 SIAM JOURNAL ON OPTIMIZATION
, 1999
"... A critical disadvantage of primaldual interiorpoint methods against dual interiorpoint methods for large scale SDPs (semidefinite programs) has been that the primal positive semidefinite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamenta ..."
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Cited by 102 (31 self)
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A critical disadvantage of primaldual interiorpoint methods against dual interiorpoint methods for large scale SDPs (semidefinite programs) has been that the primal positive semidefinite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidefinite variable matrix into an SDP having multiple but smaller size positive semidefinite variable matrices to which we can effectively apply any interiorpoint method for SDPs employing a standard blockdiagonal matrix data structure. The other way is an incorporation of our method into primaldual interiorpoint methods which we can apply directly to a given SDP. In Part II of this article, we wi...
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 89 (3 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
Group symmetry in interiorpoint methods for semidefinite programming
 Optimization and Engineering
, 1970
"... Abstract A class of group symmetric SemiDefinite Program (SDP) is introduced by using the framework of group representation theory. It is proved that the central path and several search directions of primaldual interiorpoint methods are group symmetric. Preservation of group symmetry along the se ..."
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Cited by 14 (2 self)
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Abstract A class of group symmetric SemiDefinite Program (SDP) is introduced by using the framework of group representation theory. It is proved that the central path and several search directions of primaldual interiorpoint methods are group symmetric. Preservation of group symmetry along the search direction theoretically guarantees that the numerically obtained optimal solution is group symmetric. As an illustrative example, we show that the optimization problem of a symmetric truss under frequency constraints can be formulated as a group symmetric SDP. Numerical experiments using an interiorpoint algorithm demonstrate convergence to strictly group symmetric solutions.
A sensitivity analysis and a convergence result for a sequential semidefinite programming method, Numerical Analysis Manuscript No.
, 2003
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Characterization of the Limit Point of the Central Path in Semidefinite Programming
, 2002
"... In linear programming, the central path is known to converge to the analytic center of the set of optimal solutions. Recently, it has been shown that this is not necessarily true for linear semidefinite programming in the absence of strict complementarity. The present paper deals with the formulatio ..."
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Cited by 7 (0 self)
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In linear programming, the central path is known to converge to the analytic center of the set of optimal solutions. Recently, it has been shown that this is not necessarily true for linear semidefinite programming in the absence of strict complementarity. The present paper deals with the formulation of a convex problem whose solution defines the limit point of the central path. This problem is closely related to the analytic center problem for the set of optimal solutions. In the strict
An Infeasible InteriorPoint Algorithm with Full NesterovTodd Step for Semidefinite Programming
"... This paper proposes an infeasible interiorpoint algorithm with full NesterovTodd step for semidefinite programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110– 1136, 2006). The polynomial bound coincides with that of infeasible interiorpoint methods for linear programmi ..."
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Cited by 5 (2 self)
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This paper proposes an infeasible interiorpoint algorithm with full NesterovTodd step for semidefinite programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110– 1136, 2006). The polynomial bound coincides with that of infeasible interiorpoint methods for linear programming, namely, O(n log n/ε).
Primaldual Algorithms and Infinitedimensional Jordan Algebras of finite rank
 MATH. PROGRAMMING, SER. B
, 2002
"... We consider primaldual algorithms for certain types of infinitedimensional optimization problems. Our approach is based on the generalization of the technique of finitedimensional Euclidean Jordan algebras to the case of infinitedimensional JBalgebras of finite rank. This generalization enables ..."
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Cited by 4 (2 self)
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We consider primaldual algorithms for certain types of infinitedimensional optimization problems. Our approach is based on the generalization of the technique of finitedimensional Euclidean Jordan algebras to the case of infinitedimensional JBalgebras of finite rank. This generalization enables us to develop polynomialtime primaldual algorithms for “infinitedimensional secondorder cone programs.” We consider as an example a longstep primaldual algorithm based on the NesterovTodd direction. It is shown that this algorithm can be generalized along with complexity estimates to the infinitedimensional situation under consideration. An application is given to an important problem of control theory: multicriteria analytic design of the linear regulator. The calculation of the NesterovTodd direction requires in this case solving one matrix differential Riccati equation plus solving a finitedimensional system of linear algebraic equations on each iteration. The size of this algebraic system is m +1 by m + 1, where m is a number of quadratic performance criteria.