Results 1  10
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21
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 463 (16 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.
Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results
"... In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primaldual interiorpoint methods. This framework is based on some results about po ..."
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Cited by 28 (14 self)
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In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primaldual interiorpoint methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two di#erent ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primaldual interiorpoint method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over di#erent classes of SDPs show that these methods can be very e#cient for some problems. Keywords: Semidefinite programming; Primaldual interiorpoint method; Matrix completion problem; Clique tree; Numerical results. # Department of Applied Physics, The University of Tokyo, 731 Hongo, Bunkyoku, Tokyo 1138565 Japan (nakata@zzz.t.utokyo.ac.jp ). + Department of Architecture and Architectural Systems, Kyoto University, Kyoto 6068501 Japan (fujisawa@ismj.archi.kyotou.ac.jp). # Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 1528552 Japan (mituhiro@is.titech.ac.jp). The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Toky...
Sensitivity analysis in linear programming and semidefinite programming using interiorpoint methods
 Cornell University
, 1999
"... We analyze perturbations of the righthand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the norm of the perturbations that allow interiorpoint methods to recover feasible and nearoptimal solutions in a single interiorpoint i ..."
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Cited by 13 (2 self)
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We analyze perturbations of the righthand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the norm of the perturbations that allow interiorpoint methods to recover feasible and nearoptimal solutions in a single interiorpoint iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interiorpoint methods compare nicely with the bounds arising from the simplex method. We also present explicit bounds for SDP using the AHO, H..K..M, and NT directions.
Semidefinite programming for discrete optimization and matrix completion problems
 Discrete Appl. Math
, 2002
"... Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y ..."
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Cited by 10 (5 self)
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Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y
A New Class of Polynomial PrimalDual Methods for Linear and Semidefinite Optimization
, 1999
"... We propose a new class of primaldual methods for linear optimization (LO). By using some new analysis tools, we prove that the large update method for LO based on the new search direction has a polynomial complexity O i n 4 4+ae log n " j iterations where ae 2 [0; 2] is a parameter used in t ..."
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Cited by 8 (5 self)
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We propose a new class of primaldual methods for linear optimization (LO). By using some new analysis tools, we prove that the large update method for LO based on the new search direction has a polynomial complexity O i n 4 4+ae log n " j iterations where ae 2 [0; 2] is a parameter used in the system defining the search direction. If ae = 0, our results reproduce the well known complexity of the standard primal dual Newton method for LO. At each iteration, our algorithm needs only to solve a linear equation system. An extension of the algorithms to semidefinite optimization is also presented. Keywords: Linear Optimization, Semidefinite Optimization, Interior Point Method, PrimalDual Newton Method, Polynomial Complexity. AMS Subject Classification: 90C05 1 Introduction Interior point methods (IPMs) are among the most effective methods for solving wide classes of optimization problems. Since the seminal work of Karmarkar [7], many researchers have proposed and analyzed various ...
Selfregular proximities and new search directions for linear and semidefinite optimization
 Mathematical Programming
, 2000
"... In this paper, we first introduce the notion of selfregular functions. Various appealing properties of selfregular functions are explored and we also discuss the relation between selfregular functions and the wellknown selfconcordant functions. Then we use such functions to define selfregular p ..."
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Cited by 8 (5 self)
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In this paper, we first introduce the notion of selfregular functions. Various appealing properties of selfregular functions are explored and we also discuss the relation between selfregular functions and the wellknown selfconcordant functions. Then we use such functions to define selfregular proximity measure for pathfollowing interior point methods for solving linear optimization (LO) problems. Any selfregular proximity measure naturally defines a primaldual search direction. In this way a new class of primaldual search directions for solving LO problems is obtained. Using the appealing properties of selfregular functions, we prove that these new largeupdate pathfollowing methods for LO enjoy a polynomial, O n q+1 2q log n iteration bound, where q ≥ 1 is the socalled barrier degree of the selfregular ε proximity measure underlying the algorithm. When q increases, this � bound approaches the √n n best known complexity bound for interior point methods, namely O log. Our unified �√n ε n analysis provides also the O log best known iteration bound of smallupdate IPMs. ε At each iteration, we need only to solve one linear system. As a byproduct of our results, we remove some limitations of the algorithms presented in [24] and improve their complexity as well. An extension of these results to semidefinite optimization (SDO) is also discussed.
Solving Semidefinite Programs using Preconditioned Conjugate Gradients
 Optim. Methods Softw
, 2003
"... The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primaldual interiorpoint technique which uses an inexact GaussNewton approach with a matrix free preconditioned conjugate gradient method. This approach a ..."
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Cited by 8 (3 self)
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The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primaldual interiorpoint technique which uses an inexact GaussNewton approach with a matrix free preconditioned conjugate gradient method. This approach avoids the illconditioning pitfalls that result from symmetrization and from forming the socalled normal equations, while maintaining the primaldual framework.
Recent Developments In InteriorPoint Methods
, 1999
"... 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 3 (1 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. Interiorpoint methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and columngeneration algorithms for large linear programs. We review some core developments in the area and discuss their impact on these other problem areas.
SEMIDEFINITE AND LAGRANGIAN RELAXATIONS FOR HARD COMBINATORIAL PROBLEMS
"... Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this ..."
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Cited by 3 (3 self)
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Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this
Exploiting special structure in semidefinite programming: A survey of theory and applications
 European Journal of Operational Research
"... 2009 Semidefinite Programming (SDP) may be seen as a generalization of Linear Programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation. We survey three types of special struc ..."
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Cited by 3 (0 self)
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2009 Semidefinite Programming (SDP) may be seen as a generalization of Linear Programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation. We survey three types of special structures in SDP data: 1. a common ‘chordal ’ sparsity pattern of all the data matrices. This structure arises in applications in graph theory, and may also be used to deal with more general sparsity patterns in a heuristic way. 2. low rank of all the data matrices. This structure is common in SDP relaxations of combinatorial optimization problems, and SDP approximations of polynomial optimization problems. 3. the situation where the data matrices are invariant under the action of a permutation group, or, more generally, where the data matrices belong to a low dimensional matrix algebra. Such problems arise in truss topology optimization, particle physics, coding theory, computational geometry, and graph theory. We will give an overview of existing techniques to exploit these structures in the data. Most of the paper will be devoted to the third situation, since it has received the least attention in the literature so far.