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A Study of Search Directions in PrimalDual InteriorPoint Methods for Semidefinite Programming
, 1998
"... We discuss several di#erent search directions which can be used in primaldual interiorpoint methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primaldual symmetry, and whether they always generate welldefined directions. Among ..."
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Cited by 35 (1 self)
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We discuss several di#erent search directions which can be used in primaldual interiorpoint methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primaldual symmetry, and whether they always generate welldefined directions. Among the directions satisfying all but at most two of these desirable properties are the AlizadehHaeberlyOverton, HelmbergRendl VanderbeiWolkowicz/KojimaShindohHara/Monteiro, NesterovTodd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also. Key words: semidefinite programming, search direction, invariance properties. AMS Subject classification: 90C05. Abbreviated title: Search directions in SDP 1 Introduction This paper is concerned with interiorpoint methods for semidefinite programming (SDP) problems and in particular the various search directions they use and ...
On Commutative Class of Search Directions for Linear Programming over Symmetric Cones
 Journal of Optimization Theory and Applications
, 2000
"... The Commutative Class of search directions for semidefinite programming is first proposed by Monteiro and Zhang [13]. In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including lin ..."
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Cited by 9 (2 self)
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The Commutative Class of search directions for semidefinite programming is first proposed by Monteiro and Zhang [13]. In this paper, we investigate the corresponding class of search directions for linear programming over symmetric cones, which is a class of convex optimization problems including linear programming, secondorder cone programming, and semidefinite programming as special cases. Complexity results are established for short, semilong, and long step algorithms. We then propose a subclass of Commutative Class of search directions which has polynomial complexity even in semilong and long step settings. The last subclass still contains the NT and HRVW/KSH/M directions. An explicit formula to calculate any member of the class is also given. Key words: Symmetric Cone, Primaldual InteriorPoint Method, Jordan Algebra, Polynomial Complexity A#liation: Department of Computer Science, The University of ElectroCommunications 1 1. Introduction In this paper, we consider linear ...
Some New Search Directions for PrimalDual Interior Point Methods in Semidefinite Programming
"... Search directions for primaldual pathfollowing methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict complementarity assumptions, the Jacobian matrix of the associated symmetrized Newton equation has bounded co ..."
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Cited by 7 (3 self)
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Search directions for primaldual pathfollowing methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict complementarity assumptions, the Jacobian matrix of the associated symmetrized Newton equation has bounded condition number along the central path in the limit as the barrier parameter tends to zero; (2) the Schur complement matrix of the symmetrized Newton equation is symmetric and the cost for computing this matrix is 2mn 3 + 0:5m 2 n 2 ops, where n and m are the dimension of the matrix and vector variables of the SDP, respectively. These two properties imply that a pathfollowing method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by Alizadeh, Haeberly, and Overton (currently the best search direction in terms of accuracy), but each iteration requires at most half the amount of flops (to leading order).
High Accuracy Algorithms for the Solutions of Semidefinite Linear Programs
, 2001
"... hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, i ..."
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Cited by 3 (0 self)
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hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii Abstract We present a new family of search directions and of corresponding algorithms to solve conic linear programs. The implementation is specialized to semidefinite programs but the algorithms described handle both nonnegative orthant and Lorentz cone problems and Cartesian products of these sets. The primary objective is not to develop yet another interiorpoint algorithm with polynomial time complexity. The aim is practical and addresses an often neglected aspect of the current research in the area, accuracy. Secondary goals, tempered by the first, are numerical efficiency and proper handling of sparsity. The main search direction, called GaussNewton, is obtained as a leastsquares solution to the optimality condition of the logbarrier problem. This motivation ensures that the direction is welldefined everywhere and that the underlying Jacobian is wellconditioned under standard assumptions. Moreover, it is invariant under affine transformation of the space and under orthogonal transformation of the constraining cone. The GaussNewton direction, both in the special cases of linear programming and on the central path of semidefinite programs, coincides with the search directions used in practical implementations. Finally, the MonteiroZhang family of search directions can be derived as scaled projections of the GaussNewton direction. iv
On Two Homogeneous SelfDual Systems for Linear Programming and Its Extensions
, 1998
"... We investigate the relation between interiorpoint algorithms applied to two homogeneous selfdual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of pathfollowing methods (the centerin ..."
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Cited by 1 (1 self)
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We investigate the relation between interiorpoint algorithms applied to two homogeneous selfdual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of pathfollowing methods (the centering parameter for the first approach needs to be equal to zero or larger than one half), whereas complete equivalence of potentialreduction methods can be shown. The results extend to selfscaled conic programming and to semidefinite programming using the usual search directions. Abbreviated title: On two homogeneous systems for LP 1 Introduction Ye, Todd, and Mizuno [23] presented a homogeneous and selfdual interiorpoint algorithm for solving linear programming (LP) problems. The algorithm can start from arbitrary (infeasible) interior points and achieves the best known complexity in term of the number of iterations without using a big initial constant. Recently, Nesterov, Todd, and Ye [...
SOLVING SECOND ORDER CONE PROGRAMMING VIA A REDUCED AUGMENTED SYSTEM APPROACH
"... Abstract. The standard Schur complement equation based implementation of interiorpoint methods for second order cone programming may encounter stability problems in the computation of search directions, and as a consequence, accurate approximate optimal solutions are sometimes not attainable. Based ..."
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Abstract. The standard Schur complement equation based implementation of interiorpoint methods for second order cone programming may encounter stability problems in the computation of search directions, and as a consequence, accurate approximate optimal solutions are sometimes not attainable. Based on the eigenvalue decomposition of the (1, 1) block of the augmented equation, a reduced augmented equation approach is proposed to ameliorate the stability problems. Numerical experiments show that the new approach can achieve more accurate approximate optimal solutions than the Schur complement equation based approach. Key words. second order cone programming, augmented equation, NesterovTodd direction, stability
Solving
"... second order cone programming via a reduced augmented system approach ..."
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