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88
PrimalDual InteriorPoint Methods for SelfScaled Cones
 SIAM Journal on Optimization
, 1995
"... In this paper we continue the development of a theoretical foundation for efficient primaldual interiorpoint algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are selfscaled (see [9]). The class of problems under consideration includes li ..."
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Cited by 200 (12 self)
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In this paper we continue the development of a theoretical foundation for efficient primaldual interiorpoint algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are selfscaled (see [9]). The class of problems under consideration includes linear programming, semidefinite programming and quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affinescaling and centering directions. We present efficiency estimates for several symmetric primaldual methods that can loosely be classified as pathfollowing methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
A unified approach to interior point algorithms for linear complementarity problems
 Lecture Notes in Computer Science 538 (SpringerVerlag
, 1991
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PrimalDual PathFollowing Algorithms for Semidefinite Programming
 SIAM Journal on Optimization
, 1996
"... This paper deals with a class of primaldual interiorpoint algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primaldual search directions that generalizes the one used in algorithms for linear programmin ..."
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Cited by 157 (10 self)
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This paper deals with a class of primaldual interiorpoint algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primaldual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X 1=2 S \Gamma1=2 . They study three primaldual algorithms based on this family of search directions: a shortstep pathfollowing method, a feasible potentialreduction method and an infeasible potentialreduction method. However, they were not able to provide an algorithm which generalizes the longstep pathfollowing algorithm introduced by Kojima, Mizuno and Yoshise [10]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present: 1) a simplified polynomial convergence proof for one of their shortstep pathfollowing ...
Smoothing Methods for Convex Inequalities and Linear Complementarity Problems
 Mathematical Programming
, 1993
"... A smooth approximation p(x; ff) to the plus function: maxfx; 0g, is obtained by integrating the sigmoid function 1=(1 + e \Gammaffx ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization probl ..."
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Cited by 72 (6 self)
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A smooth approximation p(x; ff) to the plus function: maxfx; 0g, is obtained by integrating the sigmoid function 1=(1 + e \Gammaffx ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for ff sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite ff. Speedup over MINOS 5.4 was as high as 515 times for linear inequalities of size 1000 \Theta 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCP's with as many as 400 variables, the proposed approach was as much as 85 times faster than Lemke's method. Key Words: Smo...
Polynomial Convergence of PrimalDual Algorithms for Semidefinite Programming Based on Monteiro and Zhang Family of Directions
 School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332
, 1997
"... This paper establishes the polynomialconvergence of the class of primaldual feasible interiorpoint algorithms for semidefinite programming (SDP) based on Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work, no condition is imposed on the scaling matrix that ..."
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Cited by 69 (10 self)
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This paper establishes the polynomialconvergence of the class of primaldual feasible interiorpoint algorithms for semidefinite programming (SDP) based on Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work, no condition is imposed on the scaling matrix that determines the search direction. We show that the polynomial iterationcomplexity bounds of two wellknown algorithms for linear programming, namely the shortstep pathfollowing algorithm of Kojima et al. and Monteiro and Adler, and the predictorcorrector algorithm of Mizuno et al., carry over to the context of SDP. Since Monteiro and Zhang family of directions includes the Alizadeh, Haeberly and Overton direction, we establish for the first time the polynomial convergence of algorithms based on this search direction. Keywords: Semidefinite programming, interiorpoint methods, polynomial complexity, pathfollowing methods, primaldual methods. AMS 1991 subject classification: 65K05, 90C25, 90C...
Local Convergence of PredictorCorrector InfeasibleInteriorPoint Algorithms for SDPs and SDLCPs
 Mathematical Programming
, 1997
"... . An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the genera ..."
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Cited by 58 (4 self)
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. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A MizunoToddYe type predictorcorrector infeasibleinteriorpoint algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, InfeasibleInteriorPoint Method, PredictorCorrectorMethod, Superlinear Convergence, PrimalDual Nondegeneracy Abbreviated Title. InteriorPoint Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 152, Japa...
Symmetric PrimalDual Path Following Algorithms for Semidefinite Programming
, 1996
"... In this paper a symmetric primaldual transformation for positive semidefinite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primaldual tran ..."
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Cited by 56 (10 self)
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In this paper a symmetric primaldual transformation for positive semidefinite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primaldual transformation is a well known fact. Based on this symmetric primaldual transformation we derive Newton search directions for primaldual pathfollowing algorithms for semidefinite programming. In particular, we generalize: (1) the short step path following algorithm, (2) the predictorcorrector algorithm and (3) the largest step algorithm to semidefinite programming. It is shown that these algorithms require at most O( p n j log ffl j) main iterations for computing an ffloptimal solution. The symmetric primaldual transformation discussed in this paper can be interpreted as a specialization of the scalingpoint concept introduced by Nesterov and Todd [12] for selfscaled conic problems. The ...
HOMOTOPY CONTINUATION METHODS FOR NONLINEAR COMPLEMENTARITY PROBLEMS
, 1991
"... A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y ffx)). Under the assumption that the mapping f is ..."
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Cited by 36 (3 self)
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A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y ffx)). Under the assumption that the mapping f is a P,,function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) 8 0 until the parameter t> 0 attains 0. Here (a, b) denotes a 2ndimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method.
A PredictorCorrector InteriorPoint Algorithm for the Semidefinite Linear Complementarity Problem Using the AlizadehHaeberlyOverton Search Direction
, 1996
"... This paper proposes a globally convergent predictorcorrector infeasibleinteriorpoint algorithm for the monotone semidefinite linear complementarity problem using the AlizadehHaeberlyOverton search direction, and shows its quadratic local convergence under the strict complementarity condition. ..."
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Cited by 34 (3 self)
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This paper proposes a globally convergent predictorcorrector infeasibleinteriorpoint algorithm for the monotone semidefinite linear complementarity problem using the AlizadehHaeberlyOverton search direction, and shows its quadratic local convergence under the strict complementarity condition.
Polynomial Convergence of a New Family of PrimalDual Algorithms for Semidefinite Programming
, 1996
"... This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 ..."
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Cited by 30 (8 self)
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This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 SP \GammaT )(P T XP ) 1=2 \Gamma I = 0; where P is a nonsingular matrix. Specifically, we show that the shortstep path following algorithm based on the Frobenius norm neighborhood and the semilongstep path following algorithm based on the operator 2norm neighborhood have O( p nL) and O(nL) iterationcomplexity bounds, respectively. When P = I, this yields the first polynomially convergent semilongstep algorithm based on a pure Newton direction. Restricting the scaling matrix P at each iteration to a certain subset of nonsingular matrices, we are able to establish an O(n 3=2 L) iterationcomplexity for the longstep path following method. The resulting subclass of search direct...