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152
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 181 (11 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.
LOQO: An interior point code for quadratic programming
, 1994
"... ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex ..."
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Cited by 156 (9 self)
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ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1.
A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems: A Summary
 Research ReportRJ7493 (70008), IBM Almaden Research Center
, 1990
"... This note summarizes a report with the same title, where a study was carried out regarding a unified approach, proposed by Kojima, Mizuno and Yoshise, for interior point algorithms for the linear complementarily problem with a positive semidefinite matrix. This approach is extended to nonsymmetri ..."
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Cited by 146 (8 self)
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This note summarizes a report with the same title, where a study was carried out regarding a unified approach, proposed by Kojima, Mizuno and Yoshise, for interior point algorithms for the linear complementarily problem with a positive semidefinite matrix. This approach is extended to nonsymmetric matrices with nonnegative principal minors.
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 138 (9 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 ...
The Mathematics Of Eigenvalue Optimization
, 2003
"... Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemp ..."
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Cited by 92 (13 self)
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Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briey on semide nite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.
Interior Methods for Constrained Optimization
 Acta Numerica
, 1992
"... Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, includ ..."
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Cited by 83 (3 self)
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Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomialtime interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of everincreasing size. Interior methods...
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
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Cited by 70 (6 self)
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CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the
Primaldual interior methods for nonconvex nonlinear programming
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterize ..."
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Cited by 59 (5 self)
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Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primaldual system similar to that proposed for interior methods. The augmented penaltybarrier function may be interpreted as a merit function for values of the primal and dual variables. An inertiacontrolling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penaltybarrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.
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 54 (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 ...
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 dete ..."
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Cited by 47 (9 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...