Results 1  10
of
93
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 ..."
Abstract

Cited by 463 (16 self)
 Add to MetaCart
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.
Determinant maximization with linear matrix inequality constraints
 SIAM Journal on Matrix Analysis and Applications
, 1998
"... constraints ..."
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 ..."
Abstract

Cited by 146 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 83 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 70 (6 self)
 Add to MetaCart
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
A Cutting Plane Method from Analytic Centers for Stochastic Programming
 Mathematical Programming
, 1994
"... The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition ..."
Abstract

Cited by 49 (18 self)
 Add to MetaCart
The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization. 1 Introduction The study of optimization problems in the presence of uncertainty still taxes the limits of methodology and software. One of the most approachable settings is that of twostaged planning under uncertainty, in which a first stage decision has to be taken bef...
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 a P,,f ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
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.
The Many Facets of Linear Programming
, 2000
"... . We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction A ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the fortyfifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twentyfifth of the awarding of the 1975 Nobe...
Smoothed Analysis of Termination of Linear Programming Algorithms
"... We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng ..."
Abstract

Cited by 23 (4 self)
 Add to MetaCart
We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
"... ..."