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48
Monotonicity of primaldual interiorpoint algorithms for semidefinite programming problems
, 1998
"... We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly imp ..."
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Cited by 181 (34 self)
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We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.
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.
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
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 46 (3 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...
A Path to the ArrowDebreu Competitive Market Equilibrium
 MATH. PROGRAMMING
, 2004
"... We present polynomialtime interiorpoint algorithms for solving the Fisher and ArrowDebreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛequilibrium solution. If the p ..."
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Cited by 36 (7 self)
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We present polynomialtime interiorpoint algorithms for solving the Fisher and ArrowDebreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛequilibrium solution. If the problem data are rational numbers and their bitlength is L, then the bound to generate an exact solution is O(n 4 L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound O(n 8 log(1/ɛ)) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the ArrowDebreu equilibrium, similar to the central path developed for linear programming interiorpoint methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixedpoint theorem. The defining equations are bilinear and possess some primaldual structure for the application of the Newtonbased pathfollowing method.
Interior Point Methods For Optimal Control Of DiscreteTime Systems
 Journal of Optimization Theory and Applications
, 1993
"... . We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete ..."
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Cited by 31 (5 self)
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. We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete time linearquadratic regulator problem with mixed state/control constraints, and show how it can be efficiently incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interiorpoint method is the narrowbanded structure of the coefficient matrix which is factorized at each iteration. Key words. interior point algorithms, optimal control, banded linear systems. 1. Introduction. The problem of optimal control of an initial value ordinary differential equation, with Bolza objectives and mixed constraints, is min x;u Z T 0 L(x(t); u(t); t) dt + OE f (x(T )); x(t) = f(x(t); u(t); t); x(0) = x init ; (1.1) g(x(t); u(...
SelfScaled Cones and InteriorPoint Methods in Nonlinear Programming
 Working Paper, CORE, Catholic University of Louvain, LouvainlaNeuve
, 1994
"... : This paper provides a theoretical foundation for efficient interiorpoint algorithms for nonlinear programming problems expressed in conic form, when the cone and its associated barrier are selfscaled. For such problems we devise longstep and symmetric primaldual methods. Because of the special ..."
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Cited by 29 (2 self)
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: This paper provides a theoretical foundation for efficient interiorpoint algorithms for nonlinear programming problems expressed in conic form, when the cone and its associated barrier are selfscaled. For such problems we devise longstep and symmetric primaldual methods. Because of the special properties of these cones and barriers, 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. Key words: Nonlinear Programming, conical form, interior point algorithms, selfconcordant barrier, selfscaled cone, selfscaled barrier, pathfollowing algorithms, potentialreduction algorithms. AMS 1980 subject classification. Primary: 90C05, 90C25, 65Y20. CORE, Catholic University of Louvain, LouvainlaNeuve, Belgium. Email: nesterov@core.ucl.ac.be. Part of this work was done while the author was visiting the Cornell C...
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 ..."
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Cited by 25 (1 self)
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. 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...
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 25 (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.