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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 199 (35 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
 Lecture Notes in Computer Sciences, 538
, 1991
"... ..."
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 75 (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 54 (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 35 (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 34 (6 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(...
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 32 (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.
On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility an ..."
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Cited by 32 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
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 30 (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...