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10
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 207 (17 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
InfeasibleStart PrimalDual Methods And Infeasibility Detectors For Nonlinear Programming Problems
 Mathematical Programming
, 1996
"... In this paper we present several "infeasiblestart" pathfollowing and potentialreduction primaldual interiorpoint methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a selfdual homogeneous primaldual problem. The methods under considerat ..."
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Cited by 31 (5 self)
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In this paper we present several "infeasiblestart" pathfollowing and potentialreduction primaldual interiorpoint methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a selfdual homogeneous primaldual problem. The methods under consideration generate an fflsolution for an fflperturbation of an initial strictly (primal and dual) feasible problem in O( p ln fflae f ) iterations, where is the parameter of a selfconcordant barrier for the cone, ffl is a relative accuracy and ae f is a feasibility measure. We also discuss the behavior of pathfollowing methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O( p ln ae \Delta ) iterations, where ae \Delta is a primal or dual infeasibility measure. 1 Introduction Nesterov and Nemirovskii [9] first developed and investigated extensions of several classes of interiorpoint algorithms for linear programming t...
A LogBarrier Method With Benders Decomposition For Solving TwoStage Stochastic Programs
 Mathematical Programming 90
, 1999
"... An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving twostage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite numb ..."
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Cited by 16 (6 self)
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An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving twostage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomialtime. Key Words: Stochastic programming, Largescale linear programming, Barrier function, Interior point methods, Benders decomposition, Complexity. Abbreviated Title: A logbarrier method with Benders decomposition AMS(MOS) subject classifications: 90C15, 90C05, 90C06, 90C60. 1 1. Introduction In this paper we propose an algorithm for solving twostage stochastic programs, establish fundamental properties of the algorithm, and analyze the convergence. An example of a twostage stochastic program is a production planning problem. The production and demand take place in the first and second periods, resp...
Polynomiality of PrimalDual Affine Scaling Algorithms for Nonlinear Complementarity Problems
, 1995
"... This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappi ..."
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Cited by 10 (4 self)
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This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primaldual affine scaling algorithms generates an approximate solution (given a precision ffl) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial of n, ln(1=ffl) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in [13].
On InteriorPoint Methods for Fractional Programs and Their Convex Reformulation
, 1994
"... Given a selfconcordant barrier function for a convex set S, we determine a selfconcordant barrier function for the conic hull ~ S of S. As our main result, we derive an "optimal" barrier for ~ S based on the barrier function for S. Important applications of this result include the conic reformulat ..."
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Cited by 5 (1 self)
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Given a selfconcordant barrier function for a convex set S, we determine a selfconcordant barrier function for the conic hull ~ S of S. As our main result, we derive an "optimal" barrier for ~ S based on the barrier function for S. Important applications of this result include the conic reformulation of a convex problem, and the solution of fractional programs by interiorpoint methods. The problem of minimizing a convexconcave fraction over some convex set can be solved by applying an interiorpoint method directly to the original nonconvex problem, or by applying an interiorpoint method to an equivalent convex reformulation of the original problem. Our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases. Key words : Convex set; Conic hull; Barrier function; Interiorpoint method; Fractional programs; Selfconcordance; Convergence rate 1. Conic hulls and fractional programming Let S ` IR n be a given convex s...
On The Complexity Of A Practical InteriorPoint Method
"... The theory of selfconcordance has been used to analyze the complexity of interiorpoint methods based on Newton's method. For large problems, it may be impractical to use Newton's method; here we analyze a truncatedNewton method, in which an approximation to the Newton search direction is used. In ..."
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Cited by 4 (0 self)
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The theory of selfconcordance has been used to analyze the complexity of interiorpoint methods based on Newton's method. For large problems, it may be impractical to use Newton's method; here we analyze a truncatedNewton method, in which an approximation to the Newton search direction is used. In addition, practical interiorpoint methods often include enhancements such as extrapolation that are absent from the theoretical algorithms analyzed previously. We derive theoretical results that apply to such an algorithm, an algorithm similar to a sophisticated computer implementation of a barrier method. The results for a single barrier subproblem are a satisfying extension of the results for Newton's method. When extrapolation is used in the overall barrier method, however, our results are more limited. We indicate (by both theoretical arguments and examples) why more elaborate results may be difficult to obtain.
Conefree” primaldual pathfollowing and potential reduction polynomial time interiorpoint methods
 Math. Prog
, 2005
"... Abstract. We present a framework for designing and analyzing primaldual interiorpoint methods for convex optimization. We assume that a selfconcordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory an ..."
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Cited by 1 (1 self)
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Abstract. We present a framework for designing and analyzing primaldual interiorpoint methods for convex optimization. We assume that a selfconcordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interiorpoint methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primaldual interiorpoint techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.