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36
Projective transformations for interior point methods, part II: an algorithm for finding the weighted center of a polyhedral system, M.I.T. Operations Research Center working paper OR
, 1988
"... i.i ·.:.'r · ·. · i ..."
Further Development on the Interior Algorithm for Convex Quadratic Programming
 Dept. of EngineeringEconomic Systems, Stanford University
, 1987
"... The interior trust region algorithm for convex quadratic programming is further developed. This development is motivated by the barrier function and the "center" pathfollowing methods, which create a sequence of primal and dual interior feasible points converging to the optimal solution. ..."
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The interior trust region algorithm for convex quadratic programming is further developed. This development is motivated by the barrier function and the "center" pathfollowing methods, which create a sequence of primal and dual interior feasible points converging to the optimal solution. At each iteration, the gap between the primal and dual objective values (or the complementary slackness value) is reduced at a global convergence ratio (1 \Gamma 1 4 p n ), where n is the number of variables in the convex QP problem. A safeguard line search technique is also developed to relax the smallstepsize restriction in the original path following algorithm. Key words: Convex Quadratic Programming, Primal and Dual, Complementarity Slackness, Polynomial Interior Algorithm. Abbreviated title: Interior Algorithm for Convex Quadratic Programming Since Karmarkar proposed the new polynomial algorithm (Karmarkar [19]), several developments have been made to the growing literature on interior a...
Multiparameter surfaces of analytic centers and longstep pathfollowing interior point methods
 RESEARCH REPORT 2, OPTIMIZATION LABORATORY, FACULTY OF INDUSTRIAL ENGINEERING AND MANAGEMENT, TECHNION  ISRAEL INSTITUTE OF TECHNOLOGY
, 1994
"... We develop a longstep polynomial time version of the Method of Analytic Centers for nonlinear convex problems. The method traces a multiparameter surface of analytic centers rather than the usual path, which allows to handle cases with noncentered and possibly infeasible starting point. ..."
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Cited by 9 (3 self)
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We develop a longstep polynomial time version of the Method of Analytic Centers for nonlinear convex problems. The method traces a multiparameter surface of analytic centers rather than the usual path, which allows to handle cases with noncentered and possibly infeasible starting point.
Outputsensitive algorithms for Tukey depth and related problems
, 2006
"... The Tukey depth (Tukey 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms ..."
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Cited by 9 (2 self)
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The Tukey depth (Tukey 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.
A globally convergent primaldual interior point algorithm for convex programming
 Mathematical Programming 64
, 1994
"... for convex programming ..."
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Barrier Functions and InteriorPoint Algorithms for Linear Programming with Zero, One, or TwoSided Bounds on the Variables
, 1993
"... This study examines two different barrier functions and their use in both pathfollowing and potentialreduction interiorpoint algorithms for solving a linear program of the form: minimize c T x subject to Ax = b and ` x u, where components of ` and u can be nonfinite, so the variables x j can ..."
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This study examines two different barrier functions and their use in both pathfollowing and potentialreduction interiorpoint algorithms for solving a linear program of the form: minimize c T x subject to Ax = b and ` x u, where components of ` and u can be nonfinite, so the variables x j can have 0\Gamma; 1\Gamma;or 2sided bounds, j = 1; :::; n: The barrier functions that we study include an extension of the standard logarithmic barrier function and an extension of a barrier function introduced by Nesterov. In the case when both ` and u have all of their components finite, these barrier functions are \Psi(x) = X j f\Gamma ln(u j \Gamma x j ) \Gamma ln(x j \Gamma ` j )g and \Psi(x) = X j f\Gamma ln(minfu j \Gamma x j ; x j \Gamma ` j g) + minfu j \Gamma x j ; x j \Gamma ` j g=((u j \Gamma ` j )=2)g: Each of these barrier functions gives rise to suitable primal and dual metrics that are used to develop both pathfollowing and potentialreduction interiorpoint algorithms ...
Efficient algorithm for approximating maximum inscribed sphere in high dimensional polytope
 In SCG ’06: Proceedings of the twentysecond annual symposium on Computational geometry
, 2006
"... In this paper, we consider the problem of computing a maximum inscribed sphere inside a high dimensional polytope formed by a set of halfspaces (or linear constraints), and present an efficient algorithm for computing a (1 − ǫ)approximation of the sphere. More specifically, given any bounded polyto ..."
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In this paper, we consider the problem of computing a maximum inscribed sphere inside a high dimensional polytope formed by a set of halfspaces (or linear constraints), and present an efficient algorithm for computing a (1 − ǫ)approximation of the sphere. More specifically, given any bounded polytope P defined by n ddimensional halfspaces, an interior point O of P, and a constant ǫ> 0, our algorithm computes in O(nd/ǫ 3) time a sphere inside P with a radius no less than (1 − ǫ)Ropt, where Ropt is the radius of a maximum inscribed sphere of P. Our algorithm is based on the coreset concept and a number of interesting geometric observations. Our result settles an open problem posted by Khachiyan and Todd [4] for the case of spheres. 1
On the worst case complexity of potential reduction algorithms for linear programming
, 1997
"... There are several classes of interior point algorithms that solve linear programming problems in O(x/ffL) iterations. Among them, several potential reduction algorithms combine both theoretical (o(vr~L) iterations) and practical efficiency as they allow the flexibility of line searches in the potent ..."
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There are several classes of interior point algorithms that solve linear programming problems in O(x/ffL) iterations. Among them, several potential reduction algorithms combine both theoretical (o(vr~L) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(v/'ffL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least ll(v/ffL) iterations to find an optimal solution.
On the simulatability condition in key generation over a nonauthenticated public channel,” Submitted to
 IEEE Trans. Inform. Theory
, 2014
"... Abstract—Simulatability condition is a fundamental concept in studying key generation over a nonauthenticated public channel, in which Eve is active and can intercept, modify and falsify messages exchanged over the nonauthenticated public channel. Using this condition, Maurer and Wolf showed a rem ..."
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Abstract—Simulatability condition is a fundamental concept in studying key generation over a nonauthenticated public channel, in which Eve is active and can intercept, modify and falsify messages exchanged over the nonauthenticated public channel. Using this condition, Maurer and Wolf showed a remarkable “all or nothing ” result: if the simulatability condition does not hold, the key capacity over the nonauthenticated public channel will be the same as that of the case with a passive Eve, while the key capacity over the nonauthenticated channel will be zero if the simulatability condition holds. However, two questions remain open so far: 1) For a given joint probability mass function (PMF), are there efficient algorithms (polynomial complexity algorithms) for checking whether the simulatability condition holds or not?; and 2) If the simulatability condition holds, are there efficient algorithms for finding the corresponding attack strategy? In this paper, we answer these two open questions affirmatively. In particular, for a given joint PMF, we construct a linear programming (LP) problem and show that the simulatability condition holds if and only if the optimal value obtained from the constructed LP is zero. Furthermore, we construct another LP and show that the minimizer of the newly constructed LP is a valid attack strategy. Both LPs can be solved with a polynomial complexity.
Two InteriorPoint Algorithms for a Class of Convex Programming Problems
, 1994
"... This paper describes two algorithms for the problem of minimizing a linear function over the intersection of an affine set and a convex set which is required to be the closure of the domain of a strongly selfconcordant barrier function. One algorithm is a pathfollowing method, while the other is a ..."
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This paper describes two algorithms for the problem of minimizing a linear function over the intersection of an affine set and a convex set which is required to be the closure of the domain of a strongly selfconcordant barrier function. One algorithm is a pathfollowing method, while the other is a primal potentialreduction method. We give bounds on the number of iterations necessary to attain a given accuracy.