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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 ..."
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Cited by 83 (3 self)
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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...
A Continuous Approach to Inductive Inference
 Mathematical Programming
, 1992
"... In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g ..."
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Cited by 38 (2 self)
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In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g using outputs obtained by applying a limited number of random inputs to the hidden function. Given this inputoutput sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used...
On the Riemannian geometry defined by selfconcordant barriers and interiorpoint methods
 Found. Comput. Math
"... We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interiorpoint methods for optimizing a linear function over the intersection of the set with ..."
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Cited by 27 (0 self)
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We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interiorpoint methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primaldual central path are in some sense close to optimal. The same is true for methods that follow the shifted primaldual central path among certain infeasibleinteriorpoint methods. We also compute the geodesics in several simple sets. ∗ Copyright (C) by SpringerVerlag. Foundations of Computational Mathewmatics 2 (2002), 333–361.
On the curvature of the central path of linear programming theory
 Foundations of Computational Mathematics
, 2003
"... Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming pro ..."
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Cited by 13 (2 self)
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Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming problem defined on a compact polytope contained in R n, the total curvature of the central path is less than or
Iteration Algorithm for Computing Bounds in Quadratic Optimization Problems
 Complexity in Numerical Optimization
, 1993
"... We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NPhard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem for minimizi ..."
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Cited by 6 (0 self)
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We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NPhard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem for minimizing the trace of a matrix subject to positive definiteness condition. We then propose an interiorpoint method to solve this problem. We show that the algorithm takes no more than O(nL) iterations (where L is the the number of bits required to represent the input). The algorithm does two matrix inversions in each iteration . Keywords: Bounds, complexity, quadratic optimization, interior point methods. 1 Outline The second section of the paper shall introduce the problem of computing upper bounds on a quadratic optimization problem. We shall also motivate an interior point approach to solving the problem. The third section gives an interior point method for solving the problem. The algorith...
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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Cited by 5 (4 self)
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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 ...
An Information Geometric Approach to Polynomialtime Interiorpoint Algorithms — Complexity Bound via Curvature Integral —
, 2007
"... In this paper, we study polynomialtime interiorpoint algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for con ..."
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Cited by 1 (0 self)
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In this paper, we study polynomialtime interiorpoint algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for conic linear programs introduced by selfconcordant barrier functions, and develop a precise iterationcomplexity estimate of the polynomialtime interiorpoint algorithm based on an integral of (embedding) curvature of the central trajectory in a rigorous differential geometrical sense. We further study implication of the theory applied to classical linear programming, and establish a surprising link to the strong “primaldual curvature ” integral bound established by Monteiro and Tsuchiya, which is based on the work of Vavasis and Ye of the layeredstep interiorpoint algorithm. By using these results, we can show that the total embedding curvature of the central trajectory, i.e., the aforementioned integral over the whole central trajectory, is bounded by O(n3.5 log(¯χ ∗ A + n)) where ¯χ ∗ A is a condition number of the coefficient matrix A and n is the number of nonnegative variables. In particular, the integral is bounded by O(n4.5m) for combinatorial linear programs including network flow problems where m is the number of constraints. We also provide a complete differentialgeometric characterization of the primaldual curvature in the primaldual algorithm. Finally, in view of this integral bound, we observe that the primal (or dual) interiorpoint algorithm requires fewer number of iterations than the primaldual interiorpoint algorithm at least in the case of linear programming.
Central paths in semidefinite programming, generalized proximalpoint method and Cauchy trajectories in Riemannian manifolds
 J. Optim. Theory Appl
"... The relationships among central path in the context of semidefinite programming, generalized proximal point method and Cauchy trajectory in Riemannian manifolds is studied in this paper. First it is proved that the central path associated to the general function is well defined. The convergence and ..."
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Cited by 1 (0 self)
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The relationships among central path in the context of semidefinite programming, generalized proximal point method and Cauchy trajectory in Riemannian manifolds is studied in this paper. First it is proved that the central path associated to the general function is well defined. The convergence and characterization of its limit point is established for functions satisfying a certain continuous property. Also, the generalized proximal point method is considered, and it is proved that the corresponding generated sequence is contained in the central path. As a consequence, both converge to the same point. Finally, it is proved that the central path coincides with the Cauchy trajectory in the Riemannian manifold.
Information Geometry and PrimalDual Interiorpoint Algorithms
, 2009
"... In this paper, we study polynomialtime interiorpoint algorithms in view of information geometry. We introduce an information geometric structure for a conic linear program based on a selfconcordant barrier function. Riemannian metric is defined with the Hessian of the barrier function. We introdu ..."
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In this paper, we study polynomialtime interiorpoint algorithms in view of information geometry. We introduce an information geometric structure for a conic linear program based on a selfconcordant barrier function. Riemannian metric is defined with the Hessian of the barrier function. We introduce two connections ∇ and ∇ ∗ which roughly corresponds to the primal and the dual problem. The dual feasible region is embedded in the primal cone and thus we consider the primal and dual problems in the same space. Characterization of the central trajectory and its property in view of the curvature is studied. A predictorcorrector primal pathfollowing algorithm is represented based on this geometry and (its asymptotic) iterationcomplexity is related to an integral involving the embedding curvature. Then we focus on the classical linear program and primaldual algorithm. We will study an integral over the central trajectory which represents the number of iterations of the MizunoToddYe predictorcorrector (MTYPC) algorithm. We will show that this integral admits a rigorous differential geometric expression involving the embedding curvature. Connecting this expression to an integral bound previously obtained by Monteiro and Tsuchiya in relation to the layered interiorpoint algorithm by Vavasis and Ye, we prove