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25
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 80 (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...
ON PROJECTED NEWTON BARRIER METHODS FOR LINEAR PROGRAMMING AND AN EQUIVALENCE TO KARMARKAR'S PROJECTIVE METHOD
, 1986
"... Interest in linear programming has been intensified recently by Karmarkar's publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrierfunction methods for nonlinear programming based on applying a logarithmic tra ..."
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Cited by 64 (8 self)
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Interest in linear programming has been intensified recently by Karmarkar's publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrierfunction methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a "projected Newton barrier" method. This method is shown to be equivalent to Karmarkar's projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several nontrivial test problems, and the implications for future developments in linear programming are discussed.
Convex Nondifferentiable Optimization: A Survey Focussed On The Analytic Center Cutting Plane Method.
, 1999
"... We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct pr ..."
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Cited by 51 (2 self)
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We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
InteriorPoint Methods for Linear Optimization
, 2000
"... Everyone with some background in Mathematics knows how to solve a system of linear equalities, since it is the basic subject in Linear Algebra. In many practical problems, however, also inequalities play a role. For example, a budget usually may not be larger than some specified amount. In such situ ..."
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Cited by 18 (6 self)
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Everyone with some background in Mathematics knows how to solve a system of linear equalities, since it is the basic subject in Linear Algebra. In many practical problems, however, also inequalities play a role. For example, a budget usually may not be larger than some specified amount. In such situations one may end up with a system of linear relations that not only contains equalities but also inequalities. Solving such a system requires methods and theory that go beyond the standard Mathematical knowledge. Nevertheless the topic has a rich history and is tightly related to the important topic of Linear Optimization, where the object is to nd the optimal (minimal or maximal) value of a linear function subject to linear constraints on the variables; the constraints may be either equality or inequality constraints. Both from a theoretical and computational point of view both topics are equivalent. In this chapter we describe the ideas underlying a new class of solution methods...
BranchandCut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem
 Comput. Optim. Appl
, 1999
"... The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years. This inequality permits to reduce in a elegant way various problems of robust control into its form. However, on ..."
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Cited by 9 (1 self)
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The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years. This inequality permits to reduce in a elegant way various problems of robust control into its form. However, on the contrary of the Linear Matrix Inequality (LMI) which can be solved by interiorpointmethods, the BMI is a computationally difficult object in theory and in practice. This article improves the branchandbound algorithm of Goh, Safonov and Papavassilopoulos (1995) by applying a better convex relaxation of the BMI Eigenvalue Problem (BMIEP), and proposes new BranchandBound and BranchandCut Algorithms. Numerical experiments were conducted in a systematic way over randomly generated problems, and they show the robustness and the efficiency of the proposed algorithms. Keywords: Bilinear Matrix Inequality, BranchandCut Algorithm, Convex Relaxation, Cut Polytope. y Author supported b...
A Generic PathFollowing Algorithm With a Sliding Constraint and Its Application to Linear Programming and the Computation of Analytic Centers
, 1996
"... We propose a generic pathfollowing scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic pr ..."
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Cited by 8 (5 self)
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We propose a generic pathfollowing scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic procedure in use, primal, dual or primaldual. We show convergence in O( p n) iterations. We verify that the primal, dual and primaldual algorithms satisfy the local quadratic convergence property. The method can be applied to solve the linear programming problem (with a feasible start) and to compute the analytic center of a bounded polytope. The generic pathfollowing scheme easily extends to the logarithmic penalty barrier approach. Keywords Interior point method, method of centers, pathfollowing, linear programming. This work has been completed with the support from the Swiss National Foundation for Scientific Research, grant 1234002.92. 1 Introduction Shortly after Karmarkar's s...
Monotone Semidefinite Complementarity Problems
, 1996
"... . In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditi ..."
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Cited by 7 (1 self)
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. In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditions. Especially, for the monotone semidefinite linear complementarity problem, the trajectory converges to an analytic center of the solution set, provided that there exists a strictly complementary solution. Finally, we propose the globally convergent infeasibleinteriorpoint algorithm for the SDCP. Key words Monotone Semidefinite Complementarity Problem, Trajectory, Interior Point Algorithm Research Report B312 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. Let M(n) and S(n) denote the class of n2n real matrices and the class of n2n symmetric real matrices, respectively. Assume that A; B 2 M(n)....
Centers of Monotone Generalized Complementarity Problems
 Math. Oper. Res
, 1996
"... . Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity prob ..."
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Cited by 7 (4 self)
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. Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with C and M: Find (x; y) 2 M " (C 2C 3 ) such that hx; yi = 0. Here C 3 = fy 2 E : hx; yi 0 for all x 2 Cg denotes the dual cone of C. The main result of the paper unifies and extends some results established for monotone complementarity problems in Euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices. Key words Central Trajectory, Path of Centers, Complementarity Problem, Interior Point Algorithm, Linear Program Research Report B303 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. The central trajectory or the path of centers is known ...
Interior Point Methods for Nondifferentiable Optimization
, 1998
"... We describe the analytic center cutting plane method and its relationship to classical methods of nondifferentiable optimization and column generation. Implementation issues are also discussed, and current applications listed. Keywords Projective Algorithm, Analytic Center, Cutting Plane Method. T ..."
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Cited by 6 (2 self)
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We describe the analytic center cutting plane method and its relationship to classical methods of nondifferentiable optimization and column generation. Implementation issues are also discussed, and current applications listed. Keywords Projective Algorithm, Analytic Center, Cutting Plane Method. This work has been completed with support from the Fonds National Suisse de la Recherche Scientifique, grant 1242503.94, from the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152 and from the FCAR of Quebec. GERAD/Faculty of Management, McGill University, 1001, Sherbrooke West, Montreal, Que., H3A 1G5, Canada. Email: ma56@musica.mcgill.ca. LOGILAB/Management Studies, University of Geneva, 102, Bd CarlVogt, CH1211 Gen`eve 4, Switzerland. Email: jpvial@hec.unige.ch. 1 Introduction Nondifferentiable convex optimization may be deemed an arcane topic in the field of optimization. Truly enough, many a times problems that are formulated as nondiffere...