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26
A trust region method based on interior point techniques for nonlinear programming
 Mathematical Programming
, 1996
"... Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direc ..."
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Cited by 103 (17 self)
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Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primaldual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, SQP iteration, barrier method, trust region method.
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...
Interior methods for nonlinear optimization
 SIAM Review
, 2002
"... Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their ..."
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Cited by 76 (4 self)
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Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomialtime interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.
Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
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Cited by 65 (14 self)
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We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
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 and compleme ..."
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Cited by 24 (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...
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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Why a Pure Primal Newton Barrier Step May Be Infeasible
 SIAM Journal on Optimization
, 1993
"... Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when ..."
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Cited by 21 (3 self)
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Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when the barrier parameter is reduced at a reasonable rate. We present local analysis showing why a pure Newton step in a longstep barrier method for nonlinearly constrained optimization may be seriously infeasible, even when taken from an apparently favorable point. The features described are illustrated numerically and connected to known theoretical results for convex problems satisfying selfconcordancy assumptions. We also indicate the contrasting nature of an approximate step to the desired minimizer of the barrier function. 1. Introduction 1.1. Background Interior methods, most commonly based on barrier functions, have been applied with great practical success in recent years to many con...
Line Search Procedures for the Logarithmic Barrier Function
 SIAM Journal on Optimization
, 1991
"... Barrier methods for constrained optimization, widely applied in the 1960's and 1970's, have recently enjoyed a revival of popularity. For problems involving nonlinear functions, some modern interior methods include a line search with respect to a logarithmic barrier function or related potential fun ..."
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Cited by 17 (5 self)
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Barrier methods for constrained optimization, widely applied in the 1960's and 1970's, have recently enjoyed a revival of popularity. For problems involving nonlinear functions, some modern interior methods include a line search with respect to a logarithmic barrier function or related potential function. Standard line search procedures tend to be inefficient in this context for two reasons: an inappropriate choice of initial trial step, and poor approximation of the barrier function by loworder polynomial interpolants. This paper discusses line search strategies specifically designed for the logarithmic barrier function. 1. Barrier Methods for Constrained Optimization Beginning in 1984 with the work of Karmarkar [Kar84], a resurgence of interest has taken place in barrier methods for constrained optimization. Classical barrier methods, which were popular in the 1960's and early 1970's, treat inequality constraints by creating a transformed function containing a positive singularity ...
Solving the SumofRatios Problem by an InteriorPoint Method
 Journal of Global Optimization
, 1999
"... . We consider the problem of minimizing the sum of a convex function and of p 1 fractions subject to convex constraints. The numerators of the fractions are positive convex functions, and the denominators are positive concave functions. Thus, each fraction is quasiconvex. We give a brief discussio ..."
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Cited by 17 (0 self)
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. We consider the problem of minimizing the sum of a convex function and of p 1 fractions subject to convex constraints. The numerators of the fractions are positive convex functions, and the denominators are positive concave functions. Thus, each fraction is quasiconvex. We give a brief discussion of the problem and prove that in spite of its special structure, the problem is NPcomplete even when only p = 1 fraction is involved. We then show how the problem can be reduced to the minimization of a function of p variables where the function values are given by the solution of certain convex subproblems. Based on this reduction, we propose an algorithm for computing the global minimum of the problem by means of an interiorpoint method for convex programs. Keywords: fractional programming, sum of ratios, global optimum, convex subproblem, interiorpoint method, NPcompleteness, knapsack problem 1. Introduction Nonlinear programming problems often involve objective functions that can...
A feasible BFGS interior point algorithm for solving strongly convex minimization problems
 SIAM J. OPTIM
, 2000
"... We propose a BFGS primaldual interior point method for minimizing a convex function on a convex set defined by equality and inequality constraints. The algorithm generates feasible iterates and consists in computing approximate solutions of the optimality conditions perturbed by a sequence of posit ..."
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Cited by 13 (1 self)
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We propose a BFGS primaldual interior point method for minimizing a convex function on a convex set defined by equality and inequality constraints. The algorithm generates feasible iterates and consists in computing approximate solutions of the optimality conditions perturbed by a sequence of positive parameters µ converging to zero. We prove that it converges qsuperlinearly for each fixed µ. We also show that it is globally convergent to the analytic center of the primaldual optimalset when µ tends to 0 and strict complementarity holds.