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21
On the Implementation of an InteriorPoint Filter LineSearch Algorithm for LargeScale Nonlinear Programming
, 2004
"... We present a primaldual interiorpoint algorithm with a filter linesearch method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration ph ..."
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Cited by 294 (6 self)
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We present a primaldual interiorpoint algorithm with a filter linesearch method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, secondorder corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several linesearch options, and a comparison is provided with two stateoftheart interiorpoint codes for nonlinear programming.
Interior methods for nonlinear optimization
 SIAM REVIEW
, 2002
"... 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 ..."
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Cited by 127 (6 self)
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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.
An interior point algorithm for largescale nonlinear . . .
, 2002
"... Nonlinear programming (NLP) has become an essential tool in process engineering, leading to prot gains through improved plant designs and better control strategies. The rapid advance in computer technology enables engineers to consider increasingly complex systems, where existing optimization codes ..."
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Cited by 64 (3 self)
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Nonlinear programming (NLP) has become an essential tool in process engineering, leading to prot gains through improved plant designs and better control strategies. The rapid advance in computer technology enables engineers to consider increasingly complex systems, where existing optimization codes reach their practical limits. The objective of this dissertation is the design, analysis, implementation, and evaluation of a new NLP algorithm that is able to overcome the current bottlenecks, particularly in the area of process engineering. The proposed algorithm follows an interior point approach, thereby avoiding the combinatorial complexity of identifying the active constraints. Emphasis is laid on exibility in the computation of search directions, which allows the tailoring of the method to individual applications and is mandatory for the solution of very large problems. In a fullspace version the method can be used as general purpose NLP solver, for example in modeling environments such as Ampl. The reduced space version, based on coordinate decomposition, makes it possible to tailor linear algebra
A PRIMALDUAL TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION
, 2003
"... This paper concerns general (nonconvex) nonlinear optimization when first and second derivatives of the objective and constraint functions are available. The proposed method is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. T ..."
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Cited by 21 (3 self)
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This paper concerns general (nonconvex) nonlinear optimization when first and second derivatives of the objective and constraint functions are available. The proposed method is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved using a secondderivative Newtontype method that employs a combined trust region and line search strategy to ensure global convergence. It is shown that the trustregion step can be computed by factorizing a sequence of systems with diagonallymodified primaldual structure, where the inertia of these systems can be determined without recourse to a special factorization method. This has the benefit that offtheshelf linear system software can be used at all times, allowing the straightforward extension to largescale problems. Numerical results are given for problems in the COPS test collection.
Global and local convergence of line search filter methods for nonlinear programming
, 1521
"... Line search methods for nonlinear programming using Fletcher and Leyffer’s filter method, which replaces the traditional merit function, are proposed and their global and local convergence properties are analyzed. Previous theoretical work on filter methods has considered trust region algorithms and ..."
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Cited by 18 (4 self)
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Line search methods for nonlinear programming using Fletcher and Leyffer’s filter method, which replaces the traditional merit function, are proposed and their global and local convergence properties are analyzed. Previous theoretical work on filter methods has considered trust region algorithms and only the question of global convergence. The presented framework is applied to barrier interior point and active set SQP algorithms. Under mild assumptions it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible, and that there exists at least one limit point that is a stationary point for the problem under consideration. Furthermore, it is shown that the proposed methods do not suffer from the Maratos effect if the search directions are improved by second order corrections, so that fast local convergence to strict local solutions is achieved. A new alternative filter approach employing the Lagrangian function instead of the objective function with identical global convergence properties is briefly discussed.
Generalized stationary points and an interiorpoint method for mathematical programs with equilibrium constraints
 Industrial Engineering & Management Sciences, Northwestern University
, 2005
"... Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier proble ..."
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Cited by 15 (0 self)
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Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPECLICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interiorpoint algorithm failed to find a stationary point. Key words: Global convergence, interiorpoint methods, mathematical programming with equilibrium constraints, stationary point
2008, Dynamic updates of the barrier parameter in primaldual methods for nonlinear programming,Computational Optimization and Applications
"... Abstract. We introduce a framework in which updating rules for the barrier parameter in primaldual interiorpoint methods become dynamic. The original primaldual system is augmented to incorporate explicitly an updating function. A Newton step for the augmented system gives a primaldual Newton s ..."
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Cited by 11 (3 self)
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Abstract. We introduce a framework in which updating rules for the barrier parameter in primaldual interiorpoint methods become dynamic. The original primaldual system is augmented to incorporate explicitly an updating function. A Newton step for the augmented system gives a primaldual Newton step and also a step in the barrier parameter. Based on local information and a linesearch, the decrease of the barrier parameter is automatically adjusted. We analyze local convergence properties, report numerical experiments on a standard collection of nonlinear problems and compare our results to a stateoftheart interiorpoint implementation. In many instances, the adaptive algorithm reduces the number of iterations and of function evaluations. Its design guarantees a better fit between the magnitudes of the primaldual residual and of the barrier parameter along the iterations.
Effects Of FinitePrecision Arithmetic On InteriorPoint Methods For Nonlinear Programming
 Preprint ANL/MCSP7050198, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne
, 1998
"... We show that the effects of finiteprecision arithmetic in forming and solving the linear system that arises at each iteration of primaldual interiorpoint algorithms for nonlinear programming are benign. When we replace the standard assumption that the active constraint gradients are independentby ..."
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Cited by 10 (2 self)
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We show that the effects of finiteprecision arithmetic in forming and solving the linear system that arises at each iteration of primaldual interiorpoint algorithms for nonlinear programming are benign. When we replace the standard assumption that the active constraint gradients are independentby the weaker MangasarianFromovitz constraint qualifiation, rapid convergence usually is attainable, even when cancellation and roundoff errors occur during the calculations. In deriving our main results, we proveakey technical result about the size of the exact primaldual step. This result can be used to modify existing analysis of primaldual interiorpoint methods for convex programming, making it possible to extend the superlinear local convergence results to the nonconvex case.
Constrained optimization in seismic reflection tomography: an SQP augmented Lagrangian approach, in "Geophysical Journal International
, 2005
"... Geophysical methods for imaging a complex geological subsurface in petroleum exploration requires the determination of an accurate wave propagation velocity model. Seismic reflection tomography turns out to be an efficient method for doing this: it determines the ..."
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Cited by 9 (1 self)
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Geophysical methods for imaging a complex geological subsurface in petroleum exploration requires the determination of an accurate wave propagation velocity model. Seismic reflection tomography turns out to be an efficient method for doing this: it determines the
InteriorPoint l_2Penalty Methods for Nonlinear Programming with Strong Global Convergence Properties
 Math. Programming
, 2004
"... We propose two line search primaldual interiorpoint methods that have a generic barrierSQP outer structure and approximately solve a sequence of equality constrained barrier subproblems. To enforce convergence for each subproblem, these methods use an # 2 exact penalty function eliminating the n ..."
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Cited by 9 (0 self)
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We propose two line search primaldual interiorpoint methods that have a generic barrierSQP outer structure and approximately solve a sequence of equality constrained barrier subproblems. To enforce convergence for each subproblem, these methods use an # 2 exact penalty function eliminating the need to drive the corresponding penalty parameter to infinity when finite multipliers exist. Instead of directly decreasing an equality constraint infeasibility measure, these methods attain feasibility by forcing this measure to zero whenever the steps generated by the methods tend to zero. Our analysis shows that under standard assumptions, our methods have strong global convergence properties. Specifically, we show that if the penalty parameter remains bounded, any limit point of the iterate sequence is either a KKT point of the barrier subproblem, or a FritzJohn (FJ) point of the original problem that fails to satisfy the MangasarianFromovitz constraint qualification (MFCQ); if the penalty parameter tends to infinity, there is a limit point that is either an infeasible FJ point of the inequality constrained feasibility problem (an infeasible stationary point of the infeasibility measure if slack variables are added) or a FJ point of the original problem at which the MFCQ fails to hold. Numerical results are given that illustrate these outcomes.