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113
Benchmarking Optimization Software with Performance Profiles
, 2001
"... We propose performance profiles  distribution functions for a performance metric  as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation. 1 Introduction The benchmarking of optimi ..."
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Cited by 367 (8 self)
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We propose performance profiles  distribution functions for a performance metric  as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation. 1 Introduction The benchmarking of optimization software has recently gained considerable visibility. Hans Mittlemann's [13] work on a variety of optimization software has frequently uncovered deficiencies in the software and has generally led to software improvements. Although Mittelmann's efforts have gained the most notice, other researchers have been concerned with the evaluation and performance of optimization codes. As recent examples, we cite [1, 2, 3, 4, 6, 12, 17]. The interpretation and analysis of the data generated by the benchmarking process are the main technical issues addressed in this paper. Most benchmarking efforts involve tables displaying the performance of each solver on each problem for a set of metrics such...
On the Implementation of an InteriorPoint Filter LineSearch Algorithm for LargeScale Nonlinear Programming
, 2004
"... We present a primaldual interiorpoint algorithm with a lter 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 phas ..."
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Cited by 240 (6 self)
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We present a primaldual interiorpoint algorithm with a lter 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
"... 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 114 (5 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.
On Augmented Lagrangian methods with general lowerlevel constraints
, 2005
"... Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. In ..."
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Cited by 74 (7 self)
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Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. Inexact resolution of the lowerlevel constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot. All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lowerlevel set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.
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 59 (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 Globally Convergent PrimalDual InteriorPoint Filter Method for Nonlinear Programming
, 2002
"... In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primaldual interiorpoint algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primaldual step obtained from the p ..."
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Cited by 46 (4 self)
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In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primaldual interiorpoint algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primaldual step obtained from the perturbed firstorder necessary conditions into a normal and a tangential step, whose sizes are controlled by a trustregion type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to firstorder critical points is proved for the new primaldual interiorpoint filter algorithm.
Some properties of regularization and penalization schemes for MPECs
 Optimization Methods and Software
, 2004
"... Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a secondorde ..."
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Cited by 31 (2 self)
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Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a secondorder sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. Existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
Interiorpoint algorithms, penalty methods and equilibrium problems
, 2003
"... Abstract. In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)—as nonlinear programs, using an interiorpoint approach. These problems pose theoretical difficultie ..."
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Cited by 29 (4 self)
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Abstract. In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)—as nonlinear programs, using an interiorpoint approach. These problems pose theoretical difficulties for nonlinear solvers, including interiorpoint methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interiorpoint algorithms encounter in general. 1.
The N k Problem in Power Grids: New Models, Formulations and Numerical Experiments
, 2008
"... Given a power grid modeled by a network together with equations describing the power flows, power generation and consumption, and the laws of physics, the socalled N − k problem asks whether there exists a set of k or fewer arcs whose removal will cause the system to fail. The case where k is small ..."
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Cited by 26 (1 self)
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Given a power grid modeled by a network together with equations describing the power flows, power generation and consumption, and the laws of physics, the socalled N − k problem asks whether there exists a set of k or fewer arcs whose removal will cause the system to fail. The case where k is small is of practical interest. We present theoretical and computational results involving a mixedinteger model and a continuous nonlinear model related to this question.
An exact primal—dual penalty method approach to warmstarting interiorpoint methods for linear programming
 Comput. Optim. Appl
"... Abstract. One perceived deficiency of interiorpoint methods in comparison to active set methods is their inability to efficiently reoptimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primaldual penalty approach to overcome this problem. We p ..."
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Cited by 26 (2 self)
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Abstract. One perceived deficiency of interiorpoint methods in comparison to active set methods is their inability to efficiently reoptimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primaldual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems. 1.