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25
An Algorithm for Nonlinear Optimization Using Linear Programming and Equality Constrained Subproblems
, 2003
"... This paper describes an activeset algorithm for largescale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the activ ..."
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Cited by 41 (12 self)
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This paper describes an activeset algorithm for largescale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ` 1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program.
KNITRO: An integrated package for nonlinear optimization
 Large Scale Nonlinear Optimization, 35–59, 2006
, 2006
"... This paper describes Knitro 5.0, a Cpackage for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving largescale, smooth nonlinear programming problems ..."
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Cited by 38 (3 self)
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This paper describes Knitro 5.0, a Cpackage for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving largescale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an activeset method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings. 1
Steering Exact Penalty Methods for Nonlinear Programming
, 2007
"... This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. I ..."
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Cited by 11 (0 self)
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This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential quadratic programming (SQP) and sequential linearquadratic programming (SLQP) methods that use trust regions to promote convergence. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.
Methods for nonlinear constraints in optimization calculations
 THE STATE OF THE ART IN NUMERICAL ANALYSIS
, 1996
"... ..."
A proximal method for composite minimization
, 2008
"... Abstract. We consider minimization of functions that are compositions of proxregular functions with smooth vector functions. A wide variety of important optimization problems can be formulated in this way. We describe a subproblem constructed from a linearized approximation to the objective and a r ..."
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Cited by 7 (2 self)
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Abstract. We consider minimization of functions that are compositions of proxregular functions with smooth vector functions. A wide variety of important optimization problems can be formulated in this way. We describe a subproblem constructed from a linearized approximation to the objective and a regularization term, investigating the properties of local solutions of this subproblem and showing that they eventually identify a manifold containing the solution of the original problem. We propose an algorithmic framework based on this subproblem and prove a global convergence result.
Active set identification in Nonlinear Programming
 SIAM Journal on Optimization
, 2006
"... Abstract. Techniques that identify the active constraints at a solution of a nonlinear programming problem from a point near the solution can be a useful adjunct to nonlinear programming algorithms. They have the potential to improve the local convergence behavior of these algorithms, and in the bes ..."
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Cited by 6 (1 self)
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Abstract. Techniques that identify the active constraints at a solution of a nonlinear programming problem from a point near the solution can be a useful adjunct to nonlinear programming algorithms. They have the potential to improve the local convergence behavior of these algorithms, and in the best case can reduce an inequality constrained problem to an equality constrained problem with the same solution. This paper describes several techniques that do not require good Lagrange multiplier estimates for the constraints to be available a priori, but depend only on function and first derivative information. Computational tests comparing the effectiveness of these techniques on a variety of test problems are described. Many tests involve degenerate cases, in which the constraint gradients are not linearly independent and/or strict complementarity does not hold.
An activeset algorithm for nonlinear programming using linear programming and equality constrained subproblems
, 2002
"... This paper describes an activeset algorithm for largescale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [9]. The step computation is performed in two stages. In the rst stage a linear program is solved to estimate the active set ..."
Abstract

Cited by 5 (1 self)
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This paper describes an activeset algorithm for largescale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [9]. The step computation is performed in two stages. In the rst stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the `1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active atthesolution of the linear program. The EQP incorporates a trustregion constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1] test set.
A Sequential Quadratic Programming Algorithm with an Additional Equality Constrained Phase
, 2008
"... A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods. It avoids the difficulties associated with indefinite quadratic programming subproblems by defining this subproblem to be always convex. The novel feature of the appr ..."
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Cited by 5 (1 self)
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A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods. It avoids the difficulties associated with indefinite quadratic programming subproblems by defining this subproblem to be always convex. The novel feature of the approach is the addition of an equality constrained phase that promotes fast convergence and improves performance in the presence of ill conditioning. This equality constrained phase uses exact second order information and can be implemented using either a direct solve or an iterative method. The paper studies the global and local convergence properties of the new algorithm and presents a set of numerical experiments to illustrate its practical performance.
On the Convergence of Successive Linear Programming Algorithms
, 2003
"... We analyze the global convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more speci cally the SLPEQP approach presented in [1]. Every iteration requires the solution of two trust region subproblems inv ..."
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Cited by 4 (1 self)
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We analyze the global convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more speci cally the SLPEQP approach presented in [1]. Every iteration requires the solution of two trust region subproblems involving linear and quadratic models, respectively. The interaction between the trust regions of these subproblems requires careful consideration. It is shown under mild assumptions that there exist an accumulation point which is a critical point for the penalty function.
Steering Exact Penalty Methods
, 2004
"... This paper reviews the development of exact penalty methods for nonlinear optimization and discusses their increasingly important role in optimization algorithms and software. In their most recent stage of development, penalty methods adjust the penalty parameter dynamically. By controlling the deg ..."
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Cited by 4 (2 self)
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This paper reviews the development of exact penalty methods for nonlinear optimization and discusses their increasingly important role in optimization algorithms and software. In their most recent stage of development, penalty methods adjust the penalty parameter dynamically. By controlling the degree of linear feasibility achieved at every iteration, these methods balance progress toward optimality and feasibility. The choice of the penalty parameter thus ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential linearquadratic penalty methods, and is then extended to sequential quadratic programming. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.