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19
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 39 (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
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation
 SIAM Journal on Scientific Computing
, 2010
"... Abstract. We propose a fast algorithm for solving the ℓ1regularized minimization problem minx∈R n µ‖x‖1 + ‖Ax − b ‖ 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a ..."
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Cited by 24 (7 self)
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Abstract. We propose a fast algorithm for solving the ℓ1regularized minimization problem minx∈R n µ‖x‖1 + ‖Ax − b ‖ 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a firstorder iterative method called “shrinkage ” yields an estimate of the subset of components of x likely to be nonzero in an optimal solution. Restricting the decision variables x to this subset and fixing their signs at their current values reduces the ℓ1norm ‖x‖1 to a linear function of x. The resulting subspace problem, which involves the minimization of a smaller and smooth quadratic function, is solved in the second phase. Our code FPC AS embeds this basic twostage algorithm in a continuation (homotopy) approach by assigning a decreasing sequence of values to µ. This code exhibits stateoftheart performance both in terms of its speed and its ability to recover sparse signals. It can even recover signals that are not as sparse as required by current compressive sensing theory.
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 9 (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.
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.
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 6 (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.
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.
A Filter ActiveSet TrustRegion Method
, 2007
"... 2.1 Sequential LinearQuadratic Programming Methods.............. 3 2.2 Difficulties with the LP/TR Step Computation................ 4 ..."
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Cited by 3 (0 self)
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2.1 Sequential LinearQuadratic Programming Methods.............. 3 2.2 Difficulties with the LP/TR Step Computation................ 4
A Line Search Exact Penalty Method Using Steering Rules
, 2009
"... Line search algorithms for nonlinear programming must include safeguards to enjoy global convergence properties. This paper describes an exact penalization approach that extends the class of problems that can be solved with line search SQP methods. In the new algorithm, the penalty parameter is adju ..."
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Cited by 3 (1 self)
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Line search algorithms for nonlinear programming must include safeguards to enjoy global convergence properties. This paper describes an exact penalization approach that extends the class of problems that can be solved with line search SQP methods. In the new algorithm, the penalty parameter is adjusted at every iteration to ensure sufficient progress in linear feasibility and to promote acceptance of the step. A trust region is used to assist in the determination of the penalty parameter (but not in the step computation). It is shown that the algorithm enjoys favorable global convergence properties. Numerical experiments illustrate the behavior of the algorithm on various difficult situations. 1
On the Use of Piecewise Linear Models in Nonlinear Programming
, 2010
"... This paper presents an activeset algorithm for largescale optimization that occupies the middle ground between sequential quadratic programming (SQP) and sequential linearquadratic programming (SLQP) methods. It consists of two phases. The algorithm first minimizes a piecewise linear approximati ..."
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Cited by 1 (0 self)
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This paper presents an activeset algorithm for largescale optimization that occupies the middle ground between sequential quadratic programming (SQP) and sequential linearquadratic programming (SLQP) methods. It consists of two phases. The algorithm first minimizes a piecewise linear approximation of the Lagrangian, subject to a linearization of the constraints, to determine a working set. Then, an equality constrained subproblem based on this working set and using second derivative information is solved in order to promote fast convergence. A study of the local and global convergence properties of the algorithm highlights the importance of the placement of the interpolation points that determine the piecewise linear model of the Lagrangian. 1