A new trust region algorithm for bound constrained minimization

Friedlander, Ana; Martínez, José Mário; Santos, Sandra A.

doi:10.1007/bf01183013

Cited by 116 publications

(93 citation statements)

References 8 publications

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“…Since the product of two strongly semismooth functions in again strongly semismooth, the first statement follows from Lemma 4.1 together with our general smoothness assumption on the mapping f . The remaining statements follow directly from the definition of the B-subdifferential, see Definition 2.3 (note, however, that, usually, (19), (20) contain more elements than those belonging to ∂ B G(x)).…”

Section: Affine-scaling Interior-point Newton Methodsmentioning

confidence: 99%

“…Taking into account the definition of M (x * ) in (19), (20), it follows after some elementary calculations that the ith column vector A i of A := M (x * ) T is given by…”

Section: Local Convergence Analysismentioning

confidence: 99%

“…The optimization problem (P) has attracted quite a few researchers during the last 15 years, and a number of different methods for its solution may be found in [4,5,9,10,15,16,20,23,24,25,31]. The approach we follow in this work is typically called the affine-scaling interior-point Newton method.…”

Section: Introductionmentioning

confidence: 99%

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On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints

Kanzow

Klug

2006

Comput Optim Applic

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Abstract. A class of new affine-scaling interior-point Newton-type methods are considered for the solution of optimization problems with bound constraints. The methods are shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict complementarity of the solution. The new methods differ from previous ones by Coleman and Li [Mathematical Programming, 67 (1994), pp. 189-224] and Heinkenschloss, Ulbrich, and Ulbrich [Mathematical Programming, 86 (1999), pp. 615-635] mainly in the choice of the scaling matrix. The scaling matrices used here have stronger smoothness properties and allow the application of standard results from nonsmooth analysis in order to obtain a relatively short and elegant local convergence result. An important tool for the definition of the new scaling matrices is the correct identification of the degenerate indices. Some illustrative numerical results with a comparison of the different scaling techniques are also included.

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Section: Affine-scaling Interior-point Newton Methodsmentioning

confidence: 99%

“…Taking into account the definition of M (x * ) in (19), (20), it follows after some elementary calculations that the ith column vector A i of A := M (x * ) T is given by…”

Section: Local Convergence Analysismentioning

confidence: 99%

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On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints

Kanzow

Klug

2006

Comput Optim Applic

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“…This is a Fortran double-precision code for solving nonlinear programming problems, based on augmented Lagrangian [19], trust region [13] and projected gradients combined with a mild active set strategy [6]. It is available at http://www.ime.unicamp.br/ ∼ martinez.…”

Section: A Simple Instance and An Alternative Formulationmentioning

confidence: 99%

Box-constrained minimization reformulations of complementarity problems in second-order cones

et al. 2007

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Reformulations of a generalization of a second-order cone complementarity problem (GSOCCP) as optimization problems are introduced, that preserve differentiability. Equivalence results are proved in the sense that the global minimizers of the reformulations with zero objective value are solutions to the GSOCCP and vice versa. Since the optimization problems involved include only simple constraints, a whole range of minimization algorithms may be used to solve the equivalent problems. Taking into account that optimization algorithms usually seek stationary points, a theoretical result is established that ensures equivalence between stationary points of the reformulation and solutions to the GSOCCP. Numerical experiments are presented that illustrate the advantages and disadvantages of the reformulations.

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“…The minimization of a nonlinear function subject to bounds on the variables has been the subject of intense previous work, along many possible avenues. Major classes of algorithms for bound-constrained problems include the ones based on: active or -active set methods (see, e.g., [1,13,32] and more recently [18] for a short review on active set methods); trust-region methods (see, e.g., [6,7,14,22,24]); interior-point methods (see, e.g., [5,10,19]); line-search projected gradient methods (see, e.g., [2] and the references therein; see also [3,25,35] for a limited memory BFGS method); and filter type methods (see [31]). The approach proposed and analyzed in this paper belongs to the trust-region class but also shares the flavor of projected gradient methods.…”

Section: Introductionmentioning

confidence: 99%

Globally convergent DC trust-region methods

et al. 2014

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In this paper, we investigate the use of DC (Difference of Convex functions) models and algorithms in the application of trust-region methods to the solution of a class of nonlinear optimization problems where the constrained set is closed and convex (and, from a practical point of view, where projecting onto the feasible region is computationally affordable). We consider DC local models for the quadratic model of the objective function used to compute the trust-region step, and apply a primal-dual subgradient method to the solution of the corresponding trust-region subproblems.One is able to prove that the resulting scheme is globally convergent for first-order stationary points. The theory requires the use of exact second-order derivatives but, in turn, the computation of the trust-region step asks only for one projection onto the feasible region (in comparison to the calculation of the generalized Cauchy point which may require more).The numerical efficiency and robustness of the proposed new scheme when applied to bound-constrained problems is measured by comparing its performance against some of the current state-of-the-art nonlinear programming solvers on a vast collection of test problems.

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A new trust region algorithm for bound constrained minimization

Cited by 116 publications

References 8 publications

On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints

On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints

Box-constrained minimization reformulations of complementarity problems in second-order cones

Globally convergent DC trust-region methods

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