A nonlinear programming algorithm based on non-coercive penalty functions

Gonzaga, Clóvis C.; Castillo, R. A.

doi:10.1007/s10107-002-0332-z

Cited by 39 publications

(53 citation statements)

References 8 publications

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“…We propose a purely primal method based on appropriate smoothing of the l 1 -penalty function and give update rules for the penalty and smoothing parameters. Especially, for the case of inequality constraints only there exists a variety of proposed algorithms using different smoothing and exact penalty functions, see for example [9,16,30,35,37,40]. Our work extends the results in [37] and compared with the work of [9,35,40] our approach differs in the choice of the smoothing kernel, the consideration of general equality constraints and by the estimates on the update rules for parameters of our introduced algorithm.…”

Section: Introductionmentioning

confidence: 62%

“…Our work extends the results in [37] and compared with the work of [9,35,40] our approach differs in the choice of the smoothing kernel, the consideration of general equality constraints and by the estimates on the update rules for parameters of our introduced algorithm. In contrast to [16] or [30] we present a completely different algorithm including also box constraints and equality constraints, respectively. Further, there are special methods known, that are capable to deal with non-differentiable penalty functions, see for example [11].…”

Section: Introductionmentioning

confidence: 96%

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Smoothed penalty algorithms for optimization of nonlinear models

et al. 2007

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We introduce an algorithm for solving nonlinear optimization problems with general equality and box constraints. The proposed algorithm is based on smoothing of the exact l 1 -penalty function and solving the resulting problem by any box-constraint optimization method. We introduce a general algorithm and present theoretical results for updating the penalty and smoothing parameter. We apply the algorithm to optimization problems for nonlinear traffic network models and report on numerical results for a variety of network problems and different solvers for the subproblems.

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Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 96%

Smoothed penalty algorithms for optimization of nonlinear models

et al. 2007

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“…Liuzzi and Lucidi [20] develop convergence theory for the smoothed-∞ penalty function in the context of pattern search methods and also handle the linear constraints explicitly. Kaplan [13] and Gonzaga and Castillo [7] develop general convergence theory for 1 smoothing functions in the context of nonlinear programming. Polyak [29] provides an extensive discussion on the general properties of smoothing techniques in the context of the minimax function.…”

Section: Theoretical Underpinningsmentioning

confidence: 99%

Nonlinearly-constrained optimization using asynchronous parallel generating set search.

Griffin¹,

Kolda²

2007

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Many optimization problems in computational science and engineering (CS&E) are characterized by expensive objective and/or constraint function evaluations paired with a lack of derivative information. Direct search methods such as generating set search (GSS) are well understood and efficient for derivative-free optimization of unconstrained and linearly-constrained problems. This paper addresses the more difficult problem of general nonlinear programming where derivatives for objective or constraint functions are unavailable, which is the case for many CS&E applications. We focus on penalty methods that use GSS to solve the linearly-constrained problems, comparing different penalty functions. A classical choice for penalizing constraint violations is 2 2 , the squared 2 norm, which has advantages for derivative-based optimization methods. In our numerical tests, however, we show that exact penalty functions based on the 1, 2, and ∞ norms converge to good approximate solutions more quickly and thus are attractive alternatives. Unfortunately, exact penalty functions are discontinuous and consequently introduce theoretical problems that degrade the final solution accuracy, so we also consider smoothed variants. Smoothed-exact penalty functions are theoretically attractive because they retain the differentiability of the original problem. Numerically, they are a compromise between exact and 2 2 , i.e., they converge to a good solution somewhat quickly without sacrificing much solution accuracy. Moreover, the smoothing is parameterized and can potentially be adjusted to balance the two considerations. Since many CS&E optimization problems are characterized by expensive function evaluations, reducing the number of function evaluations is paramount, and the results of this paper show that exact and smoothed-exact penalty functions are well-suited to this task.3

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“…In several recent publications [1-4, 8, 12, 14, 15, 17] methods have been analysed and discussed. We are interested in an approach presented in [4,13,16] for linear-quadratic finite-dimensional problems and extended in [6] and [7] to the infinite dimensional case. Numerical results have also been presented on a similar method in [5].…”

Section: Introductionmentioning

confidence: 99%

Infinite Penalization for Optimal Control Problems: An infinite‐dimensional optimization method for constrained optimization problems

Gugat

Herty

2013

Proc Appl Math and Mech

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We present results on a method for infinite dimensional constrained optimization problems. In particular, we are interested in state constrained optimal control problems and discuss an algorithm based on penalization and smoothing. The algorithm contains update rules for the penalty and the smoothing parameter that depend on the constraint violation. Theoretical as well as numerical results are given.

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A nonlinear programming algorithm based on non-coercive penalty functions

Cited by 39 publications

References 8 publications

Smoothed penalty algorithms for optimization of nonlinear models

Smoothed penalty algorithms for optimization of nonlinear models

Nonlinearly-constrained optimization using asynchronous parallel generating set search.

Infinite Penalization for Optimal Control Problems: An infinite‐dimensional optimization method for constrained optimization problems

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