Steering exact penalty methods for nonlinear programming

Byrd, Richard H.; Nocedal, Jorge; Waltz, Richard A.

doi:10.1080/10556780701394169

Cited by 76 publications

(67 citation statements)

References 36 publications

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“…Filter methods (see [12] and references therein, and-for the nondifferentiable case- [18,34]) employ feasibility restoration steps to reduce the value of a constraint violation function. In [4] dynamical steering of exact penalty methods toward feasibility and optimality is reviewed and analysed. While these references consider infeasibility within traditional nonlinear programming (inspired) algorithms, our work is devoted to the study of corresponding issues within subgradient based methods applied to Lagrange duals.…”

Section: Outline and Main Resultsmentioning

confidence: 99%

Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems

et al. 2016

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Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal-dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which-in the inconsistent case-the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional ε-subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal-dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved. Keywords Mathematics Subject Classification

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Section: Outline and Main Resultsmentioning

confidence: 99%

Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems

et al. 2016

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“…These two components are far from trivial. The linear system is highly ill conditioned, nonsymmetric and indefinite, and the choice of µ has been the topic of several recent papers; see, e.g., [4,15]. In the next section we discuss the solution of the linear system.…”

Section: B)mentioning

confidence: 99%

Multilevel Algorithms for Large-Scale Interior Point Methods

Benzi¹,

Haber²,

Taralli³

2010

SIAM J. Sci. Comput.

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Abstract. We develop and compare multilevel algorithms for solving bound constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and iterative solution. Furthermore, we show how a multilevel continuation strategy can be used to obtain good initial guesses ("hot starts") for each nonlinear iteration. A minimal surface problem is used to illustrate the various approaches.

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“…Penalty and Barrier functions have been widely used and studied in the last years, for example by Byrd, Nocedal, and Waltz (2008), Goldfarb (2006), Fletcher (1997), Gould, Orban, and Toint (2003), Leyffer, Calva, and Nocedal (2006), Klatte and Kummer (2002), Mongeau and Sartenaer (1995) and Zaslavski (2005), due to its ability to deal with Degenerated Problems.…”

Section: Introductionmentioning

confidence: 99%

“…They were also used in Constrained NonLinear Programming to assure the admissibility of sub-problems and the iteration reliability by Byrd et al (2008) and Chen and Goldfarb (2006).…”

Section: Introductionmentioning

confidence: 99%

Adaptive Penalty and Barrier function based on Fuzzy Logic

Matias

Correia

Mestre

et al. 2015

Expert Systems with Applications

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Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twentyeight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.

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Steering exact penalty methods for nonlinear programming

Cited by 76 publications

References 36 publications

Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems

Ergodic, primal convergence in dual subgradient schemes for convex programming, II: the case of inconsistent primal problems

Multilevel Algorithms for Large-Scale Interior Point Methods

Adaptive Penalty and Barrier function based on Fuzzy Logic

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