Decreasing Functions with Applications to Penalization

Rubinov, A. M.; Glover, B. M.; Yang, Xiaoqi

doi:10.1137/s1052623497326095

Cited by 81 publications

(45 citation statements)

References 10 publications

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“…tolerance = 10 −6 is chosen for the Newton iteration To conclude this section, we comment that from Tables 2, 3 and 4, it is clear that the l 2 penalty method needs a much smaller penalty parameter than those used in l 1/2 and l 1 penalty methods in order to archive a comparable accuracy to those from the latter two methods. This confirms the theoretical results in the precious section as well in [8]. Furthermore, the computed convergence rates for the l 2 penalty method are respectively one and two order higher than those of of the l 1 and l 1/2 penalty methods, as predicted by the theoretical result in Theorem 4.…”

Section: Penalty Methodssupporting

confidence: 88%

“…In the case of k = 2, this is the l 2 penalty method which is also a so-called lower order penalty method ( [8]). In this case, (12) becomes…”

Section: Penalty Methodsmentioning

confidence: 99%

“…In this section, we use the fitted finite volume discretization in space and the two level time stepping scheme in time with a splitting parameter θ for (5) and (8). For brevity, we omit the derivation of the fitted finite volume method which can be found in ( [11,9]) and only list the final discretized forms.…”

Section: Discretizationmentioning

confidence: 99%

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Numerical performance of penalty method for American option pricing

Zhang

Yang

Wang

et al. 2010

Optimization Methods and Software

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This paper is devoted to studying the numerical performance of a power penalty method for a linear parabolic complementarity problem arising from American option valuation. The penalized problem is a nonlinear parabolic partial differential equation (PDE). A fitted finite volume method and an implicit time-stepping scheme are used for respectively the spatial and time discretizations of the PDE. The rate of convergence of the penalty methods with respect to the penalty parameters is investigated both theoretically and numerically. The numerical robustness and computational effectiveness of the penalty method with respect to the market parameters are also studied and compared with those from an existing popular method, PSOR.

show abstract

Section: Penalty Methodssupporting

confidence: 88%

“…In the case of k = 2, this is the l 2 penalty method which is also a so-called lower order penalty method ( [8]). In this case, (12) becomes…”

Section: Penalty Methodsmentioning

confidence: 99%

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Numerical performance of penalty method for American option pricing

Zhang

Yang

Wang

et al. 2010

Optimization Methods and Software

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“…The terminology "nonlinear" refers to the nonlinearity of the objective function of the transformed problems with respect to the objective function of the original constrained optimization problem. The exact penalization result for nonconvex inequality constrained single objective optimization was obtained under a generalized calmness condition in [18]. It is worth noting that the early study on the nonlinear Lagrangian can be found in the work [23].…”

Section: Introductionmentioning

confidence: 99%

“…The condition used is the lower semicontinuity of the functions involved, which is much weaker than the continuity assumption in [17]. Moreover, the conditions for exact penalization are a generalization of the ones for single objective optimization in [3,4,15,18]. It is worth noting that nonlinear Lagrangian dual problems studied in this paper provide new models for convex composite optimization problems studied in [6,7,21].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Lagrangian for Multiobjective Optimization and Applications to Duality and Exact Penalization

Huang¹,

Yang²

2002

SIAM J. Optim.

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Abstract. Duality and penalty methods are popular in optimization. The study on duality and penalty methods for nonconvex multiobjective optimization problems is very limited. In this paper, we introduce vector-valued nonlinear Lagrangian and penalty functions and formulate nonlinear Lagrangian dual problems and nonlinear penalty problems for multiobjective constrained optimization problems. We establish strong duality and exact penalization results. The strong duality is an inclusion between the set of infimum points of the original multiobjective constrained optimization problem and that of the nonlinear Lagrangian dual problem. Exact penalization is established via a generalized calmness-type condition.

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Efficiency and Approachability of Nonconvex Bicriteria Programs

Huang

Yang²

2001

Journal of Mathematical Analysis and Applications

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In this paper we investigate equivalences between an efficient solution of a bicriteria program and a lower envelope point of a certain image set of the bicriteria program. We also employ various kinds of approachabilities to characterize efficiency and proper efficiency for nonconvex bicriteria programs. In particular, nonlinear Lagranian functions are applied to construct dual problems and to study stability for the corresponding constrained scalar optimization problems. Under certain conditions we show that the finite approachability, proper efficiency, stability, and exact penalization of the relevant constrained scalar optimization problems are equivalent.

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Decreasing Functions with Applications to Penalization

Cited by 81 publications

References 10 publications

Numerical performance of penalty method for American option pricing

Numerical performance of penalty method for American option pricing

Nonlinear Lagrangian for Multiobjective Optimization and Applications to Duality and Exact Penalization

Efficiency and Approachability of Nonconvex Bicriteria Programs

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