Description:
This study concentrates on optimizing a multi-state weighted k-out-of-n system consisting of different types of reparable components. An optimization model is developed to decide upon both optimal redundancy allocation and best maintenance activities for reliability allocation. The optimization model is to maximize system reliability and minimize system cost simultaneously. To reach this goal, a financial view has been employed to estimate the system reliability, and combine it with system cost. The proposed model determines the number of each component type in the system, and the maintenance actions and the time at which they should be applied. Also, the model is programmed by MATLAB, and Genetic Algorithm is used to solve a practical case. (C) 2016 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
Description:
This paper is a kind of biased survey of the most representative and recent results on stability for the linear optimization problem. Qualitative and quantitative approaches are considered in this survey, and some infinite-dimensional extensions of the main results to more general problems are also included. In particular the paper deals with the lower/upper semicontinuity of the feasible/optimal set mappings, different types of ill-posedness, distance to ill-posedness, Lipschitz properties of these mappings under different types of perturbations, and estimates of the associated Lipschitz bounds.
Description:
We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.
Description:
We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.
Description:
In this paper, we study the problem of best Chebyshev approximation by linear splines. We construct linear splines as a max - min of linear functions. Then we apply nonsmooth optimisation techniques to analyse and solve the corresponding optimisation problems. This approach allows us to identify and introduce a new important property of linear spline knots (regular and irregular). Using this property, we derive a necessary optimality condition for the case of regular knots. This condition is stronger than those existing in the literature. We also present a numerical example which demonstrates the difference between the old and the new optimality conditions.
Description:
The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal–dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.
Description:
The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504-1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.
Description:
The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504-1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.
Description:
In this paper, we propose a new algorithm for global minimization of functions represented as a difference of two convex functions. The proposed method is a derivative free method and it is designed by adapting the extended cutting angle method. We present preliminary results of numerical experiments using test problems with difference of convex objective functions and box-constraints. We also compare the proposed algorithm with a classical one that uses prismatical subdivisions.
Description:
Clusterwise linear regression consists of finding a number of linear regression functions each approximating a subset of the data. In this paper, the clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem and an algorithm based on an incremental approach and on the discrete gradient method of nonsmooth optimization is designed to solve it. This algorithm incrementally divides the whole dataset into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate good starting points for solving global optimization problems at each iteration of the incremental algorithm. The algorithm is compared with the multi-start Spath and the incremental algorithms on several publicly available datasets for regression analysis.
Description:
The Australian Adverse Drug Reaction Advisory Committee (ADRAC) database has been collected and maintained by the Therapeutic Goods Administration. In this paper we study a part of his database (Card2) which contains records having just reactions from the Cardiovascular group. Drug-reaction relationships are presented by a vector of degrees which shows the degree of association of a drug with each class of reactions. In this work we examine these relationships in the classification of reaction outcomes. A modified version of the fuzzy derivative method (FDM2) is used for classification.
Description:
In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.
Description:
In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.