Applying the canonical dual theory in optimal control problems
- Authors: Zhu, Jinghao , Wu, Dan , Gao, David
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of global optimization Vol. 54, no. 2 (2012), p. 221-233
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- Description: This paper presents some applications of the canonical dual theory in optimal control problems. The analytic solutions of several nonlinear and nonconvex problems are investigated by global optimizations. It turns out that the backward differential flow defined by the KKT equation may reach the globally optimal solution. The analytic solution to an optimal control problem is obtained via the expression of the co-state. Some examples are illustrated.
Codimension two bifurcation in a delayed neural network with unidirectional coupling
- Authors: He, Xing , Li, Chuandong , Huang, Tingwen , Li, Chaojie
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis : Real World Applications Vol. 14, no. 2 (2012), p. 1191-1202
- Full Text: false
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- Description: In this paper, a delayed neural network model with unidirectional coupling is considered. Zero-Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm form at the zero-Hopf singularity and show that the model can exhibit pitchfork, Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Some numerical simulations are given to support the analytic results. © 2012 Elsevier Ltd. All rights reserved.
Comments on : Stability in linear optimization and related topics. A personal tour
- Authors: Kruger, Alexander
- Date: 2012
- Type: Text , Journal article
- Relation: TOP Vol. 20, no. 2 (2012), p. 255-257
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- Description: The article presents a report on a wonderful tour in the area of stability analysis of linear (and not only linear) optimization undertaken in the last 15 years by the author and his team of collaborators. 15 years is a very short period for developing a mathematical theory. Nevertheless the scope of achievement presented in the article and the level of development of the theory are really impressive. The tour is full of attractions and the route is very carefully marked. Now the tour is on offer, and the author is eager to share its highlights with interested travelers.
Comparison of metaheuristic algorithms for pump operation optimization
- Authors: Bagirov, Adil , Ahmed, S. T. , Barton, Andrew , Mala-Jetmarova, Helena , Al Nuaimat, Alia , Sultanova, Nargiz
- Date: 2012
- Type: Text , Conference paper
- Relation: 14th Water Distribution Systems Analysis Conference 2012, WDSA 2012 Vol. 2; Adelaide, Australia; 24th-27th September 2012; p. 886-896
- Relation: http://purl.org/au-research/grants/arc/LP0990908
- Full Text: false
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- Description: Pumping cost constitutes the main part of the overall operating cost of water distribution systems. There are different optimization formulations of the pumping cost minimization problem including those with application of continuous and integer programming approaches. To date mainly various metaheuristics have been applied to solve this problem. However, the comprehensive comparison of those metaheuristics has not been done. Such a comparison is important to identify strengths and weaknesses of different algorithms which reflects on their performance. In this paper, we present a methodology for comparative analysis of widely used metaheuristics for solving the pumping cost minimization problem. This methodology includes the following comparison criteria: (a) the "optimal solution" obtained; (b) the efficiency; and (c) robustness. Algorithms applied are: particle swarm optimization, artificial bee colony and firefly algorithms. These algorithms were applied to one test problem available in the literature. The results obtained demonstrate that the artificial bee colony is the most robust and the firefly is the most efficient and accurate algorithm for this test problem. Funding :ARC
Computing optimal electronic and mathematical properties of Buckyball nanoparticle using graph algorithms
- Authors: Khataee, H. , Ibrahim, Yousef , Sourchi, S. , Eskandari, L. , Teh Noranis, M.
- Date: 2012
- Type: Text , Journal article
- Relation: COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 31, no. 2 (2012), p. 387-400
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- Description: Purpose – One of the significant underlying principles of nanorobotic systems deals with the understanding and conceptualization of their respective complex nanocomponents. This paper introduces a new methodology to compute a set of optimal electronic and mathematical properties of Buckyball nanoparticle using graph algorithms based on dynamic programming and greedy algorithm. Design/methodology/approach – Buckyball, C60, is composed of sixty equivalent carbon atoms arranged as a highly symmetric hollow spherical cage in the form of a soccer ball. At first, Wiener, hyper-Wiener, Harary and reciprocal Wiener indices were computed using dynamic programming and presented them as: W(Buckyball)=11870.4, WW(Buckyball)=52570.9, Ha(Buckyball)=102.2 and RW(Buckyball)=346.9. The polynomials of Buckyball, Hosoya and hyper-Hosoya, which are in relationship with Buckyball's indices, have also been computed. The relationships between Buckyball's indices and polynomials were then computed and demonstrated a good agreement with their mathematical equations. Also, a graph algorithm based on greedy algorithms was used to find some optimal electronic aspects of Buckyball's structure by computing the Minimum Weight Spanning Tree (MWST) of Buckyball. Findings – The computed MWST was indicated that for connecting sixty carbon atoms of Buckyball together: the minimum numbers of double bonds were 30; the minimum numbers of single bonds were 29; and the minimum numbers of electrons were 178. These results also had good agreement with the principles of the authors' used greedy algorithm. Originality/value – This paper has used the graph algorithms for computing the optimal electronic and mathematical properties of BB. It has focused on mathematical properties of BB including Wiener, hyper-Wiener, Harary and reciprocal Wiener indices as well as Hosoya and Hyper-Hosoya polynomials and computerized them with dynamic programming graph algorithms.
Controllability and observability of switched linear systems with continuous-time and discrete-time subsystems
- Authors: Liu, Chao , Li, ChengYong , Li, Chaojie
- Date: 2012
- Type: Text , Journal article
- Relation: IET Control Theory and Applications Vol. 6, no. 6 (2012), p. 855-863
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- Description: In this study, the authors focus on the controllability and observability of switched linear systems composed by continuous-time and discrete-time subsystems. Necessary and sufficient conditions for controllability and observability are obtained. A simple example is proposed to illustrate the effectiveness of the current theoretical results. © 2012 The Institution of Engineering and Technology.
- Description: 2003010568
Global optimal solutions to a class of quadrinomial minimization problems with one quadratic constraint
- Authors: Yuan, Y. B. , Fang, Shucherng , Gao, David
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 52, no. 2 (2012), p. 195-209
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- Description: This paper studies the canonical duality theory for solving a class of quadri- nomial minimization problems subject to one general quadratic constraint. It is shown that the nonconvex primal problem in Rn can be converted into a concave maximization dual problem over a convex set in R2 , such that the problem can be solved more efficiently. The existence and uniqueness theorems of global minimizers are provided using the triality theory. Examples are given to illustrate the results obtained. © 2011 Springer Science+Business Media, LLC.
Limited memory discrete gradient bundle method for nonsmooth derivative-free optimization
- Authors: Karmitsa, Napsu , Bagirov, Adil
- Date: 2012
- Type: Text , Journal article
- Relation: Optimization Vol. 61, no. 12 (2012), p. 1491-1509
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- Description: Typically, practical nonsmooth optimization problems involve functions with hundreds of variables. Moreover, there are many practical problems where the computation of even one subgradient is either a difficult or an impossible task. In such cases derivative-free methods are the better (or only) choice since they do not use explicit computation of subgradients. However, these methods require a large number of function evaluations even for moderately large problems. In this article, we propose an efficient derivative-free limited memory discrete gradient bundle method for nonsmooth, possibly nonconvex optimization. The convergence of the proposed method is proved for locally Lipschitz continuous functions and the numerical experiments to be presented confirm the usability of the method especially for medium size and large-scale problems. © 2012 Copyright Taylor and Francis Group, LLC.
- Description: 2003010398
Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions
- Authors: Adly, Samir , Hantoute, Abderrahim , Théra, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Theory, Methods & Applications Vol. 75, no. 3 (February, 2012), p. 985-1008
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- Description: The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.
On Hölder calmness of solution mappings in parametric equilibrium problems
- Authors: Anh, Lam Quoc , Kruger, Alexander , Thao, Nguyen
- Date: 2012
- Type: Text , Journal article
- Relation: TOP Vol. 22, no. 1 (2012), p. 331-342
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- Description: We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces.
On regular coderivatives in parametric equilibria with non-unique multipliers
- Authors: Henrion, René , Outrata, Jiri , Surowiec, Thomas
- Date: 2012
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 136, no. 1 (December 2012), p. 111-131
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- Description: This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations (GEs). The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such GEs. The advantages are illustrated by means of examples.
- Description: C1
Predicting default probabilities in emerging markets by new conic generalized partial linear models and their optimization
- Authors: Weber, Gerhard-Wilhelm , Çavu , Özmen, Ay
- Date: 2012
- Type: Text , Journal article
- Relation: Optimization Vol. 61, no. 4 (2012), p. 443-457
- Full Text: false
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- Description: Nowadays, the importance of financial crises and defaults of countries are becoming clear due to the globalization in the economic area and investments. Generalized partial linear model (GPLM) is a combination of two different regression models connecting with the mean of the dependent variable with the help of a link function. It is adequate to high-dimensional, non-normal data sets having the flexibility to reflect all anomalies effectively. The nonlinear patterns are also easily explained by the nonparametric component of the model. In this study, we introduce a newly developed conic GPLM (CGPLM) to predict default probabilities of 45 emerging markets using the contribution of a continuous model CMARS and a discrete model logistic regression. We present its application results on a data set with 13 macroeconomic variables in 25 years' time. To predict debt crises, CGPLM gives better results than a single CMARS and a single logistic regression. In fact, we have 91.81% and 89.31% accuracy rates, computed according to the correctness of the model output, for training and validation sample, respectively. This improvement in prediction of crises can contribute to new prospects and developments in financial mathematics to make more accurate previsions for investments and to take measures due to coming risks. © 2012 Copyright Taylor and Francis Group, LLC.
Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems
- Authors: Cánovas, Maria , López, Marco , Mordukhovich, Borris , Parra, Juan
- Date: 2012
- Type: Text , Journal article
- Relation: TOP Vol. 20, no. 2 (2012), p. 310-327
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- 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.
Some remarks on stability of generalized equations
- Authors: Henrion, René , Kruger, Alexander , Outrata, Jiri
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 159, no. 3 (2012), p. 681-697
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the solution map to a class of generalized equations, where the multivalued term amounts to the regular normal cone to a (possibly nonconvex) set given by C 2 inequalities. Instead of the linear independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian-Fromovitz and the constant rank qualification conditions. Based on the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constraints are derived, and a workable characterization of the isolated calmness of the considered solution map is provided. © 2012 Springer Science+Business Media, LLC.
Stability in linear optimization and related topics. A personal tour
- Authors: López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: TOP Vol. 20, no. 2 (2012), p. 217-244
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- 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.
Stationarity and regularity of infinite collections of sets
- Authors: Kruger, Alexander , López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 154, no. 2 (2012), p. 339-369
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC.
Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization
- Authors: Kruger, Alexander , López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 155, no. 2 (2012), p. 390-416
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements-normals and/or subdifferentials.
Subdifferential of the closed convex hull of a function and integration with nonconvex data in general normed spaces
- Authors: López, Marco , Volle, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 390, no. 1 (2012), p. 307-312
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this paper we approach the study of the subdifferential of the closed convex hull of a function and the related integration problem without the usual assumption of epi-pointedness. The key tool is, as in Hiriart-Urruty et al. (2011) [7], the concept of ε-subdifferential. Some other assumptions which are standard in the literature are also removed.
Subgradient Method for Nonconvex Nonsmooth Optimization
- Authors: Bagirov, Adil , Jin, L. , Karmitsa, Napsu , Al Nuaimat, A. , Sultanova, Nargiz
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol.157, no.2 (2012), p.416–435
- Full Text: false
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- Description: In this paper, we introduce a new method for solving nonconvex nonsmooth optimization problems. It uses quasisecants, which are subgradients computed in some neighborhood of a point. The proposed method contains simple procedures for finding descent directions and for solving line search subproblems. The convergence of the method is studied and preliminary results of numerical experiments are presented. The comparison of the proposed method with the subgradient and the proximal bundle methods is demonstrated using results of numerical experiments. © 2012 Springer Science+Business Media, LLC.
Sufficient conditions for global optimality of semidefinite optimization
- Authors: Quan, Jing , Wu, Zhiyou , Li, Guoquan , Wu, Ou
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Inequalities and Applications Vol. 2012, no. 108
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- Description: In this article, by using the Lagrangian function, we investigate the sufficient global optimality conditions for a class of semi-definite optimization problems, where the objective function are general nonlinear, the variables are mixed integers subject to linear matrix inequalities (LMIs) constraints as well as bounded constraints. In addition, the sufficient global optimality conditions for general nonlinear programming problems are derived, where the variables satisfy LMIs constraints and box constraints or bivalent constraints. Furthermore, we give the sufficient global optimality conditions for standard semi-definite programming problem, where the objective function is linear, the variables satisfy linear inequalities constraints and box constraints. © 2012 Quan et al.