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
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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.
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
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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.
The effect of regularization on drug-reaction relationships
- Authors: Mammadov, Musa , Zhao, L. , Zhang, Jianjun
- Date: 2012
- Type: Text , Journal article
- Relation: Optimization Vol. 61, no. 4 (2012), p. 405-422
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- Description: The least-squares method is a standard approach used in data fitting that has important applications in many areas in science and engineering including many finance problems. In the case when the problem under consideration involves large-scale sparse matrices regularization methods are used to obtain more stable solutions by relaxing the data fitting. In this article, a new regularization algorithm is introduced based on the Karush-Kuhn-Tucker conditions and the Fisher-Burmeister function. The Newton method is used for solving corresponding systems of equations. The advantages of the proposed method has been demonstrated in the establishment of drug-reaction relationships based on the Australian Adverse Drug Reaction Advisory Committee database. © 2012 Copyright Taylor and Francis Group, LLC.
The new robust conic GPLM method with an application to finance : prediction of credit default
- Authors: Özmen, Ay , Weber, Gerhard-Wilhelm , Çavu , Defterli, Özlem
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol.56, no. 2 (2012), p. 233–249
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- Description: This paper contributes to classification and identification in modern finance through advanced optimization. In the last few decades, financial misalignments and, thereby, financial crises have been increasing in numbers due to the rearrangement of the financial world. In this study, as one of the most remarkable of these, countries' debt crises, which result from illiquidity, are tried to predict with some macroeconomic variables. The methodology consists of a combination of two predictive regression models, logistic regression and robust conic multivariate adaptive regression splines (RCMARS), as linear and nonlinear parts of a generalized partial linear model. RCMARS has an advantage of coping with the noise in both input and output data and of obtaining more consistent optimization results than CMARS. An advanced version of conic generalized partial linear model which includes robustification of the data set is introduced: robust conic generalized partial linear model (RCGPLM). This new model is applied on a data set that belongs to 45 emerging markets with 1,019 observations between the years 1980 and 2005. © 2012 Springer Science+Business Media, LLC.
An integral function and vector sequence method for unconstrained global optimization
- Authors: Yang, Yongjian , Bai, Fusheng
- Date: 2011
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 50, no. 2 (2011), p. 293-311
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- Description: An integral function and a vector sequence are constructed in this paper. Their theoretical and numerical properties are investigated. Based on the integral function and the vector sequence, an algorithm is proposed for solving a class of unconstrained global optimization problems. For the algorithm, convergence to a global minimizer is discussed under some conditions. Some typical examples are tested to illustrate the efficiency of the algorithm. © Springer Science+Business Media, LLC. 2010.
Codifferential method for minimizing nonsmooth DC functions
- Authors: Bagirov, Adil , Ugon, Julien
- Date: 2011
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 50, no. 1 (2011), p. 3-22
- Relation: http://purl.org/au-research/grants/arc/DP0666061
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- Description: In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (DC functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that DC decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given. © 2010 Springer Science+Business Media, LLC.
Distance to ill-posedness for linear inequality systems under block perturbations : Convex and infinite-dimensional cases
- Authors: Cánovas, Maria , López, Marco , Parra, Juan , Toledoa, Javier
- Date: 2011
- Type: Text , Journal article
- Relation: Optimization Vol. 60, no. 7 (2011), p. 925-946
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article extends some results of Cá novas et al. [M.J. Cá novas, M.A. Ló pez, J. Parra, and F.J. Toledo, Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems, Math. Prog. Ser. A 103 (2005), pp. 95-126.] about distance to ill-posedness (feasibility/ infeasibility) in three directions: From individual perturbations of inequalities to perturbations by blocks, from linear to convex inequalities and from finite- to infinite-dimensional (Banach) spaces of variables. The second of the referred directions, developed in the finite-dimensional case, was the original motivation of this article. In fact, after linearizing a convex system via the Fenchel-Legendre conjugate, affine perturbations of convex inequalities translate into block perturbations of the corresponding linearized system. We discuss the key role played by constant perturbations as an extreme case of block perturbations. We emphasize the fact that constant perturbations are enough to compute the distance to ill-posedness in the infinite-dimensional setting, as shown in the last part of this article, where some remarkable differences of infinite- versus finite-dimensional systems are presented. Throughout this article, the set indexing the constraints is arbitrary, with no topological structure. Accordingly, the functional dependence of the system coefficients on the index has no qualification at all.
Global descent methods for unconstrained global optimization
- Authors: Wu, Zhiyou , Li, Duan , Zhang, Lian-Sheng
- Date: 2011
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 50, no. 3 (2011), p. 379-3976
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- Description: We propose in this paper novel global descent methods for unconstrained global optimization problems to attain the global optimality by carrying out a series of local minimization. More specifically, the solution framework consists of a two-phase cycle of local minimization: the first phase implements local search of the original objective function, while the second phase assures a global descent of the original objective function in the steepest descent direction of a (quasi) global descent function. The key element of global descent methods is the construction of the (quasi) global descent functions which possess prominent features in guaranteeing a global descent. © 2010 Springer Science+Business Media, LLC.
Global optimization over a box via canonical dual function
- Authors: Zhu, Jinghao , Wang, Chao , Gao, David
- Date: 2011
- Type: Text , Journal article
- Relation: Journal of Computational and Applied Mathematics Vol. 235, no. 5 (January 2011), p. 1141-1147
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- Description: In this paper, we study global concave optimization by the canonical dual function. A differential flow on the dual feasible space is introduced. We show that the flow reaches a global minimizer of the concave function over a box. An example is illustrated.
A necessary optimality condition for free knots linear splines: Special cases
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 2, Suppl. 1 (2010), p. 305-317
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- 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.
A new local and global optimization method for mixed integer quadratic programming problems
- Authors: Li, G. Q. , Wu, Zhiyou , Quan, Jing
- Date: 2010
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 217, no. 6 (2010), p. 2501-2512
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- Description: In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable. Crown Copyright © 2010.
A quasisecant method for minimizing nonsmooth functions
- Authors: Bagirov, Adil , Ganjehlou, Asef Nazari
- Date: 2010
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 25, no. 1 (2010), p. 3-18
- Relation: http://purl.org/au-research/grants/arc/DP0666061
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- Description: We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.
Alexander Rubinov - An outstanding scholar
- Authors: Bagirov, Adil
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 2, Suppl. 1 (2010), p. 203-209
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Canonical dual least square method for solving general nonlinear systems of quadratic equations
- Authors: Ruan, Ning , Gao, David
- Date: 2010
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 47, no. (2010), p. 335-347
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- Description: This paper presents a canonical dual approach for solving general non- linear algebraic systems. By using least square method, the nonlinear system of m -quadratic equations in n -dimensional space is first formulated as a nonconvex opti- mization problem. We then proved that, by the canonical duality theory developed by the second author, this nonconvex problem is equivalent to a concave maximization problem in R, which can be solved easily by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.
- Description: C1
Internet security applications of Grobner-Shirvov bases
- Authors: Kelarev, Andrei , Yearwood, John , Watters, Paul
- Date: 2010
- Type: Text , Journal article
- Relation: Asian-European Journal of Mathematics Vol. 3, no. 3 (2010), p. 435-442
- Relation: http://purl.org/au-research/grants/arc/DP0211866
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Outer approximation schemes for generalized semi-infinite variational inequality problems
- Authors: Burachik, Regina , Lopes, J.
- Date: 2010
- Type: Text , Journal article
- Relation: Optimization Vol. 59, no. 4 (2010), p. 601-617
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- Description: We introduce and analyse outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems generalized semi-infinite variational inequality problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.
Separation properties via connectedness of topological convexity spaces
- Authors: Sharikov, Evgeny
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 2, Suppl. 1 (2010), p. 227-241
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- Description: For a given collection H of subsets of a set X we examine the convexity on X generated by H. We use a special type of connectedness of H and X for investigation of separation of convex sets by elements of H. In particular, we give a description of convex sets, which can be represented as the intersection of a subfamily of H. As an application, we give a description of abstract convex functions and sets. We also describe the abstract convex hull of a finite union of abstract convex sets.