Implementation of novel methods of global and nonsmooth optimization : GANSO programming library
- Authors: Beliakov, Gleb , Ugon, Julien
- Date: 2007
- Type: Text , Journal article
- Relation: Optimization Vol. 56, no. 5-6 (2007), p. 543-546
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- Description: We discuss the implementation of a number of modern methods of global and nonsmooth continuous optimization, based on the ideas of Rubinov, in a programming library GANSO. GANSO implements the derivative-free bundle method, the extended cutting angle method, dynamical system-based optimization and their various combinations and heuristics. We outline the main ideas behind each method, and report on the interfacing with Matlab and Maple packages.
- Description: C1
- Description: 2003004865
Weak stationarity : Eliminating the gap between necessary and sufficient conditions
- Authors: Kruger, Alexander
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 2 (Apr 2004), p. 147-164
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- Description: Starting from known necessary extremality conditions in terms of strict subdifferentials and normals the notion of weak stationarity is introduced. It is defined in terms of initial space elements. The necessary conditions become necessary and sufficient (for stationarity).
- Description: 2003000887
Monotonic analysis over cones : I
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 2 (2004), p. 129-146
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- Description: In this article, we study increasing and positively homogeneous functions defined on convex cones of locally convex spaces. This work is the first part in a series of studies to have a general view of the emerging area of Monotonic Analysis. We develop a general notion of so-called elementary functions, so that the generalized increasing and positively homogeneous functions can be represented as upper-envelopes of families of such functions. We also study many other associated properties like the description of support sets and normal and co-normal sets in a very general setting.
- Description: C1
- Description: 2003000930
Non-smooth optimization methods for computation of the conditional value-at-risk and portfolio optimization
- Authors: Beliakov, Gleb , Bagirov, Adil
- Date: 2006
- Type: Text , Journal article
- Relation: Optimization Vol. 55, no. 5-6 (2006), p. 459-479
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- Description: We examine numerical performance of various methods of calculation of the Conditional Value-at-risk (CVaR), and portfolio optimization with respect to this risk measure. We concentrate on the method proposed by Rockafellar and Uryasev in (Rockafellar, R.T. and Uryasev, S., 2000, Optimization of conditional value-at-risk. Journal of Risk, 2, 21-41), which converts this problem to that of convex optimization. We compare the use of linear programming techniques against a non-smooth optimization method of the discrete gradient, and establish the supremacy of the latter. We show that non-smooth optimization can be used efficiently for large portfolio optimization, and also examine parallel execution of this method on computer clusters. © 2006 Taylor & Francis.
- Description: C1
- Description: 2003002156
Degrees of maximums of linear mappings
- Authors: Hajilarov, Eldar
- Date: 2008
- Type: Text , Journal article
- Relation: Optimization Vol. 57, no. 4 (2008), p. 505-514
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- Description: In this article, we introduce the notion of (topological) degree for maximums of linear mappings (MLM) and study its properties. It is shown, that in E2 the only possible values for the degrees of MLMs are 0 and ± 1, whereas for En with n > 2 the degrees can take arbitrary integer values. © 2008 Taylor & Francis.
- Description: C1
Abstract convexity for nonconvex optimization duality
- Authors: Nedic, A. , Ozdaglar, A. , Rubinov, Alex
- Date: 2007
- Type: Text , Journal article
- Relation: Optimization Vol. 56, no. 5-6 (2007), p. 655-674
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- Description: In this article, we use abstract convexity results to study augmented dual problems for (nonconvex) constrained optimization problems. We consider a nonincreasing function f that is lower semicontinuous at 0 and establish its abstract convexity at 0 with respect to a set of elementary functions defined by nonconvex augmenting functions. We consider three different classes of augmenting functions: nonnegative augmenting functions, bounded-below augmenting functions, and unbounded augmenting functions. We use the abstract convexity results to study augmented optimization duality without imposing boundedness assumptions.
- Description: C1
Monotonic analysis over cones : II
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 5-6 (2004), p. 529-547
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- Description: In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).
- Description: C1
- Description: 2003000931
Comparative analysis of the cutting angle and simulated annealing methods in global optimization
- Authors: Bagirov, Adil , Zhang, Jiapu
- Date: 2003
- Type: Text , Journal article
- Relation: Optimization Vol. 52, no. 4-5 (2003), p. 363-378
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- Description: This article presents a comparative analysis of two methods of global optimization: the simulated annealing method and a method based on a combination of the cutting angle method and a local search. This analysis is carried out using results of numerical experiments. These results demonstrate that the combined method is more effective than the simulated annealing method.
- Description: C1
- Description: 2003000436
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.
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.
Special Issue on recent advances in continuous optimization on the occasion of the 25th European conference on Operational Research (EURO XXV 2012)
- Authors: Weber, Gerhard-Wilhelm , Kruger, Alexander , Martinez-Legaz, Juan , Mordukhovich, Boris , Sakalauskas, Leonidas
- Date: 2014
- Type: Text , Journal article
- Relation: Optimization Vol. 63, no. 1 (2014), p. 1-5
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Error bounds and metric subregularity
- Authors: Kruger, Alexander
- Date: 2015
- Type: Text , Journal article
- Relation: Optimization Vol. 64, no. 1 (2015), p. 49-79
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.
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
A new auxiliary function method for general constrained global optimization
- Authors: Wu, Zhiyou , Bai, Fusheng , Yang, Yongjian , Mammadov, Musa
- Date: 2013
- Type: Text , Journal article
- Relation: Optimization Vol. 62, no. 2 (2013), p. 193-210
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- Description: In this article, we first propose a method to obtain an approximate feasible point for general constrained global optimization problems (with both inequality and equality constraints). Then we propose an auxiliary function method to obtain a global minimizer or an approximate global minimizer with a required precision for general global optimization problems by locally solving some unconstrained programming problems. Some numerical examples are reported to demonstrate the efficiency of the present optimization method. © 2013 Taylor & Francis.
- Description: 2003011103
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
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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.
Hyperbolic smoothing function method for minimax problems
- Authors: Bagirov, Adil , Al Nuaimat, Alia , Sultanova, Nargiz
- Date: 2013
- Type: Text , Journal article
- Relation: Optimization Vol. 62, no. 6 (2013), p. 759-782
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- Description: In this article, an approach for solving finite minimax problems is proposed. This approach is based on the use of hyperbolic smoothing functions. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in general algebraic modelling system. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem. © 2013 Copyright Taylor and Francis Group, LLC.
- Description: 2003011099
Globally convergent algorithms for solving unconstrained optimization problems
- Authors: Taheri, Sona , Mammadov, Musa , Seifollahi, Sattar
- Date: 2013
- Type: Text , Journal article
- Relation: Optimization Vol. , no. (2013), p. 1-15
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- Description: New algorithms for solving unconstrained optimization problems are presented based on the idea of combining two types of descent directions: the direction of anti-gradient and either the Newton or quasi-Newton directions. The use of latter directions allows one to improve the convergence rate. Global and superlinear convergence properties of these algorithms are established. Numerical experiments using some unconstrained test problems are reported. Also, the proposed algorithms are compared with some existing similar methods using results of experiments. This comparison demonstrates the efficiency of the proposed combined methods.
On computation of limiting coderivatives of the normal-cone mapping to inequality systems and their applications
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2016
- Type: Text , Journal article
- Relation: Optimization Vol. 65, no. 4 (2016), p. 671-700
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: The paper concerns the computation of the limiting coderivative of the normal-cone mapping related to inequality constraints under weak qualification conditions. The obtained results are applied to verify the Aubin property of solution maps to a class of parameterized generalized equations.