Global optimality conditions for some classes of optimization problems
- Authors: Wu, Zhiyou , Rubinov, Alex
- Date: 2009
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 145, no. 1 (2009), p. 164-185
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- Description: We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity. © 2009 Springer Science+Business Media, LLC.
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
- Full Text: false
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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.
Solutions to quadratic minimization problems with box and integer constraints
- Authors: Gao, David , Ruan, Ning
- Date: 2010
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 47, no. 3 (2010), p. 463-484
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Stability of error bounds for convex constraints systems in Banach spaces
- Authors: Thera, Michel , Van Ngai, Huynh , Kruger, Alexander
- Date: 2010
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 20, no. 6 (2010), p. 3280-3296
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- Description: This paper studies stability of error bounds for convex constraints in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered.
- Description: C1
Uniform approximation by the highest defect continuous polynomial splines : Necessary and sufficient optimality conditions and their generalisations
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 147, no. 2 (2010), p. 378-394
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- Description: In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval. © 2010 Springer Science+Business Media, LLC.
Vallee poussin theorem and remez algorithm in the case of generalised degree polynomial spline approximation
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 1 (2010), p. 103-114
- Full Text: false
- Description: The classical Remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval. In this paper the classical Remez algorithm is generalised to the problem of polynomial spline (piece-wise polynomial) approximation with the spline defect equal to the spline degree. Also, the values of the splines in the end points of the approximation interval may be fixed Polynomial splines combine simplicity of polynomials and flexibility, which allows one to significantly decrease the degree of the corresponding polynomials and oscillations of deviation functions. Therefore polynomial splines are a powerful tool for function and data approximation. The generalisation of the Remez algorithm developed in this research has been tested on several approximation problems. The results of the numerical experiments are presented.
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
- Full Text: false
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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
- Full Text: false
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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 complementarity partition theorem for multifold conic systems
- Authors: Peña, Javier , Roshchina, Vera
- Date: 2012
- Type: Text , Journal article
- Relation: Mathematical Programming Vol.142 , no.1-2 (2012), p.579-589
- Full Text: false
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- Description: Consider a homogeneous multifold convex conic system {Mathematical expression}and its alternative system {Mathematical expression}, where K 1,..., K r are regular closed convex cones. We show that there is a canonical partition of the index set {1,..., r} determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming. © 2012 Springer and Mathematical Optimization Society.
Anticipating synchronization through optimal feedback control
- Authors: Huang, Tingwen , Gao, David , Li, Chuandong , Xiao, MingQing
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 52, no. 2 (2012), p. 281-290
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- Description: In this paper, we investigate the anticipating synchronization of a class of coupled chaotic systems through discontinuous feedback control. The stability criteria for the involved error dynamical system are obtained by means of model transformation incorporated with Lyapunov functional and linear matrix inequality. Also, we discuss the optimal designed controller based on the obtained criteria. The numerical simulation is presented to demonstrate the theoretical results. © 2011 Springer Science+Business Media, LLC.