Generalized Fenchel's conjugation formulas and duality for abstract convex functions
- Authors: Jeyakumar, Vaithilingam , Rubinov, Alex , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 132, no. 3 (Mar 2007), p. 441-458
- Full Text: false
- Reviewed:
- Description: In this paper, we present a generalization of Fenchel's conjugation and derive infimal convolution formulas, duality and subdifferential (and epsilon-subdifferential) sum formulas for abstract convex functions. The class of abstract convex functions covers very broad classes of nonconvex functions. A nonaffine global support function technique and an extended sum- epiconjugate technique of convex functions play a crucial role in deriving the results for abstract convex functions. An additivity condition involving global support sets serves as a constraint qualification for the duality.
- Description: C1
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
- Full Text: false
- Reviewed:
- 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.
Global optimization of marginal functions with applications to economic equilibrium
- Authors: Bagirov, Adil , Rubinov, Alex
- Date: 2001
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 20, no. 3-4 (Aug 2001), p. 215-237
- Full Text: false
- Reviewed:
- Description: We discuss the applicability of the cutting angle method to global minimization of marginal functions. The search of equilibrium prices in the exchange model can be reduced to the global minimization of certain functions, which include marginal functions. This problem has been approximately solved by the cutting angle method. Results of numerical experiments are presented and discussed.
Hadamard type inequality for quasiconvex functions in higher dimensions
- Authors: Rubinov, Alex , Dutta, J.
- Date: 2002
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 270, no. 1 (2002), p. 80-91
- Full Text: false
- Reviewed:
- Description: In this article we study a Hadamard type inequality for nonnegative evenly quasiconvex functions. The approach of our study is based on the notion of abstract convexity. We also provide an explicit calculation to evaluate the asymptotically sharp constant associated with the inequality over a unit square in the two-dimensional plane. © 2002 Elsevier Science (USA). All rights reserved.
- Description: 2003000149
Hermite-Hadamard-type inequalities for increasing convex-along-rays function
- Authors: Rubinov, Alex , Dragomir, S. S , Dutta, J.
- Date: 2004
- Type: Text , Journal article
- Relation: Analysis Vol. 24, no. 2 (2004), p. 171-181
- Full Text: false
- Reviewed:
- Description: C1
- Description: 2003000933
Hidden abstract convex functions
- Authors: Rubinov, Alex , Wu, Zhiyou , Li, Duan
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Nonlinear and Convex Analysis Vol. 6, no. 1 (2005), p. 203-216
- Full Text: false
- Reviewed:
- Description: C1
- Description: 2003001424
Increasing quasiconcave co-radiant functions with applications in mathematical economics
- Authors: Martinez-Legaz, Juan , Rubinov, Alex , Schaible, Siegfried
- Date: 2005
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 61, no. 2 (2005), p. 261-280
- Full Text: false
- Reviewed:
- Description: We study increasing quasiconcave functions which are co-radiant. Such functions have frequently been employed in microeconomic analysis. The study is carried out in the contemporary framework of abstract convexity and abstract concavity. Various properties of these functions are derived. In particular we identify a small "natural" infimal generator of the set of all coradiant quasiconcave increasing functions. We use this generator to examine two duality schemes for these functions: classical duality often used in microeconomic analysis and a more recent duality concept. Some possible applications to the theory of production functions and utility functions are discussed. © Springer-Verlag 2005.
- Description: C1
- Description: 2003001423
Lagrange-type functions in constrained non-convex optimization
- Authors: Rubinov, Alex , Yang, Xiao
- Date: 2003
- Type: Text , Book
- Full Text: false
- Reviewed:
- Description: A1
- Description: 2003000355
Lagrange-type functions in constrained optimization
- Authors: Rubinov, Alex , Yang, Xiao , Bagirov, Adil , Gasimov, Rafail
- Date: 2003
- Type: Text , Journal article
- Relation: Journal of Mathematical Sciences Vol. 115, no. 4 (2003), p. 2437-2505
- Full Text: false
- Reviewed:
- Description: We examine various kinds of nonlinear Lagrange-type functions for constrained optimization problems. In particular, we study the weak duality, the zero duality gap property, and the existence of an exact parameter for these functions. The paper contains a detailed survey of results in these directions and comparison of different methods proposed by different authors. Some new results are also given.
- Description: C1
- Description: 2003000358
Local optimization method with global multidimensional search
- Authors: Bagirov, Adil , Rubinov, Alex , Zhang, Jiapu
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 32, no. 2 (2005), p. 161-179
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- Reviewed:
- Description: This paper presents a new method for solving global optimization problems. We use a local technique based on the notion of discrete gradients for finding a cone of descent directions and then we use a global cutting angle algorithm for finding global minimum within the intersection of the cone and the feasible region. We present results of numerical experiments with well-known test problems and with the so-called cluster function. These results confirm that the proposed algorithms allows one to find a global minimizer or at least a deep local minimizer of a function with a huge amount of shallow local minima. © Springer 2005.
- Description: C1
- Description: 2003001351
Methods for global optimization of nonsmooth functions with applications
- Authors: Rubinov, Alex
- Date: 2006
- Type: Text , Journal article
- Relation: Applied and Computational Mathematics Vol. 5, no. 1 (2006), p. 3-15
- Full Text: false
- Reviewed:
- Description: In this survey paper we present some results obtained in the Centre for Informatics and Applied Optimization (CIAO) at University of Ballarat, Australia, in the area of numerical global optimization. We describe a conceptual scheme of two methods developed in CIAO and present results of numerical experiments with some real world problems. The paper is based on a plenary lecture given by the author at the First International Conference on Control and Optimization with Industrial Applications, Baku, Azerbaijan, 2005.
- Description: C1
- Description: 2003001547
Metric projection onto a closed set : Necessary and sufficient conditions for the global minimum
- Authors: Mohebi, Hossein , Rubinov, Alex
- Date: 2006
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 31, no. 1 (2006), p. 124-132
- Full Text: false
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- Description: Necessary and sufficient conditions for a local minimum form a well-developed chapter of optimization theory. Determination of such conditions for the global minimum is a challenging problem. Useful conditions are currently known only for a few classes of nonconvex optimization problems. It is important to find different classes of problems for which the required conditions can be obtained. In this paper we examine one of these classes: the minimization of the distance to an arbitrary closed set in a class of ordered normed spaces. We use the structure of the objective function in order to present necessary and sufficient conditions that give a clear understanding of the structure of a global minimizer and can be easily verified for some problems under consideration. © 2006 INFORMS.
- Description: C1
- Description: 2003001835
Minimization of the sum of minima of convex functions and its application to clustering
- Authors: Rubinov, Alex , Soukhoroukova, Nadejda , Ugon, Julien
- Date: 2005
- Type: Text , Book chapter
- Relation: Continuous Optimization Chapter p. 409-434
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- Description: We study functions that can be represented as the sum of minima of convex functions. Minimization of such functions can be used for approximation of finite sets and their clustering. We suggest to use the local discrete gradient (DG) method [Bag99] and the hybrid method between the cutting angle method and the discrete gradient method (DG+CAM) [BRZ05b] for the minimization of these functions. We report and analyze the results of numerical experiments.
- Description: 2003004082
Monotonic analysis : Convergence of sequences of monotone functions
- Authors: Rubinov, Alex
- Date: 2003
- Type: Text , Journal article
- Relation: Optimization Vol. 52, no. 6 (2003), p. 673-692
- Full Text: false
- Reviewed:
- Description: In this article we examine various kinds of convergence of sequences of increasing positively homogeneous (IPH) functions and nonnegative decreasing functions defined on the interior of a pointed closed solid convex cone K. We show that five different types of convergency (including pointwise and epi-convergence) coincide for IPH functions. If the space under consideration is finite dimensional then the sixth type can be added: uniform convergence on bounded subsets of int K. Using IPH functions, we study epi-convergence of sequences of lower semi-continuous (lsc) nonnegative decreasing functions.
- Description: C1
- Description: 2003000356
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
- Full Text: false
- Reviewed:
- 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
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
- Full Text: false
- Reviewed:
- 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
Monotonic analysis over cones : III
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 15, no. 3 (2008), p. 561-579
- Full Text: false
- Reviewed:
- Description: This paper studies the class of increasing and co-radiant (ICR) functions over a cone equipped with an order relation which agrees with the conic structure. In particular, a representation of ICR functions as abstract convex functions is provided. This representation suggests the introduction of some polarity notions between sets. The relationship between ICR functions and increasing positively homogeneous functions is also shown.
- Description: C1
Multipliers and general Lagrangians
- Authors: Penot, Jean Paul , Rubinov, Alex
- Date: 2005
- Type: Text , Journal article
- Relation: Optimization Vol. 54, no. 4-5 (2005), p. 443-467
- Full Text: false
- Reviewed:
- Description: We combine a Lagrangian approach inspired by convex and quasiconvex dualities with a penalization approach to mathematical programming. We use the ideas of abstract convexity. We focus our attention on the set of multipliers. We look for an interpretation of multipliers as elements of generalized subdifferentials of the performance function associated with a dualizing parameterization of the given problem. © 2005 Taylor & Francis Group Ltd.
- Description: C1
- Description: 2003001422
Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions
- Authors: Jeyakumar, Vaithilingam , Rubinov, Alex , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 110, no. 3 (2007), p. 521-541
- Full Text: false
- Reviewed:
- Description: In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. © Springer-Verlag 2007.
- Description: C1
On a class of abstract convex functions
- Authors: Rubinov, Alex , Hajilarov, Eldar
- Date: 2004
- Type: Text , Conference paper
- Relation: Paper presented at ICOTA6: 6th International Conference on Optimization - Techniques and Applications, Ballarat, Victoria : 9th December, 2004
- Full Text: false
- Reviewed:
- Description: E1
- Description: 2003000928