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54
Unsupervised and Supervised Data Classification via Nonsmooth and Global Optimization
 Top
, 2003
"... We examine various methods for data clustering and data classification that are based on the minimization of the socalled cluster function and its modifications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a... ..."
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Cited by 17 (7 self)
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We examine various methods for data clustering and data classification that are based on the minimization of the socalled cluster function and its modifications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a...
Maxplus convex geometry
 of Lecture Notes in Comput. Sci
, 2006
"... Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theore ..."
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Cited by 14 (9 self)
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Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices. 1
The zero duality gap property and lower semicontinuity of the perturbation function
"... We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrangetype dual scheme. The necessary and su#cient conditions for the zero duality gap property to hold are established in terms of the lower se ..."
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Cited by 11 (4 self)
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We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrangetype dual scheme. The necessary and su#cient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.
A Geometric Framework for Nonconvex Optimization Duality using Augmented Lagrangian Functions,” forthcoming in Journal of Global Optimization
, 2006
"... We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use this separation result to provide necessary and ..."
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Cited by 9 (3 self)
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We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use this separation result to provide necessary and sufficient conditions for establishing strong duality between geometric primal and dual problems. Using the primal function of a constrained optimization problem, we apply our results both in the analysis of duality schemes constructed using augmented Lagrangian functions, and in establishing necessary and sufficient conditions for the convergence of penalty methods. 1
Hidden structures in the class of convex functions and a new duality transform
 J. EUR. MATH. SOC. 13, 975–1004
, 2011
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About regularity of set systems
 University of Ballarat
"... Abstract. Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stati ..."
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Cited by 8 (3 self)
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Abstract. Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle) are presented for the case of an Asplund space. 1.
On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian
 Journal of Global Optimization
, 2006
"... We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condit ..."
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Cited by 8 (2 self)
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We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the stepsize parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.
Semismoothness and directional subconvexity of functions, preprint
, 2003
"... The relationships between the semismoothness of a function and the submonotonicity of its subdifferentials at some given point are studied. A notion of approximate starshapedness at that point is introduced and compared with these properties. Some criteria ensuring that different subdifferentials c ..."
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Cited by 8 (3 self)
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The relationships between the semismoothness of a function and the submonotonicity of its subdifferentials at some given point are studied. A notion of approximate starshapedness at that point is introduced and compared with these properties. Some criteria ensuring that different subdifferentials coincide at that point are obtained. Key words: approximate convexity, nonsmooth analysis, semismoothness, starshaped function, subconvexity, subdifferential.
Information and Content
 Blackwell Guide to the Philosophy of Information and Computing, Basil
, 2004
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