20 results
Search Results
2. Gap Function for Set-Valued Vector Variational-Like Inequalities.
- Author
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Mishra, S. K., Wang, S. Y., and Lai, K. K.
- Subjects
VARIATIONAL inequalities (Mathematics) ,MATHEMATICAL optimization ,NONSMOOTH optimization ,SELECTION theorems ,MATHEMATICAL mappings ,SET-valued maps ,MATHEMATICAL analysis ,MATHEMATICAL programming ,SET theory ,TOPOLOGICAL spaces - Abstract
Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
3. Globally convergent limited memory bundle method for large-scale nonsmooth optimization.
- Author
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Haarala, Napsu, Miettinen, Kaisa, and Mäkelä, Marko M.
- Subjects
MATHEMATICAL optimization ,MATHEMATICAL variables ,MATHEMATICAL analysis ,STOCHASTIC convergence ,NONSMOOTH optimization - Abstract
Many practical optimization problems involve nonsmooth (that is, not necessarily differentiable) functions of thousands of variables. In the paper [Haarala, Miettinen, Mäkelä, Optimization Methods and Software, 19, (2004), pp. 673–692] we have described an efficient method for large-scale nonsmooth optimization. In this paper, we introduce a new variant of this method and prove its global convergence for locally Lipschitz continuous objective functions, which are not necessarily differentiable or convex. In addition, we give some encouraging results from numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
4. Optimizing Preventive Maintenance Models.
- Author
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Bartholomew-Biggs, Michael, Christianson, Bruce, and Ming Zuo
- Subjects
MAINTENANCE ,ALGORITHMS ,FAILURE time data analysis ,SYSTEM failures ,MATHEMATICAL optimization ,NONSMOOTH optimization ,MATHEMATICAL analysis ,NUMERICAL analysis ,MAINTENANCE costs - Abstract
We deal with the problem of scheduling preventive maintenance (PM) for a system so that, over its operating life, we minimize a performance function which reflects repair and replacement costs as well as the costs of the PM itself. It is assumed that a hazard rate model is known which predicts the frequency of system failure as a function of age. It is also assumed that each PM produces a step reduction in the effective age of the system. We consider some variations and extensions of a PM scheduling approach proposed by Lin et al. [6]. In particular we consider numerical algorithms which may be more appropriate for hazard rate models which are less simple than those used in [6] and we introduce some constraints into the problem in order to avoid the possibility of spurious solutions. We also discuss the use of automatic differentiation (AD) as a convenient tool for computing the gradients and Hessians that are needed by numerical optimization methods. The main contribution of the paper is a new problem formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. This formulation involves the global minimization of a non-smooth performance function. In our numerical tests this is done via the algorithm DIRECT proposed by Jones et al. [19]. We show results for a number of examples, involving different hazard rate models, to give an indication of how PM schedules can vary in response to changes in relative costs of maintenance, repair and replacement. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
5. Functions and Sets of Smooth Substructure: Relationships and Examples.
- Author
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Hare, W. L.
- Subjects
SMOOTHNESS of functions ,NONSMOOTH optimization ,MATHEMATICAL optimization ,SET theory ,MATHEMATICAL decomposition ,MATHEMATICAL analysis ,CONJUGATE gradient methods ,PROBABILITY theory ,NUMERICAL analysis - Abstract
The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., ‘amenable functions’, ‘partly smooth functions’, and ‘g o F decomposable Junctions’. Along with these classes a number of structural properties have been proposed, e.g., ‘identifiable suifaces’. ‘fast tracks’, and ‘primal-dual gradient structures’. In this paper we examine the relationships between these various classes of functions and their smooth substructures. In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g o F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
6. A class of constraint qualifications in nonsmooth optimization.
- Author
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Guohui Tian, Xiaofeng Li, and Tianxia Zhao
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,CONSTRAINT programming ,THEORY of constraints ,MATHEMATICAL analysis ,SIMULATION methods & models ,SYSTEM analysis - Abstract
In this paper, we give a generalized Abadie constraint qualification (CQ) for a class of nonsmooth vector-valued optimization. Under this CQ, we obtained the Kuhn-Tucker-type optimality condition where the multipliers associated with the objective functions are all positive. [ABSTRACT FROM AUTHOR]
- Published
- 1999
- Full Text
- View/download PDF
7. Constrained Nonconvex Nonsmooth Optimization via Proximal Bundle Method.
- Author
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Yang, Yang, Pang, Liping, Ma, Xuefei, and Shen, Jie
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,MATHEMATICAL analysis ,MATHEMATICS ,MAXIMA & minima - Abstract
In this paper, we consider a constrained nonconvex nonsmooth optimization, in which both objective and constraint functions may not be convex or smooth. With the help of the penalty function, we transform the problem into an unconstrained one and design an algorithm in proximal bundle method in which local convexification of the penalty function is utilized to deal with it. We show that, if adding a special constraint qualification, the penalty function can be an exact one, and the sequence generated by our algorithm converges to the KKT points of the problem under a moderate assumption. Finally, some illustrative examples are given to show the good performance of our algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
8. Lipschitz behavior of solutions to nonconvex semi-infinite vector optimization problems.
- Author
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Huy, N. and Kim, D.
- Subjects
LIPSCHITZ spaces ,FUNCTION spaces ,NONSMOOTH optimization ,MATHEMATICAL optimization ,MATHEMATICAL analysis - Abstract
This paper is devoted to developing new applications from the limiting subdifferential in nonsmooth optimization and variational analysis to the study of the Lipschitz behavior of the Pareto solution maps in parametric nonconvex semi-infinite vector optimization problems (SIVO for brevity). We establish sufficient conditions for the Aubin Lipschitz-like property of the Pareto solution maps of SIVO under perturbations of both the objective function and constraints. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
9. Convexification of Nonsmooth Monotone Functions.
- Author
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Sun, X. L., Luo, H. Z., and Li, D.
- Subjects
MATHEMATICAL optimization ,NONSMOOTH optimization ,MATHEMATICAL analysis ,CONVEX functions ,CONCENTRATION functions ,SUBDIFFERENTIALS ,REAL variables ,COMPLEX variables ,MATHEMATICS - Abstract
We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
10. Minimization of the Hausdorff distance between convex polyhedrons.
- Author
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Lakhtin, A. S. and Ushakov, V. N.
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,HAUSDORFF measures ,MATHEMATICAL analysis ,POLYHEDRA - Abstract
In this paper, we solve the problem on the optimization of the mutual location of two convex polyhedrons. Our study is based on methods of nonsmooth analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
11. First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems.
- Author
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Ye, J. J. and Wu, S. Y.
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,MATHEMATICAL programming ,ALGORITHMS ,MATHEMATICAL functions ,MATHEMATICAL models ,MATHEMATICAL analysis - Abstract
We study first-order optimality conditions for the class of generalized semi-infinite programming problems (GSIPs). We extend various well-known constraint qualifications for finite programming problems to GSIPs and analyze the extent to which a corresponding Karush-Kuhn-Tucker (KKT) condition depends on these extensions. It is shown that in general the KKT condition for GSIPs takes a weaker form unless a certain constraint qualification is satisfied. In the completely convex case where the objective of the lower-level problem is concave and the constraint functions are quasiconvex, we show that the KKT condition takes a sharper form. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
12. A note on an approximate lagrange multiplier rule.
- Author
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Dutta, Joydeep, Pattanaik, Suvendu Ranjan, and Théra, Michel
- Subjects
MATHEMATICAL optimization ,LAGRANGE spaces ,LIPSCHITZ spaces ,NONSMOOTH optimization ,CALCULUS ,MATHEMATICAL analysis - Abstract
In this note we show that for a large class of optimization problems, the Lagrange multiplier rule can be derived from the approximate multiplier rule. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
13. A primal dual modified subgradient algorithm with sharp Lagrangian.
- Author
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Burachik, Regina S., Iusem, Alfredo N., and Melo, Jefferson G.
- Subjects
ALGORITHMS ,MATHEMATICAL optimization ,LAGRANGE equations ,MATHEMATICAL analysis ,NONCONVEX programming - Abstract
We apply a modified subgradient algorithm (MSG) for solving the dual of a nonlinear and nonconvex optimization problem. The dual scheme we consider uses the sharp augmented Lagrangian. A desirable feature of this method is primal convergence, which means that every accumulation point of a primal sequence (which is automatically generated during the process), is a primal solution. This feature is not true in general for available variants of MSG. We propose here two new variants of MSG which enjoy both primal and dual convergence, as long as the dual optimal set is nonempty. These variants have a very simple choice for the stepsizes. Moreover, we also establish primal convergence when the dual optimal set is empty. Finally, our second variant of MSG converges in a finite number of steps. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
14. Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization.
- Author
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Bagirov, A. M., Karasözen, B., and Sezer, M.
- Subjects
MATHEMATICAL optimization ,NONSMOOTH optimization ,MATHEMATICAL functions ,MATHEMATICAL analysis ,SUBDIFFERENTIALS ,CONVEX functions ,MAXIMA & minima - Abstract
A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with the nonsmooth optimization solver DNLP from CONOPT-GAMS and the derivative-free optimization solver CONDOR are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
15. Higher-order optimality conditions for strict local minima.
- Author
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Jiménez, Bienvenido and Novo, Vicente
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,MAXIMA & minima ,MATHEMATICAL analysis ,LAGRANGIAN functions ,OPERATIONS research - Abstract
In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
16. Optimality Criteria for Nonsmooth Continuous-Time Problems of Multiobjective Optimization.
- Author
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Nobakhtian, S. and Pouryayevali, M. R.
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,MATHEMATICAL analysis ,CONVEXITY spaces ,CONVEX domains ,MATHEMATICAL models ,OPERATIONS research ,MATHEMATICAL models of decision making ,MATHEMATICAL programming - Abstract
A nonsmooth multiobjective continuous-time problem is introduced. We establish the necessary and sufficient optimality conditions under generalized convexity assumptions on the functions involved. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
17. Lexicographic differentiation of nonsmooth functions.
- Author
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Nesterov, Yu.
- Subjects
NONSMOOTH optimization ,SMOOTHNESS of functions ,MATHEMATICAL optimization ,MATHEMATICS ,MATHEMATICAL analysis - Abstract
We present a survey on the results related to the theory of lexicographic differentiation. This theory ensures an efficient computation of generalized ( lexicographic) derivative of a nonsmooth function belonging to a special class of lexicographically smooth functions. This class is a linear space which contains all differentiable functions, all convex functions, and which is closed with respect to component-wise composition of the members. In order to define lexicographic derivative in a unique way, it is enough to fix a basis in the space of variables. Lexicographic derivatives can be used in black-box optimization methods. We give some examples of applications of these derivatives in analysis of nonsmooth functions. It is shown that the system of lexicographic derivatives along a fixed basis correctly represents corresponding nonsmooth function (Newton-Leibnitz formula). We present nonsmooth versions of standard theorems on potentiality of nonlinear operators, on differentiation of parametric integrals and on differentiation of functional sequences. Finally, we show that an appropriately defined lexicographic subdifferential ensures a more rigorous selection of a candidate optimal solution than the subdifferential of Clarke. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
18. Nonsmooth Analysis of Singular Values. Part I: Theory.
- Author
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Adrian S. Lewis and Hristo S. Sendov
- Subjects
NONSMOOTH optimization ,MATHEMATICAL optimization ,MATRICES (Mathematics) ,MATHEMATICAL analysis - Abstract
Abstract The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumanns trace inequality for singular values. [ABSTRACT FROM AUTHOR]
- Published
- 2005
19. A General Iterative Procedure for Solving Nonsmooth Generalized Equations.
- Author
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Geoffroy, Michel H. and Pietrus, Alain
- Subjects
GENERALIZED estimating equations ,STATISTICAL correlation ,LINEAR statistical models ,NONSMOOTH optimization ,MATHEMATICAL optimization ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
We present a general iterative procedure for solving generalized equations in the nonsmooth framework. To this end, we consider a class of functions admitting a certain type of approximation and establish a local convergence theorem that one can apply to a wide range of particular problems. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
20. Contingent Derivatives of Implicit (Multi-) Functions and Stationary Points.
- Author
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Klatte, Diethard and Kummer, Bernd
- Subjects
IMPLICIT functions ,STATIONARY processes ,STOCHASTIC processes ,NONSMOOTH optimization ,MATHEMATICAL optimization ,SIMULATION methods & models ,MATHEMATICAL analysis - Abstract
For an implicit multifunction Φ (p) defined by the generally nonsmooth equation F(x, p) = 0, contingent derivative formulas are derived, being similar to the formula Φ′ = -F
x -1 Fp in the standard implicit function theorem for smooth F and Φ. This will be applied to the projection X(p) = {x ¦ 3y: (x, y) ∈ Φ(p)} of the solution set Φ(p) of the system F(x, y, p) = 0 onto the x-space. In particular settings, X(p) may be interpreted as stationary solution sets. We discuss in detail the situation in which X(p) arises from the Karush-Kuhn-Tucker system of a nonlinear program. [ABSTRACT FROM AUTHOR]- Published
- 2001
- Full Text
- View/download PDF
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