共查询到20条相似文献,搜索用时 15 毫秒
1.
G. Romano 《Applied Mathematics and Optimization》1995,32(3):213-234
Chain and addition rules of subdifferential calculus are revisited in the paper and new proofs, providing local necessary and sufficient conditions for their validity, are presented. A new product rule pertaining to the composition of a convex functional and a Young function is also established and applied to obtain a proof of Kuhn-Tucker conditions in convex optimization under minimal assumptions on the data. Applications to plasticity theory are briefly outlined in the concluding remarks.The financial support of the Italian Ministry for University and Scientific and Technological Research is gratefully acknowledged. 相似文献
2.
In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs. 相似文献
3.
Piecewise affine functions arise from Lagrangian duals of integer programming problems, and optimizing them provides good
bounds for use in a branch and bound method. Methods such as the subgradient method and bundle methods assume only one subgradient
is available at each point, but in many situations there is more information available. We present a new method for optimizing
such functions, which is related to steepest descent, but uses an outer approximation to the subdifferential to avoid some
of the numerical problems with the steepest descent approach. We provide convergence results for a class of outer approximations,
and then develop a practical algorithm using such an approximation for the compact dual to the linear programming relaxation
of the uncapacitated facility location problem. We make a numerical comparison of our outer approximation method with the
projection method of Conn and Cornuéjols, and the bundle method of Schramm and Zowe.
Received September 10, 1998 / Revised version received August 1999?Published online December 15, 1999 相似文献
4.
A. Auslender 《Journal of Optimization Theory and Applications》1992,73(3):427-449
We study dual functionals which have two fundamental properties. Firstly, they have a good asymptotical behavior. Secondly, to each dual sequence of subgradients converging to zero, one can associate a primal sequence which converges to an optimal solution of the primal problem. Furthermore, minimal conditions for the convergence of the Gauss-Seidel methods are given and applied to such kinds of functionals. 相似文献
5.
In this article we study approximate optimality in the setting of a Banach space. We study various solution concepts existing
in the literature and develop very general necessary optimality conditions in terms of limiting subdifferentials. We also
study saddle point conditions and relate them to various solution concepts.
Part of this research was carried out when the author was a post-doctoral fellow at UAB, Barcelona by the Grant No. SB99-B0771103B
of the Spanish Ministry of Education and Culture. The hospitality and the facilities provided at CODE, UAB is gratefully acknowledged. 相似文献
6.
Shunsuke Shiraishi 《Mathematical Programming》1993,58(1-3):257-262
For a real-valued convex functionf, the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivativef
(x
0;d, d) when 0+ and both values are the same. The aim of the present work is to show the converse of this result. It will be shown that upper and lower limits of the approximate second-order directional derivative are equal to the second-order upper and lower Dini derivatives, respectively. Consequently the existence of the limit of the approximate second-order directional derivative and that of second-order Dini derivative are equivalent.Dedicated to Professor N. Furukawa of Kyushu University for his 60th birthday. 相似文献
7.
César Gutiérrez 《Journal of Mathematical Analysis and Applications》2005,310(1):309-327
In this work we obtain a chain rule for the approximate subdifferential considering a vector-valued proper convex function and its post-composition with a proper convex function of several variables nondecreasing in the sense of the Pareto order. We derive an interesting formula for the conjugate of a composition in the same framework and we prove the chain rule using this formula. To get the results, we require qualification conditions since, in the composition, the initial function is extended vector-valued. This chain rule extends analogous well-known calculus rules obtained when the functions involved are finite and it gives a complementary simple expression for other chain rules proved without assuming any qualification condition. As application we deduce the well-known calculus rule for the addition and we extend the formula for the maximum of functions. Finally, we use them and a scalarization process to obtain Kuhn-Tucker type necessary and sufficient conditions for approximate solutions in convex Pareto problems. These conditions extend other obtained in scalar optimization problems. 相似文献
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10.
Minoru Siotani Dhanwant S. Gill Christian Löschcke 《Annals of the Institute of Statistical Mathematics》1978,30(1):365-375
Let an overall null hypothesisH be factored in a certain stepwise manner intok subhypotheses as
. Suppose the test statisticw forH be correspondingly expressed asw=w
1 w2…wk wherew
i is the test statistic forH
i. We consider the case where the Box method [2] is applicable for the distributions ofw andw
i's. Ifw
i's are independent underH, we obtain a stepwise test procedure forH on the basis of an approximate chi-square analysis. To demonstrate the procedure of this sort, the testing hypotheses of
equality of several convariance matrices and of the multiple independence are discussed. Finally the related asymptotic distributions
are shortly noted. 相似文献
11.
Markus Haase 《Journal of Evolution Equations》2007,7(3):529-554
Let –iA be the generator of a C
0-group U on a Banach space X. Via a transference principle we obtain results of the form
for certain functions f, provided that X is a UMD space. Special examples are
The first choice leads to easy proofs of the theorems of Monniaux and Dore–Venni, the second is related to a new proof of
Fattorini’s theorem on cosine functions. 相似文献
12.
Alfred Auslender Paulo J. S. Silva Marc Teboulle 《Computational Optimization and Applications》2007,38(3):305-327
We consider nonmonotone projected gradient methods based on non-Euclidean distances, which play the role of barrier for a
given constraint set. Our basic scheme uses the resulting projection-like maps that produces interior trajectories, and combines
it with the recent nonmonotone line search technique originally proposed for unconstrained problems by Zhang and Hager. The
combination of these two ideas leads to produce a nonmonotone scheme for constrained nonconvex problems, which is proven to
converge to a stationary point. Some variants of this algorithm that incorporate spectral steplength are also studied and
compared with classical nonmonotone schemes based on the usual Euclidean projection. To validate our approach, we report on
numerical results solving bound constrained problems from the CUTEr library collection. 相似文献
13.
A quadratically approximate framework for constrained optimization,global and local convergence 总被引:1,自引:0,他引:1
Jin Bao Jian 《数学学报(英文版)》2008,24(5):771-788
This paper presents a quadratically approximate algorithm framework (QAAF) for solving general constrained optimization problems, which solves, at each iteration, a subproblem with quadratic objective function and quadratic equality together with inequality constraints. The global convergence of the algorithm framework is presented under the Mangasarian-Fromovitz constraint qualification (MFCQ), and the conditions for superlinear and quadratic convergence of the algorithm framework are given under the MFCQ, the constant rank constraint qualification (CRCQ) as well as the strong second-order sufficiency conditions (SSOSC). As an incidental result, the definition of an approximate KKT point is brought forward, and the global convergence of a sequence of approximate KKT points is analysed. 相似文献
14.
A. A. Lazarev 《Computational Mathematics and Mathematical Physics》2009,49(2):373-386
An approach is proposed for estimating absolute errors and finding approximate solutions to classical NP-hard scheduling problems of minimizing the maximum lateness for one or many machines and makespan is minimized. The concept of a metric (distance) between instances of the problem is introduced. The idea behind the approach is, given the problem instance, to construct another instance for which an optimal or approximate solution can be found at the minimum distance from the initial instance in the metric introduced. Instead of solving the original problem (instance), a set of approximating polynomially/pseudopolynomially solvable problems (instances) are considered, an instance at the minimum distance from the given one is chosen, and the resulting schedule is then applied to the original instance. 相似文献
15.
Nadia Benakli Ashwin Satyanarayana Satyanand Singh 《International Journal of Mathematical Education in Science & Technology》2017,48(3):393-427
The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data analysis. We present nine computer experiments and suggest a few more, with applications to calculus, probability and data analysis, which engage computational thinking through simulations, visualizations and data analysis. We are using the free (open-source) statistical programming language R. Our goal is to give a taste of what R offers rather than to present a comprehensive tutorial on the R language. In our experience, these kinds of interactive computer activities can be easily integrated into a smart classroom. Furthermore, these activities do tend to keep students motivated and actively engaged in the process of learning, problem solving and developing a better intuition for understanding complex mathematical concepts. 相似文献
16.
17.
A. Auslender 《Mathematical Programming》1997,79(1-3):3-18
The aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions
which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions,
but also for constructing new methods. 相似文献
18.
Zengqin Zhao 《Applied mathematics and computation》2009,215(8):2971-2977
Existence results of fixed points for some convex operators are given by means of fixed point theorem of cone expansion and compression, then they are applied to nonlinear multi-point boundary value problems. 相似文献
19.
Gregory S. Chirikjian Alexander B. Kyatkin 《Journal of Fourier Analysis and Applications》2000,6(6):583-606
In this article we develop analytical and computational tools arising from harmonic analysis on the motion group of three-dimensional
Euclidean space. We demonstrate these tools in the context of applications in robotics and polymer science. To this end, we
review the theory of unitary representations of the motion group of three dimensional Euclidean space. The matrix elements
of the irreducible unitary representations are calculated and the Fourier transform of functions on the motion group is defined.
New symmetry and operational properties of the Fourier transform are derived. A technique for the solution of convolution
equations arising in robotics is presented and the corresponding regularized problem is solved explicity for particular functions.
A partial differential equation from polymer science is shown to be solvable using the operational properties of the Euclidean-group
Fourier transform. 相似文献
20.
Sien Deng 《Mathematical Programming》1998,83(1-3):263-276
In this paper several types of perturbations on a convex inequality system are considered, and conditions are obtained for
the system to be well-conditioned under these types of perturbations, where the well-conditionedness of a convex inequality
system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations. It is shown that
certain types of perturbations can be used to characterize the well-conditionedness of a convex inequality system, in which
either the system has a bounded solution set and satisfies the Slater condition or an associated convex inequality system,
which defines the recession cone of the solution set for the system, satisfies the Slater condition. Finally, sufficient conditions
are given for the existence of a global error bound for an analytic system. It is shown that such a global error bound always
holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative
interior of the domain of an associated convex conjugate function. 相似文献