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About: Subderivative is a(n) research topic. Over the lifetime, 8510 publication(s) have been published within this topic receiving 219039 citation(s). The topic is also known as: subdifferential & subgradient. more


Open accessBook
Ivar Ekeland1, Roger Téman2Institutions (2)
01 Jan 1976-
Abstract: Preface to the classics edition Preface Part I. Fundamentals of Convex Analysis. I. Convex functions 2. Minimization of convex functions and variational inequalities 3. Duality in convex optimization Part II. Duality and Convex Variational Problems. 4. Applications of duality to the calculus of variations (I) 5. Applications of duality to the calculus of variations (II) 6. Duality by the minimax theorem 7. Other applications of duality Part III. Relaxation and Non-Convex Variational Problems. 8. Existence of solutions for variational problems 9. Relaxation of non-convex variational problems (I) 10. Relaxation of non-convex variational problems (II) Appendix I. An a priori estimate in non-convex programming Appendix II. Non-convex optimization problems depending on a parameter Comments Bibliography Index. more

Topics: Convex analysis (73%), Duality (optimization) (70%), Variational analysis (66%) more

4,272 Citations

Open accessBook
21 Oct 1993-
Abstract: IX. Inner Construction of the Subdifferential.- X. Conjugacy in Convex Analysis.- XI. Approximate Subdifferentials of Convex Functions.- XII. Abstract Duality for Practitioners.- XIII. Methods of ?-Descent.- XIV. Dynamic Construction of Approximate Subdifferentials: Dual Form of Bundle Methods.- XV. Acceleration of the Cutting-Plane Algorithm: Primal Forms of Bundle Methods.- Bibliographical Comments.- References. more

Topics: Convex analysis (65%), Subderivative (64%), Convex set (61%) more

2,979 Citations

Journal ArticleDOI: 10.1137/0314056
Abstract: For the problem of minimizing a lower semicontinuous proper convex function f on a Hilbert space, the proximal point algorithm in exact form generates a sequence $\{ z^k \} $ by taking $z^{k + 1} $ to be the minimizes of $f(z) + ({1 / {2c_k }})\| {z - z^k } \|^2 $, where $c_k > 0$. This algorithm is of interest for several reasons, but especially because of its role in certain computational methods based on duality, such as the Hestenes-Powell method of multipliers in nonlinear programming. It is investigated here in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential $\partial f$ is replaced by an arbitrary maximal monotone operator T. Convergence is established under several criteria amenable to implementation. The rate of convergence is shown to be “typically” linear with an arbitrarily good modulus if $c_k $ stays large enough, in fact superlinear if $c_k \to \infty $. The case of $T = \partial f$ is treated in extra detail. Applicati... more

Topics: Strongly monotone (60%), Duality (optimization) (51%), Monotone polygon (51%) more

2,873 Citations

Open accessBook ChapterDOI: 10.1007/978-1-84800-155-8_7
Michael C. Grant1, Stephen Boyd1Institutions (1)
Abstract: We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interiorpoint methods for smooth or cone convex programs. more

Topics: Convex analysis (72%), Convex set (70%), Proper convex function (69%) more

2,653 Citations

Journal ArticleDOI: 10.1016/0041-5553(67)90040-7
Abstract: IN this paper we consider an iterative method of finding the common point of convex sets. This method can be regarded as a generalization of the methods discussed in [1–4]. Apart from problems which can be reduced to finding some point of the intersection of convex sets, the method considered can be applied to the approximate solution of problems in linear and convex programming. more

Topics: Convex analysis (73%), Proper convex function (71%), Convex combination (71%) more

2,456 Citations

No. of papers in the topic in previous years

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Topic's top 5 most impactful authors

Jonathan M. Borwein

57 papers, 6.9K citations

Heinz H. Bauschke

35 papers, 3.4K citations

Boris S. Mordukhovich

31 papers, 1.2K citations

Abderrahim Hantoute

30 papers, 293 citations

Nikolaos S. Papageorgiou

29 papers, 223 citations

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