Topic
Proper convex function
About: Proper convex function is a research topic. Over the lifetime, 5337 publications have been published within this topic receiving 165910 citations.
Papers published on a yearly basis
Papers
More filters
Book•
01 Jan 1976TL;DR: In this article, the authors consider non-convex variational problems with a priori estimate in convex programming and show that they can be solved by the minimax theorem.
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.
4,434 citations
Book•
21 Oct 1993
TL;DR: In this article, the cutting plane algorithm is used to construct approximate subdifferentials of convex functions, and the inner construction of the subdifferential is performed by a dual form of Bundle Methods.
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.
3,043 citations
TL;DR: Graph implementations as mentioned in this paper is a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework, which allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved.
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.
2,991 citations
TL;DR: This method can be regarded as a generalization of the methods discussed in [1–4] and applied to the approximate solution of problems in linear and convex programming.
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.
2,668 citations
TL;DR: If U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.
Abstract: We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.
2,501 citations