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1.
In this paper we first provide a general formula of inclusion for the Dini-Hadamard ε-subdifferential of the difference of two functions and show that it becomes equality in case the functions are directionally approximately starshaped at a given point and a weak topological assumption is fulfilled. To this end we give a useful characterization of the Dini-Hadamard ε-subdifferential by means of sponges. The achieved results are employed in the formulation of optimality conditions via the Dini-Hadamard subdifferential for cone-constrained optimization problems having the difference of two functions as objective. 相似文献
2.
Given an arbitrary functiong and a convex functionh, we derive the expression of the conjugate ofg —h via a simple proof. 相似文献
3.
N. S. Rozinova 《Russian Mathematics (Iz VUZ)》2010,54(10):75-78
We consider a quadratic d. c. optimization problem on a convex set. The objective function is represented as the difference
of two convex functions. By reducing the problem to the equivalent concave programming problem we prove a sufficient optimality
condition in the form of an inequality for the directional derivative of the objective function at admissible points of the
corresponding level surface. 相似文献
4.
A. S. Strekalovsky 《Computational Mathematics and Mathematical Physics》2007,47(11):1788-1801
Two control problems for a state-linear control system are considered: the minimization of a terminal functional representable as the difference of two convex functions (d.c. functions) and the minimization of a convex terminal functional with a d.c. terminal inequality contraint. Necessary and sufficient global optimality conditions are proved for problems in which the Pontryagin and Bellman maximum principles do not distinguish between locally and globally optimal processes. The efficiency of the approach is illustrated by examples. 相似文献
5.
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
6.
7.
In this paper we use the tools of the convex analysis in order to give a suitable characterization for the epigraph of the
conjugate of the pointwise maximum of two proper, convex and lower semicontinuous functions in a normed space. By using this
characterization we obtain, as a natural consequence, the formula for the biconjugate of the pointwise maximum of two functions,
provided the so-called Attouch–Brézis regularity condition holds.
相似文献
8.
In this paper we work in separated locally convex spaces where we give equivalent statements for the formulae of the conjugate function of the sum of a convex lower‐semicontinuous function and the precomposition of another convex lower‐semicontinuous function which is also K ‐increasing with a K ‐convex K ‐epi‐closed function, where K is a nonempty closed convex cone. These statements prove to be the weakest constraint qualifications given so far under which the formulae for the subdifferential of the mentioned sum of functions are valid. Then we deliver constraint qualifications inspired from them that guarantee some conjugate duality assertions. Two interesting special cases taken from the literature conclude the paper. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
Selden Y. Trimble 《Mathematische Zeitschrift》1969,109(2):112-114
10.
S. Simons 《Journal of Optimization Theory and Applications》1991,71(1):127-136
In this paper, we give a direct proof of Rockafellar's result that the subdifferential of a proper convex lower-semicontinuous function on a Banach space is maximal monotone. Our proof is simpler than those that have appeared to date. In fact, we show that Rockafellar's result can be embedded in a more general situation in which we can quantify the degree of failure of monotonicity in terms of a quotient like the one that appears in the definition of Fréchet differentiability. Our analysis depends on the concepts of the least slope of a convex function, which is related to the steepest descent of optimization theory.The author would like to express his thanks to R. R. Phelps for reading a preliminary version of this paper and making some very valuable suggestions. 相似文献
11.
Tao Sun Penghang Yin Lizhi Cheng Hao Jiang 《Advances in Computational Mathematics》2018,44(3):723-744
In this paper, we consider the minimization of a class of nonconvex composite functions with difference of convex structure under linear constraints. While this kind of problems in theory can be solved by the celebrated alternating direction method of multipliers (ADMM), a direct application of ADMM often leads to difficult nonconvex subproblems. To address this issue, we propose to convexify the subproblems through a linearization technique as done in the difference of convex functions algorithm (DCA). By assuming the Kurdyka-?ojasiewicz property, we prove that the resulting algorithm sequentially converges to a critical point. It turns out that in the applications of signal and image processing such as compressed sensing and image denoising, the proposed algorithm usually enjoys closed-form solutions of the subproblems and thus can be very efficient. We provide numerical experiments to demonstrate the effectiveness of our algorithm. 相似文献
12.
A. Seeger 《Journal of Optimization Theory and Applications》1992,74(3):527-544
Hiriart-Urruty and the author recently introduced the notions of Dupin indicatrices for nonsmooth convex surfaces and studied them in connection with their concept of a second-order subdifferential for convex functions. They noticed that second-order subdifferentials can be viewed as limit sets of difference quotients involving approximate subdifferentials. In this paper, we elaborate this point in a more detailed way and discuss some related questions.The author is grateful to the referees for their helpful comments. 相似文献
13.
14.
J.-E. Martínez-Legaz Ivan Singer 《Journal of Mathematical Analysis and Applications》2006,313(2):717-729
We give simpler proofs of some known conjugation formulas and subdifferential formulas of convex analysis and we give some new interconnections between them, showing how each of them follows from the others. 相似文献
15.
Rafael Correa Abderrahim Hantoute 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(3):1188-1201
Starting from explicit expressions for the subdifferential of the conjugate function, we establish in the Banach space setting some integration results for the so-called epi-pointed functions. These results use the ε-subdifferential and the Fenchel subdifferential of an appropriate weak lower semicontinuous (lsc) envelope of the initial function. We apply these integration results to the construction of the lsc convex envelope either in terms of the ε-subdifferential of the nominal function or of the subdifferential of its weak lsc envelope. 相似文献
16.
A. S. Strekalovsky M. V. Yanulevich 《Computational Mathematics and Mathematical Physics》2008,48(7):1119-1132
A nonconvex optimal control problem is examined for a system that is linear with respect to state and has a terminal objective functional representable as the difference of two convex functions. A new local search method is proposed, and its convergence is proved. A strategy is also developed for the search of a globally optimal control process, because the Pontryagin and Bellman principles as applied to the above problem do not distinguish between the locally and globally optimal processes. The convergence of this strategy under appropriate conditions is proved. 相似文献
17.
AbstractThe paper provides a new subdifferential characterization for Motzkin decomposable (convex) functions. This characterization leads to diverse stability properties for such a decomposability for operations like addition and composition. 相似文献
18.
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
19.
Note on the topological degree of the subdifferential of a lower semi-continuous convex function 总被引:3,自引:0,他引:3
Sergiu Aizicovici Yuqing Chen 《Proceedings of the American Mathematical Society》1998,126(10):2905-2908
The purpose of the present paper is to prove that the topological degree of the subdifferential of a coercive lower semi-continuous function on a sufficiently large ball in a reflexive Banach space is equal to one.
20.
ABSTRACTHaving a function being a difference of sublinear functions defined on a plane, we present a formula for effective calculation of sublinear functions such that their difference is equal to the given one. Moreover, these newly calculated sublinear functions are minimal and as such unique-up-to-linear-summand. We also provide examples of such functions. 相似文献