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1.
Petra Weidner 《Optimization》2018,67(7):1121-1141
Scalarization in vector optimization is often closely connected to the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are proved. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini. The results contain statements about minimizers of certain Minkowski functionals and norms. Some existence results for solutions of vector optimization problems are derived. 相似文献
2.
利用共轭函数的上图性质,引入新的约束规范条件,等价刻画了目标函数为凸函数与凸复合函数之和的复合优化问题及其Fenchel-Lagrange对偶问题之间的强对偶与稳定强对偶. 相似文献
3.
Waldemar Pompe 《Mathematical Methods in the Applied Sciences》2008,31(7):775-792
A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C→C* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ‘dual space’ C* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C*⊃X*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
4.
We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy. 相似文献
5.
A class of nonconvex minimization problems can be classified as hidden convex minimization problems. A nonconvex minimization problem is called a hidden convex minimization problem if there exists an equivalent transformation such that the equivalent transformation of it is a convex minimization problem. Sufficient conditions that are independent of transformations are derived in this paper for identifying such a class of seemingly nonconvex minimization problems that are equivalent to convex minimization problems. Thus, a global optimality can be achieved for this class of hidden convex optimization problems by using local search methods. The results presented in this paper extend the reach of convex minimization by identifying its equivalent with a nonconvex representation. 相似文献
6.
Antonio S. Granero 《Journal of Mathematical Analysis and Applications》2007,326(2):1383-1393
If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ2-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w∗-compact subset K⊂X∗∗ then and, if K∩C is w∗-dense in K, then . 相似文献
7.
M. Jimé nez Sevilla J. P. Moreno 《Proceedings of the American Mathematical Society》1998,126(7):1989-1997
We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space is reflexive if and only if it satisfies any of the following properties: (i) admits a norm with the Mazur Intersection Property and the set of all norm attaining functionals of contains an open set, (ii) the set of all norm one elements of contains a (relative) weak* open set of the unit sphere, (iii) has and contains a (relative) weak open set of the unit sphere, (iv) is , has and contains a (relative) weak open set of the unit sphere. Finally, if is separable, then is reflexive if and only if contains a (relative) weak open set of the unit sphere.
8.
Jan Eriksson 《Mathematical Programming》1980,18(1):146-154
This paper describes a method to solve large sparse maximum entropy problems with linear equality constraints using Newtons and the conjugate gradient method. A numerical example is given to introduce the reader to possible applications of entropy models and this method. Some experience from large scale problems is also reported. 相似文献
9.
A. Ben-Tal A. Ben-Israel S. Zlobec 《Journal of Optimization Theory and Applications》1976,20(4):417-437
Necessary and sufficient conditions of optimality are given for convex programming problems with no constraint qualification. The optimality conditions are stated in terms of consistency or inconsistency of a family of systems of linear inequalities and cone relations.This research was supported by Project No. NR-047-021, ONR Contract No. N00014-67-A-0126-0009 with the Center for Cybernetics Studies, The University of Texas; by NSF Grant No. ENG-76-10260 at Northwestern University; and by the National Research Council of Canada. 相似文献
10.
Antonio S. Granero Henryk Hudzik 《Proceedings of the American Mathematical Society》1996,124(12):3777-3787
In this paper we study some structural and geometric properties of the quotient Banach spaces , where is an arbitrary set, is an Orlicz function, is the corresponding Orlicz space on and , being the ideal of elements with finite support. The results we obtain here extend and complete the ones obtained by Leonard and Whitfield (Rocky Mountain J. Math. 13 (1983), 531-539). We show that is not a dual space, that , if for every , that has no smooth points, that it cannot be renormed equivalently with a strictly convex or smooth norm, that is a Grothendieck space, etc.
11.
Various characterizations are given of the exponential Orlicz space L and the Orlicz‐Lorentz space L. By way of application we give a simple proof of the celebrated theorem of Brézis and Wainger concerning a limiting case of a Sobolev imbedding theorem. 相似文献
12.
Aldo Tagliani 《Applied mathematics and computation》2000,110(2-3):105-114
The discrete maximum entropy (ME) probability distribution which can take on a finite number of values and whose first moments are assigned, is considered. The necessary and sufficient conditions for the existence of a maximum entropy solution are identical to the general ones for the finite moment problem. The entropy decreasing by adding one more moment is studied. Unstability of the distribution recovering is proved when an increasing number of moments is used. 相似文献
13.
《Journal of Functional Analysis》2023,284(1):109739
We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space. 相似文献
14.
In this paper we mainly prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak-Orlicz spaces. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions. 相似文献
15.
K. C. Kiwiel 《Journal of Optimization Theory and Applications》1998,96(1):159-173
We replace orthogonal projections in the Polyak subgradient method for nonnegatively constrained minimization with entropic projections, thus obtaining an interior-point subgradient method. Inexact entropic projections are quite cheap. Global convergence of the resulting method is established. 相似文献
16.
A. Ben-Tal M. Teboulle A. Charnes 《Journal of Optimization Theory and Applications》1988,58(2):209-223
We consider infinite-dimensional optimization problems involving entropy-type functionals in the objective function as well as as in the constraints. A duality theory is developed for such problems and applied to the reliability rate function problem in information theory.This research was supported by ONR Contracts N00014-81-C-0236 and N00014-82-K-0295 with the Center for Cybernetics Studies, University of Texas, Austin, Texas. The first author was partly supported by NSF. 相似文献
17.
N. M. Ivochkina 《Journal of Mathematical Sciences》1992,62(3):2741-2746
Problems of minimization of functionals, related to curvature operators of order m, are formulated. Existence theorems for the minimizer are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 29–37, 1990. 相似文献
18.
We study a strongly coupled system consisting of a parabolic equation and a singular Hamilton–Jacobi equation in one space dimension. This system describes the dynamics of dislocation densities in a material submitted to an exterior applied stress. Our system is a natural extension of that studied in [16] where the applied stress was set to be zero. The equations are written on a bounded interval with Dirichlet boundary conditions and require special attention to the boundary. We prove a result of global existence of a solution. The method of the proof consists in considering first a parabolic regularization of the full system, and then passing to the limit. For this regularized system, a result of global existence and uniqueness of a solution has been given in [17]. We show some uniform bounds on this solution which uses in particular an entropy estimate for the densities. 相似文献
19.
We determine the exact asymptotic behaviour of entropy and approximation numbers of the limiting restriction operator , defined by J(f)=f|Ω. Here Ω is a non-empty bounded domain in , ψ is an increasing slowly varying function, , and is the Besov space of generalized smoothness given by the function tsψ(t). Our results improve and extend those established by Leopold [Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 213, Marcel Dekker, New York, 2000, pp. 323–336]. 相似文献
20.
V. Jeyakumar 《Optimization Letters》2008,2(1):15-25
A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal
(i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it
plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality
requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality
have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical
hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas
of convex optimization such as constrained approximation and error bounds.
Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful
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