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1.
Alireza M. Ashtiani Paulo A. V. Ferreira 《Journal of Optimization Theory and Applications》2011,149(2):411-419
The paper addresses the problem of maximizing a sum of products of positive and concave real-valued functions over a convex
feasible set. A reformulation based on the image of the feasible set through the vector-valued function which describes the
problem, combined with an adequate application of convex analysis results, lead to an equivalent indefinite quadratic extremum
problem with infinitely many linear constraints. Special properties of this later problem allow to solve it by an efficient
relaxation algorithm. Some numerical tests illustrate the approach proposed. 相似文献
2.
We consider the problem of constructing the convex envelope of a lower semi-continuous function defined over a compact convex set. We formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets. In particular, we consider nonnegative functions that are products of convex and component-wise concave functions and derive closed-form expressions for the convex envelopes of a wide class of such functions. Several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques. 相似文献
3.
4.
根据向量值全纯函数和亚纯函数的理论,由向量值Plemelj公式,讨论一类局部凸空间中具有ζ-函数核的奇异积分方程与边值问题的关系,给出向量值奇异积分方程和边值问题的解及其稳定性. 相似文献
5.
We study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions. Specifically, we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach, due to McCormick. The standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently. We show that for a multilinear function having a single product term, this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero. We then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms. Finally, for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is always within a constant of the difference between the concave and convex envelopes. These results, along with numerical examples we provide, give insight into how to construct strong relaxations of multilinear functions. 相似文献
6.
The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular
function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions
and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set
functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions
and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems
that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis.
Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000 相似文献
7.
K.O Kortanek 《Journal of Mathematical Analysis and Applications》1974,46(3):725-755
It is shown that any convex or concave extremum problem possesses a subsidiary extremum problem which has certain homogeneous properties. Analogous to the given problem, the “homogenized” extremum problem seeks the minimum of a convex function or the maximum of a concave function over a convex domain. By using homogenized extremum problems, new relationships are developed between any given convex extremum problem (P) and a concave extremum problem (P1) (also having a convex domain), called the “dual” problem of (P). This is achieved by combining all possibilities in tabular form of (1) the values of the extremum functions and (2) the nature of the convex domains including perturbations of all problems (P), (P1), and each of their respective homogenized extremum problems.This detailed and refined classification is contrasted to the relationships obtainable by combining only the possible values of the extremum functions of the problems (P) and (P1) and the possible limiting values of these functions stemming from perturbations of the convex constraint domains of (P) and (P1), respectively.The extremum problems in this paper and classification results are set forth in real topologically paired vector spaces having the Hahn-Banach separation property. 相似文献
8.
证明了由m个Lμp空间产生的Banach向量空间(Lμp)m的弱Banach-Saks性质,其中m是自然数, 1 p 〈+∞.当m= 1时,这就是著名的Banach-Saks-Szlenk定理.运用该性质,还给出了定义在向量空间Rm的一个凸集上的非负连续凸函数与取值在空间(Lpμ)m的一个弱紧子集中的向量值函数的复合函数的积分不等式.当这些向量值函数属于由m个Lμ∞空间产生的积空间(Lμ∞)m的一个弱*紧子集时,类似的积分不等式还是成立的. 相似文献
9.
M. E. Shirokov 《Mathematical Notes》2007,82(3-4):395-409
We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered. 相似文献
10.
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
11.
This work is concerned with exploring the new convexity and concavity properties of the optimal value function in parametric
programming. Some convex (concave) functions are discussed and sufficient conditions for new convexity and concavity of the
optimal value function in parametric programming are given. Many results in this paper can be considered as deepen the convexity
and concavity studies of convex (concave) functions and the optimal value functions. 相似文献
12.
Using the concept of a subdifferential of a vector-valued convex mapping, we provide duality formulas for the minimization of nonconvex composite functions and related optimization problems such as the minimization of a convex function over a vectorial DC constraint. 相似文献
13.
Lagrangian Duality and Cone Convexlike Functions 总被引:1,自引:0,他引:1
In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization
for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky
Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some
of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional
optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of
so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong
Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare
the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called
image space approach initiated by Giannessi. 相似文献
14.
R. Horst 《Journal of Optimization Theory and Applications》1988,58(1):11-37
A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods. 相似文献
15.
Bappaditya Bhowmik 《Archiv der Mathematik》2010,95(6):575-581
We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions
concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe
that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability
involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear
combination of certain successive Taylor coefficients of concave functions. 相似文献
16.
It is well known that, if a vector-valued function can be written as difference of componentwise convex functions, the norm
of such function inherits this property. In this note we show that, if the norm in use is monotonic in the positive orthant
and the functions are non-negative, a sharper decomposition can be obtained. 相似文献
17.
《Operations Research Letters》1986,5(6):313-316
Necessary and sufficient conditions for a function to be representable as a sum of an increasing convex and an increasing concave function are given. Adding a complementary slackness requirement yields a uniquely determined representation. The function class has applications to the theory of majorization and provides an instance of ‘best’ decomposition of functions which are differences of convex functions. 相似文献
18.
W. Hare 《Optimization Letters》2017,11(7):1217-1227
Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide the optimization algorithm. Proving convergence of such methods often applies an assumption that the approximations form fully linear models—an assumption that requires the true objective function to be smooth. However, some recent methods have loosened this assumption and instead worked with functions that are compositions of smooth functions with simple convex functions (the max-function or the \(\ell _1\) norm). In this paper, we examine the error bounds resulting from the composition of a convex lower semi-continuous function with a smooth vector-valued function when it is possible to provide fully linear models for each component of the vector-valued function. We derive error bounds for the resulting function values and subgradient vectors. 相似文献
19.
林一星 《纯粹数学与应用数学》2010,26(5):872-880
由有界变差向量值测度的值域,通过取凸包和闭包,构造了L[0,1],L2[0,1]和C[0,1]空间上的有界变差紧凸集值测度,结果由欧氏空间推广到函数空间. 相似文献
20.
S. Schaible 《Journal of Optimization Theory and Applications》1976,19(2):347-352
Optimization problems are considered where ratios of functionals of several different types (convex/concave, concave/convex, convex/convex, and concave/concave) are to be minimized over a convex set. All these problems are related to the minimization of a quasi-convex functional or a quasi-concave functional. 相似文献