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Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.


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Book ChapterDOI
01 Jan 2014
TL;DR: A hybrid method combining DCA with an adaptive Branch and Bound is established for guaranteeing the feasibility of the BMI and QMI and a concept of partial solution of SDP via DCA is proposed to improve the convergence of the algorithm when handling more large-scale cases.
Abstract: We propose some new DC (difference of convex functions) programming approaches for solving the Bilinear Matrix Inequality (BMI) Feasibility Problems and the Quadratic Matrix Inequality (QMI) Feasibility Problems. They are both important NP-hard problems in the field of robust control and system theory. The inherent difficulty lies in the nonconvex set of feasible solutions. In this paper, we will firstly reformulate these problems as a DC program (minimization of a concave function over a convex set). Then efficient approaches based on the DC Algorithm (DCA) are proposed for the numerical solution. A semidefinite program (SDP) is required to be solved during each iteration of our algorithm. Moreover, a hybrid method combining DCA with an adaptive Branch and Bound is established for guaranteeing the feasibility of the BMI and QMI. A concept of partial solution of SDP via DCA is proposed to improve the convergence of our algorithm when handling more large-scale cases. Numerical simulations of the proposed approaches and comparison with PENBMI are also reported.

12 citations

01 Nov 1974
TL;DR: In this article, the authors put the problem of path stability in the proper perspective by discussing the much simpler problem of comparative dynamics, i.e., the determination of the "direction" of change in the optimal path of decision variables due to a change of the exogenous variables.
Abstract: Lately, there has been an increased interest in stability of growth paths, see e.g., Brock and Scheinkman [1974]. The problem has been stated in terms of properties of stationary paths. In order to appreciate the difficulty of the general stability problem, one must realize that there are two types of "time" involved in the analysis: stability "time” and path “time.” Thus, the appropriate mathematical field is that of differential equations defined on a space of functions rather than a finite dimensional space. Naturally, if one restricts one’s attention to stationary paths, then the usual stability analysis is appropriate. However, we would be then discussing the asymptotic behavior of the asymptotic state of the economy. This note strives to put the problem of path stability in the proper perspective by discussing the much simpler problem of comparative dynamics. Unfortunately this term has been used in the economic growth literature to discuss the basically comparative statics problem of comparing stationary growth paths. By comparative dynamics, we mean the determination of the “direction” of change in the optimal path of decision variables due to a change in the exogenous variables. The traditional method of deriving comparative statics results has been to use second order conditions for optimality. However, if one is willing to assume concavity, these results could be derived in a more direct way by utilizing the fact that a differentiable concave function lies below its tangent plane. We shall use this concept in deriving the main inequalities of this note. By way of motivation, we first derive two inequalities of comparative statics. Then we derive the comparative dynamics results and finally we discuss some economic theoretical examples.

12 citations

Proceedings ArticleDOI
31 Oct 2005
TL;DR: It is shown that a correlation inequality of statistical mechanics can be applied to low-density parity-check codes and it is proved that the growth rate of regular LDPC codes, can be exactly calculated by iterative methods, at least on the interval where it is a concave function of the relative weight of code words.
Abstract: It is shown that a correlation inequality of statistical mechanics can be applied to low-density parity-check codes. Thanks to this tool we prove that the growth rate of regular LDPC codes, can be exactly calculated by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. We also consider communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code and prove that (at least part of) the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation decoder. In both problems, the correlation inequality yields sharp lower bounds. We also use a non trivial extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem

12 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the maximal operator for functions taking values in a Banach space and showed that the Lp-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance.
Abstract: This paper studies a new maximal operator introduced by Hytonen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The Lp-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to - finite measure spaces with filtrations and the L p -boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques are applied to prove that a weak type inequality is sucient for L p -boundedness and also to provide a characterization by concave functions.

12 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202316
202240
202158
202049
201952
201860