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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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TL;DR: For positive semidefinite matrices A and B, Ando and Zhan as mentioned in this paper proved the inequalities |||f(A) − f(B)||| ≤ ||| f(|A − B|)|||.
Abstract: For positive semidefinite matrices A and B, Ando and Zhan proved the inequalities |||f(A) + f(B)||| ≥ |||f(A + B)||| and |||g(A) + g(B)||| ≤ |||g(A + B)|||, for any unitarily invariant norm, and for any non-negative operator monotone f on [0, ∞) with inverse function g. These inequalities have very recently been generalised to non-negative concave functions f and non-negative convex functions g, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities |||f(A) − f(B)||| ≤ |||f(|A − B|)|||, and |||g(A) − g(B)||| ≥ |||g(|A − B|)|||, obtained by Ando, for operator monotone f with inverse g, also have a similar generalisation to nonnegative concave f and convex g. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when A ≥ ||B||. In the course of this work, we introduce the novel notion of Y -dominated majorisation between the spectra of two Hermitian matrices, where Y is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.
4 citations
TL;DR: In this article, the money metric utility is concave for any preference relation representable by a concave function if and only if the indirect utility is affine in wealth, which is a new observation.
4 citations
01 Jan 1988
TL;DR: In this paper, two kinds of variational metrics are introduced between closed proper classes of convex-concave functions, and the comparison between these two distances gives rise to a metric stability result for the associated saddle-points.
Abstract: In the lines of H. Attouch and R. Wets, two kinds of variational metrics are introduced between closed proper classes of convex-concave functions. The comparison between these two distances gives rise to a metric stability result for the associated saddle-points.
4 citations
Dissertation•
01 Jan 2012
TL;DR: In this article, the authors considered the canonical duality theory for several classes of optimization problems, such as convex primal optimization, quadratic minimization, and discrete value selection.
Abstract: Duality is one of the most successful ideas in modern science [46] [91]. It is essential in natural phenomena, particularly, in physics and mathematics [39] [94] [96]. In this thesis, we consider the canonical duality theory for several classes of optimization problems. The first problem that we consider is a general sum of fourth-order polynomial minimization problem. This problem arises extensively in engineering and science, including database analysis, computational biology, sensor network communications, nonconvex mechanics, and ecology. We first show that this global optimization problem is actually equivalent to a discretized minimal potential variational problem in large deformation mechanics. Therefore, a general analytical solution is proposed by using the canonical duality theory. The second problem that we consider is a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory, the nonconvex primal problem in n-dimensional space can be converted into a one-dimensional canonical dual problem, which is either a concave maximization or a convex minimization problem with zero duality gap. Several examples are solved so as to illustrate the applicability of the theory developed. The third problem that we consider is quadratic minimization problems subjected to either box or integer constraints. Results show that these nonconvex problems can be converted into concave maximization dual problems over convex feasible spaces without duality gap and the Boolean integer programming problem is actually equivalent to a critical point problem in continuous space. These dual problems can be solved under certain conditions. Both existence and uniqueness of the canonical dual solutions are presented. A canonical duality algorithm is presented and applications are illustrated. The fourth problem that we consider is a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a quadratic 0-1 integer programming problem. The dual problem is thus constructed by using the canonical duality theory. Under appropriate conditions, this dual problem is a maximization problem of a concave function over a convex continuous space. Theoretical results show that the canonical duality theory can either provide a global optimization solution, or an optimal lower bound approximation to this NP-hard problem. Numerical simulation studies, including some relatively large scale problems, are carried out so as to demon-
4 citations
TL;DR: This work gives sufficient conditions for the existence of a time-varying state-dependent routing strategy which ensures that the closed-loop system satisfies basic network properties of positivity, conservation and interconnection while ensuring that capacity constraints are satisfied, when possible, or adjusted if a solution cannot be found.
4 citations