Concave function

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

Reducing Concave Programs with Some Linear Constraints

Abstract: The problem of maximizing a concave function over a general convex set subject to linear inequality constraints is reduced to a finite sequence of sub-problems involving linear equality constraints. This reduction can be expected to be computationally useful when there are but a few constraints, or when at most a few constraints are binding at the optimal solution of the original problem, or when prior (though possibly fallible) information is available concerning which constraints are likely to be binding. For quadratic programs the procedure specializes to an improved version of the Theil-van de Panne method. Computational considerations and experience are discussed, and a graphical example is given. The theory and viewpoint developed herein provide the foundation for related reduction procedures that may prove computationally useful even for large problems in the absence of a priori information.

Multifractal decomposition of the Sierpinski carpet

Abstract: We analyse the multifractal decomposition of a measure defined on a general Sierpinski carpet. We compute the dimension spectrum f(α) and we show that this function is real-analytic on a certain domain when it is not degenerated. Actually, we prove that f is the Legendre-Fenchel transformation of a free energy function F which is also real-analytic. These two functions have the typical behaviours. In particular, F is strictly increasing and is in general strictly concave (respectively linear in the degenerate case), and f, on its domain, has a typical shape of a strictly concave function (respectively defined in one point in the degenerate case). We associate also to the singularity sets Cα measures which are singular with respect to each other, and we see that these measures are very well fitted to these singularity sets. This work completes with (D. Simpelaere, Chaos, Solitons and Fractals 4(12), 2223–2235 (1994)) the study of the multifractal analysis of the Sierpinski carpets.

Positive solutions for nonlinear Neumann problems with concave and convex terms

Abstract: We consider a nonlinear elliptic Neumann problem driven by the p-Laplacian with a reaction that involves the combined effects of a “concave” and of a “convex” terms. The convex term (p-superlinear term) need not satisfy the Ambrosetti–Rabinowitz condition. Employing variational methods based on the critical point theory together with truncation techniques, we prove a bifurcation type theorem for the equation.

Sharp bounds of Jensen type for the generalized Sugeno integral

Abstract: In this paper we provide two-sided attainable bounds of Jensen type for the generalized Sugeno integral of {\it any} measurable function. The results extend the previous results of Roman-Flores et al. for increasing functions and Abbaszadeh et al. for convex and concave functions. We also give corrections of some results of Abbaszadeh et al. As a by-product, we obtain sharp inequalities for symmetric integral of Grabisch. To the best of our knowledge, the results in the real-valued functions context are presented for the first time here.

Concave elliptic equations and generalized Khovanskii–Teissier inequalities

Abstract: We explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii-Teissier inequalities.