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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: It is shown that indeed the logarithmic term is unnecessary and it is proved a risk bound which scales like n-4/5 up to constant factors and also extends to the case of model misspecification, where the true function may not be concave.
Abstract: A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator. The best known risk bound, as had appeared in \citet{GSvex}, scaled like $\log(en) n^{-4/5}$ under the mean squared error loss, up to a constant factor. The authors in \cite{GSvex} had conjectured that the logarithmic term may be an artifact of their proof. We show that indeed the logarithmic term is unnecessary and prove a risk bound which scales like $n^{-4/5}$ up to constant factors. Our proof technique has one extra peeling step than in a usual chaining type argument. Our risk bound holds in expectation as well as with high probability and also extends to the case of model misspecification, where the true function may not be concave.
2 citations
TL;DR: For practical large-scale problems arising from planning transportation networks on inhomogeneous surfaces defined by a digital model, a method of local optimization over a flow polytope vertex set is proposed, which is far more effective in comparison with the Gallo and Sodini method underpolytope strong degeneracy conditions.
Abstract: An optimization problem of interactive inhomogenous flows (Steiner multicommodity network flow problem) is formulated. The problem's main characteristic is a fixed charge change when combining multicommodity communications. In this paper we propose a method for solving this problem which, in order to restrict the search on the feasible domain, reduces the original problem to a concave programming problem in the form: min {f(x)|x∈X} wheref:ℝn→ℝ is a concave function, andX⊂ℝ≥0n is a flow polytope defined by network transportation constraints. For practical large-scale problems arising from planning transportation networks on inhomogeneous surfaces defined by a digital model, a method of local optimization over a flow polytope vertex set is proposed, which is far more effective in comparison with the Gallo and Sodini method under polytope strong degeneracy conditions.
2 citations
01 Jan 2008
TL;DR: Konno et al. as mentioned in this paper formulated the problem of portfolio optimization as a mixed 0−1 integer linear programming problem and solved it using a convex piecewise linear risk measure such as absolute deviation instead of variance.
Abstract: The problems to be discussed in this chapter make up a class of nonconvex financial optimization problems that can be solved within a practical amount of time using the state-of-the-art integer programming methodologies. We will first discuss mean-risk portfolio optimization problems (Elton and Gruber, 1998; Konno and Yamazaki, 1991; Markowitz, 1959) subject to nonconvex constraints such as minimal transaction unit constraints and cardinality constraints on the number of assets to be included in the portfolio (Konno and Yamamoto, 2005b). Also, we will discuss problems with piecewise linear nonconvex transaction costs (Konno and Wijayanayake, 2001, 2002; Konno and Yamamoto, 2005a, 2005b). It will be shown that fairly large-scale problems can now be solved to optimality by formulating the problem as a mixed 0−1 integer linear programming problem if we use convex piecewise linear risk measure such as absolute deviation instead of variance. The second class of problems are so-called maximal predictability portfolio optimization problems (Lo and MacKinlay, 1997), where we maximize the coefficient of determination of the portfolio using factor models. This model, though very promising, was set aside long ago, since we need to maximize the ratio of convex quadratic functions, which is not a concave function. This problem can be solved to optimality by a hyper-rectangular subdivision algorithm (Gotoh and Konno, 2001; Phong et al., 1995) or by 0−1 integer programming approach (Yamamoto and Konno, to appear; Yamamoto et al., to appear) if the number of assets is relatively small. To solve larger problems, we employ absolute deviation as a measure of variation and define the coefficient of determination as the ratio of functions defined by the sum of absolute values of linear functions. The resulting nonconvex minimization problem can be reformulated as a linear complementarity problem that can be solved by using 0−1 integer programming algorithms (Konno et al., 2007, to appear).
2 citations
TL;DR: In this article, the basic facts of the real interpolation method remain true for couples of cones obtained by intersection of the cone of concave functions with rearrangement invariant spaces, and it is shown that the same holds for the couples of convex functions.
Abstract: We prove that the basic facts of the real interpolation method remain true for couples of cones obtained by intersection of the cone of concave functions with rearrangement invariant spaces.
2 citations