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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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05 Jan 2001
TL;DR: It is shown that if the structure is restored then Multiple Sequence Alignment is achieved and an algorithm for local maximums search on proposed structure has been developed.
Abstract: Multiple sequence alignment is usually considered as an optimization problem, which has a statistical and a structural component. It is known that in the problem of protein sequence alignment a processed sample is too small and not representative in the statistical sense though this information can be sufficient if an appropriate structural model is used. In order to utilize this information a new structural description of the pairwise alignment results union has been developed. It is shown that if the structure is restored then Multiple Sequence Alignment is achieved. Introduced structure represents the set of local maximums of quasi-concave set function on a lower semi lattice, which in turn is a union of the set-theoretical intervals. This union is a set of the consistent subsets of diagonals, introduced by B. Morgenstern, A. Dress, and T. Werner (1996). Algorithm for local maximums search on proposed structure has been developed. It consists of an alternation of the Forward and Backward passes. The Backward pass in this algorithm is a rigorous while the Forward pass is based on heuristics. Multiple alignment of 5 protein sequences are used as an illustration of the proposed algorithm.

3 citations

Journal ArticleDOI
TL;DR: This proof involves elementary methods, without using any advanced theories such as Weierstrass’s Approximation Theorem, from which the technical core result of the paper comes.

3 citations

Book ChapterDOI
01 Jun 2016
TL;DR: This is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed integer nonlinear programs.
Abstract: We study valid inequalities for a set relevant for optimization models that have both binary indicator variables, which indicate positivity of associated continuous variables, and separable concave constraints. Such models reduce to a mixed-integer linear program MILP when the concave constraints are ignored, and to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms to solve such problems to global optimality, relaxations are traditionally obtained by using valid inequalities for the MILP ignoring the concave constraints, and by independently relaxing each concave constraint using the secant obtained from the bounds of the associated variable. We propose a technique to obtain valid inequalities that are based on both the MILP and the concave constraints. We begin by analyzing a low-dimensional set that contains a single binary indicator variable, a single concave constraint, and three continuous variables. Using this analysis, for the canonical Single Node Flow Set SNFS, we demonstrate how to "tilt" a given valid inequality for the SNFS to obtain additional valid inequalities that account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities on a fixed plus concave cost transportation problem. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed integer nonlinear programs.

3 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the theory of Thompson aggregators and prove a variant of their recovery theorem establishing the existence of extremal solutions to the Koopmans equation, where the least fixed point is constructed with either continuity hypothesis by the partial sum method, and the solution is a concave function whenever the aggregator is concave and norm continuous on the interior of its effective domain.
Abstract: We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio. We prove a variant of their Recovery Theorem establishing the existence of extremal solutions to the Koopmans equation. We apply the constructive Tarski-Kantorovich Fixed Point Theorem rather than the nonconstructive Tarski Theorem employed in. We also obtain additional properties of the extremal solutions. The Koopmans operator possesses two distinct order continuity properties. Each is sufficient for the application of the Tarski-Kantorovich Theorem. One version builds on the order properties of the underlying vector spaces for utility functions and commodities. The second form is topological. The Koopmans operator is continuous in Scott's induced topology. The least fixed point is constructed with either continuity hypothesis by the partial sum method. This solution is a concave function whenever the Thompson aggregator is concave and also norm continuous on the interior of its effective domain.

3 citations


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