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Precision, Local Search and unimodal functions

ALGORITHMICA 2011年第3期59卷 301-322页

作者： Dietzfelbinger, Martin Rowe, Jonathan E. Wegener, Ingo Woelfel, Philipp Univ Birmingham Sch Comp Sci Birmingham B15 2TT W Midlands England Tech Univ Ilmenau Fak Informat & Automatisierung D-98684 Ilmenau Germany Tech Univ Dortmund FB Informat D-44221 Dortmund Germany Univ Calgary Dept Comp Sci Calgary AB T2N 1N4 Canada

We investigate the effects of precision on the efficiency of various local search algorithms on 1-D unimodal functions. We present a (1+1)-EA with adaptive step size which finds the optimum in O(log n) steps, where n is the number of points used. We then consider binary (base-2) and reflected Gray code representations with single bit mutations. The standard binary method does not guarantee locating the optimum, whereas using the reflected Gray code does so in I similar to((log n)(2)) steps. A(1+1)-EA with a fixed mutation probability distribution is then presented which also runs in O((log n)(2)). Moreover, a recent result shows that this is optimal (up to some constant scaling factor), in that there exist unimodal functions for which a lower bound of Omega((log n)(2)) holds regardless of the choice of mutation distribution. For continuous multimodal functions, the algorithm also locates the global optimum in O((log n)(2)). Finally, we show that it is not possible for a black box algorithm to efficiently optimise unimodal functions for two or more dimensions (in terms of the precision used).

关键词： Local search Precision Computational complexity unimodal functions Multimodal functions

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Streams and Graphs of Dynamical Systems

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QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 2025年第1期24卷 1-1页

作者： De Leo, Roberto Yorke, James A. Howard Univ Dept Math 2400 6th St NW Washington DC 20059 USA Univ Maryland Inst Phys Sci & Technol 4254 Stadium Dr College Pk MD 20742 USA Univ Maryland Dept Math & Phys 4176 Campus Dr College Pk MD 20742 USA

While studying gradient dynamical systems, Morse introduced the idea of encoding the qualitative behavior of a dynamical system into a graph. Smale later refined Morse's idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale's vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node M to node N (we say that N is downstream from M) if the unstable manifold of M intersects the stable manifold of N. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale's construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set, introduced by Birkhoff in 1920s, were elaborated first by Auslander in early 1960s, by Conley in early 1970s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the prolongational relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations (which we call streams) containing the space of orbits of a discrete-time or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of dynamical systems. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. Our main general result is that each stream of a semi-flow with "compact dynamics" has a connected graph. The range of semi-flows covered by our theorem goes from 1-dimensional discrete-time systems like the logistic map up to infinite-dimensional continuous-time systems like the semi-flow of quasilinear parabolic reaction-diffusion partial differential equati

关键词： Qualitative theory of dynamical systems Graphs Generalized recurrence Chain-recurrence Summable chain-recurrence Non-wandering set Quasi-orders Streams Lyapunov functions unimodal functions Towers

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Optimization problems with unimodal fnctions in max-separable constraints

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Optimization 1992年第1-2期24卷 31-41页

作者： Zimmermann, K. Matematicko-Fyzikální Fakulta Univerzity Karlovy Katedra Aplikované Matematiky 11800 PRAHA 1 Malostranské nám. 25 Czech Republic

Optimization problems of the form [formula omitted] subject to [formula omitted] are considered. In these problems F, f1,⋯, fnare continuous, tʹ≦tʺ⇒ F(tʹ)≦F(tʺ) and rʹʹij’s are continuous and upper unimodal on [hj ... 详细信息

Optimization problems of the form [formula omitted] subject to [formula omitted] are considered. In these problems F, f₁,⋯, f_nare continuous, tʹ≦tʺ⇒ F(tʹ)≦F(tʺ) and rʹʹ_ij’s are continuous and upper unimodal on [h_j H_j] for all i, j. A method for solving these problems is suggested, a motivating example from service-point-location problems is described. © 1992 Gordon and Breach Science Publishers S.A.

关键词： max-separable constraints unimodal functions

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A fast method for finding the global solution of the regularized structured total least squares problem for image deblurring

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2008年第1期30卷 419-443页

作者： Beck, Amir Ben-Tal, Aharon Kanzow, Christian Technion Israel Inst Technol Dept Ind Engn MINERVA Optimizat Ctr IL-32000 Haifa Israel Univ Wurzburg Inst Math D-97074 Wurzburg Germany

Given a linear system Ax approximate to b over the real or complex field, where both A and b are subject to noise, the total least squares (TLS) problem seeks to find a correction matrix and a correction right-hand side vector of minimal norm which makes the linear system feasible. To avoid ill posedness, a regularization term is added to the objective function;this leads to the so-called regularized TLS problem. A further complication arises when the matrix A and correspondingly the correction matrix must have a specific structure. This is modeled by the regularized structured TLS (RSTLS) problem. In general this problem is nonconvex and hence difficult to solve. However, the RSTLS problem arising from image deblurring applications under reflexive or periodic boundary conditions possesses a special structure where all relevant matrices are simultaneously diagonalizable (SD). In this paper we introduce an algorithm for finding the global optimum of the RSTLS problem with this SD structure. The devised method is based on decomposing the problem into single variable problems and then transforming them into one-dimensional unimodal real-valued minimization problems which can be solved globally. Based on the uniqueness and attainment properties of the RSTLS solution we show that a constrained version of the problem possesses a strong duality result and can thus be solved via a sequence of RSTLS problems.

关键词： structured total least squares nonconvex optimization image deblurring unimodal functions simultaneously diagonalizable matrices

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The Effects of Walking Speed and Hardness of Terrain on the Foot-Terrain Interaction and Driving Torque for Planar Human Walking

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IEEE ACCESS 2019年 7卷 56174-56189页

作者： Yang, Chuanxiao Ding, Liang Wang, Kunpeng Song, Rong Tang, Dewei Gao, Haibo Deng, Zongquan Harbin Inst Technol State Key Lab Robot & Syst Harbin 150080 Heilongjiang Peoples R China CH UAV Technol Co Ltd China Acad Aerosp Aerodynam Shenzhen Peoples R China Sun Yat Sen Univ Sch Biomed Engn Shenzhen 510006 Peoples R China

The foot-terrain interaction mechanics of human is important in understanding the biped locomotion mechanism. This paper aims to investigate the effect of walking speed and the hardness of the terrain on foot-terrain interaction mechanics. In the experiments, mobile portable plantar mechanical measuring insoles with 107 sensor elements were used to gauge the plantar force and contact area in real-time. Seven subjects with healthy feet have participated in typical experiments. The statistical methods including correlation analysis, univariance analysis, and systematic identification are primarily used to obtain several main results. Bimodal functions in describing the total pressure-time relations were performed at slow and regular speeds but only one hump is left if walking at fast paces;they can be unified and summed by three subsections' unimodal functions. The forefoot produces a large peak impact force in slow and regular quasi-static walking on both hard ground and deformable terrains, while the heel absorbs more dynamic impact shocks in fast walking on hard ground. The torques generated by the ankle joints are calculated based on plantar force or its derived information such as zero moment point. Some implications are drawn, for example, the dynamically changed positions of zero moment point (ZMP) for humans are similar to that generated by reptiles wiggling through the flowable terrains;increasing the stride length on the hard ground produces more impact vibrations than quickening paces but more effective in accelerating walks on deformable terrains.

关键词： Foot-terrain interaction mechanics of human plantar force walking speed deformable terrain unimodal functions ankle torque

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Some inequalities for symmetric convex sets with applications

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ANNALS OF STATISTICS 1996年第2期24卷 753-762页

作者： Anderson, TW Department of Statistics Stanford University Stanford CA 94305 United States

Under appropriate conditions the probability of a convex symmetric set decreases as the spread or scatter of the distribution increases. This paper studies the conditions when the random vector has a symmetric unimoda... 详细信息

关键词： convex sets inequalities unimodal functions elliptically contoured distributions monotonicity of power functions

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A FIBONACCI VERSION OF KRAFTS INEQUALITY APPLIED TO DISCRETE unimodal SEARCH

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SIAM JOURNAL ON COMPUTING 1993年第4期22卷 751-777页

作者： GOLDSTEIN, AS REINGOLD, EM

A function is unimodal if it strictly increases to a unique maximum and then strictly decreases. The problem of determining the smallest possible interval containing the maximum of a unimodal function, by probing only at integer values is studied. In the finite case, the search takes place over the range 0 to N, while in the infinite case the search takes place over the nonnegative integers. The analyses are based on an unusual Fibonacci version of Kraft's inequality.

关键词： unimodal SEARCH FIBONACCI SEARCH KRAFT INEQUALITY FIBONACCI NUMBERS unimodal functions UNBOUNDED SEARCH OPTIMAL ALGORITHMS ACKERMANN FUNCTION INVERSE ACKERMANN FUNCTION

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Empirical comparison of level-wise hierarchical multi-population genetic algorithm*

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JOURNAL OF INFORMATION AND TELECOMMUNICATION 2017年第1期1卷 66-78页

作者： Hong, Tzung-Pei Peng, Yuan-Ching Lin, Wen-Yang Wang, Shyue-Liang Natl Univ Kaohsiung Dept Comp Sci & Informat Engn 700 Kaohsiung Univ Rd Kaohsiung 811 Taiwan Natl Sun Yat Sen Univ Dept Comp Sci & Engn Kaohsiung Taiwan Natl Univ Kaohsiung Dept Informat Management Kaohsiung Taiwan

Metaheuristics have recently been commonly used to solve complex problems in real applications. Some scholars used multiple populations in metaheuristic approaches to shorten the execution time in finding nearly optimal solutions. In this paper, we revisit the properties of sub-population execution for genetic algorithm (GA) and design a level-wise hierarchical sub-population architecture. We also make experiments for comparing the performance of the architecture for unimodal and multimodal functions. The experimental results show that using the hierarchical GA architecture produces a more significant improvement on multimodal functions than on unimodal ones.

关键词： Genetic algorithm hierarchical multi-population genetic algorithm unimodal functions multimodal functions sub-population

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Limited Memory, Limited Arity Unbiased Black-Box Complexity: First Insights 19

Limited Memory, Limited Arity Unbiased Black-Box Complexity:...

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Bulanova, Nina Buzdalov, Maxim ITMO Univ St Petersburg Russia

ISBN: (纸本)9781450367486

Limited arity unbiased black-box complexity was proven to be a successful tool for understanding the working principles of randomized search heuristics and delivered insights to develop new efficient algorithms. While good upper bounds for simple problems were found long time ago, there are still no matching lower bounds. On a road towards closing this gap, we introduce the notion of limited-memory, limited arity unbiased black-box complexity. We show that some efficient binary unbiased algorithms (almost) satisfy the memory-2 requirement, and present an algorithm to compute, for a given problem size, the exact lower bound on the runtime of any memory-m k-ary algorithm on any unimodal function.

关键词： Unbiased complexity memory-resticted unimodal functions

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Conditions for equality in Anderson's theorem

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STATISTICS & PROBABILITY LETTERS 2024年 209卷

作者： Bocinec, Filip Nagy, Stanislav Charles Univ Prague Fac Math & Phys Sokolovska 83 Prague 18675 Czech Republic

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson's theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson's theorem. Our results are used to characterize the independence of Gaussian -distributed random variables.

关键词： Anderson's theorem Convex sets Inequalities Quasiconcavity unimodal functions

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