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representative functions, Variational Convergence and Almost Convexity

SET-VALUED AND VARIATIONAL ANALYSIS 2024年第1期32卷 4-4页

作者： Eberhard, A. Wenczel, R. RMIT Math Sci GPO Box 2476V Melbourne Vic 3001 Australia

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal-cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

关键词： Maximal monotone operators representative functions Almost convexity Recession operators

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Surrogate-based automated hyperparameter optimization for expensive automotive crashworthiness optimization

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2025年第4期68卷 1-19页

作者： Long, Fu Xing van Stein, Niki Frenzel, Moritz Krause, Peter Gitterle, Markus Back, Thomas BMW Grp Knorrstr 147 D-80788 Munich Germany Leiden Univ LIACS Niels Bohrweg 1 NL-2333 Leiden Netherlands Altair Engn GmbH Calwer Str 7 D-71034 Boblingen Germany Divis Intelligent Solut GmbH Joseph von Fraunhofer Str 20 D-44227 Dortmund Germany Munich Univ Appl Sci Dachauer Str 98b D-80335 Munich Germany

In the automotive industry, solving crashworthiness optimization problems efficiently is crucial to minimize time and cost investment on expensive function evaluations, e.g., using simulation runs. Nonetheless, automotive crashworthiness optimization is time-consuming and challenging even with domain knowledge, due to the fact that crash problems are typically high-dimensional, nonlinear, and discontinuous. In this work, we propose an automated hyperparameter optimization (HPO) approach for expensive black-box optimization (BBO) problems that can assist practitioners to solve automotive crash problems efficiently using optimally configured optimization algorithms. Precisely, the landscape characteristics of BBO problems, e.g., quantified using exploratory landscape analysis (ELA), are analyzed to identify cheap-to-evaluate representative functions that belong to the same optimization problem class. Based on these representative functions, algorithm configurations can be optimally fine-tuned at a relatively low computational cost. Using three optimization algorithms, consisting of modular covariance matrix adaptation evolutionary strategy (CMA-ES), modular differential evolution (DE), and Bayesian optimization (BO), we evaluate the potential of our approach based on the black-box optimization benchmarking (BBOB) suite and an automotive side crash problem. Since the optimal configurations identified using our approach can perform well on most of the BBOB functions, we believe that our approach can generalize well to BBO problems with similar optimization complexity. For the automotive side crash problem, the BO configuration fine-tuned using our approach can outperform the default BO configuration as well as the conventional response surface method (RSM), in terms of the best-found-solution and convergence speed. Furthermore, better solutions can be identified using the proposed approach compared to successive RSM (SRSM), when dealing with complex crash functions and

关键词： Black-box optimization Hyperparameter optimization representative functions Bayesian optimization Exploratory landscape analysis Automotive crashworthiness

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Closure Operations of Convex Sets in X x X * and Monotone Operators

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VIETNAM JOURNAL OF MATHEMATICS 2025年第1期53卷 17-49页

作者： Eberhard, Andrew Wenczel, Robert RMIT Math Sci GPO Box 2476V Melbourne Vic 3001 Australia

We develop a number of new results on "closure operations" based on s x w*-convergence of bounded nets for convex sets and functions on X x X *. We discuss their relevance to the study of monotone operators ... 详细信息

关键词： Monotone operators representative functions Banach-Dieudonne property Graph closures

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On the FPV Property, Monotone Operator Structure and the Monotone Polar of Representable Monotone Sets

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SET-VALUED AND VARIATIONAL ANALYSIS 2021年第2期29卷 301-339页

作者： Eberhard, A. Wenczel, R. RMIT Math Sci GPO Box 2476V Melbourne Vic 3001 Australia

We study the pointwise partial ordering of representative functions for a monotone operator and in particular we focus on the bigger conjugate representative functions that represent a fixed initial (non-maximal) monotone operator. The first problem considered is that of constructing a new representative function from a given member of this class when wanting to add an additional monotonically related point. This study allows us to prove that all bigger conjugate representable monotone sets are monotonically closed. This result sheds light on the structure of the domains for maximal monotone operators and enables us to study the sum theorem for FPV operators in Banach spaces which posses a dual space that has a strictly convex renorm.

关键词： Maximal monotone operators representative functions Sum theorems FPV property

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All Maximal Monotone Operators in a Banach Space are of Type FPV

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SET-VALUED AND VARIATIONAL ANALYSIS 2014年第3期22卷 597-615页

作者： Eberhard, A. Wenczel, R. RMIT Sch Math & Geospatial Sci Melbourne Vic 3001 Australia

Within a nonzero, real Banach space we show that a monotone operator with a bounded domain that is representable by a representative function with a bigger conjugate must be maximal. This study allows us to resolve some long outstanding questions in the area. It follows that all maximal monotone operators are of type FPV and their domains have a convex closure.

关键词： Maximal monotone operators representative functions Sum theorems FPV property

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Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

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CENTRAL EUROPEAN JOURNAL OF MATHEMATICS 2011年第1期9卷 162-172页

作者： Bot, Radu Ioan Grad, Sorin-Mihai Univ Dusseldorf Inst Math D-4000 Dusseldorf Germany Tech Univ Chemnitz Fac Math Chemnitz Germany

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator S(. + p) + T(.), where p is an element of X and S and T are maximal monotone operators on the reflexive Banach space chi. Then, this is used to obtain sufficient conditions for the surjectivity of S + T and for the situation when 0 belongs to the range of S + T. Several special cases are discussed, some of them delivering interesting byproducts.

关键词： Conjugate functions Subdifferentials representative functions Maximal monotone operators Surjectivity

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Functional interpretation of logics for 'generally'

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LOGIC JOURNAL OF THE IGPL 2004年第6期12卷 627-640页

作者： Veloso, Paulo A. S. Veloso, Sheila R. M. Univ Fed Rio de Janeiro COPPE Progr Eng Sist & Comp BR-21945970 Rio De Janeiro Brazil

Logics for 'generally' are intended to express some vague notions, such as 'generally', 'several', 'many', 'most', etc., by means of the new generalized quantifier del and to re... 详细信息

Logics for 'generally' are intended to express some vague notions, such as 'generally', 'several', 'many', 'most', etc., by means of the new generalized quantifier del and to reason about assertions with 'generally' (important issues in qualitative reasoning). We introduce the idea of functional interpretation for 'generally' and show that representative functions (akin to Skolem functions) enable elimination of del and reduce consequence to classical theories. Thus, one can use proof procedures and theorem provers for classical first-order logic to reason about assertions involving 'generally'.

关键词： logics for 'generally' functional interpretation theorem proving vague notions generalized quantifiers representative functions classical first-order logic

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GENERALIZED REPRESENTATION OF THE LOW-FREQUENCY RADIAL DYNAMIC PARAMETERS OF ROLLING TIRES

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INTERNATIONAL JOURNAL OF VEHICLE DESIGN 1991年第5-6期12卷 513-524页

作者： OLATUNBOSUN, OA DUNN, JW Univ of Birmingham United Kingdom

Low frequency dynamic behaviour of tyres is studied using a non-linear lumped parameter model in which the stiffness and damping parameters vary with the test frequency and tyre operating parameters. From an analysis of the results, representative functions are proposed for describing tyre radial stiffness and damping as functions of the tyre operating parameters. These functions have been tested on a number of tyres and shown to be valid.

关键词： LOW FREQUENCY DYNAMICS NONLINEAR MODELING RADIAL PARAMETERS representative functions TYRE

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ON REPRESENTATIONS OF THE HOLOMORPH OF ANALYTIC GROUPS

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 1985年第1期95卷 135-138页

作者： DONG, HL Department of Mathematics and Statistics Case Western Reserve University Cleveland OH 44106 United States

We completely characterize those complex analytic groups whose holomorph groups admit finite-dimensional faithful complex analytic representations.

关键词： Holomorph representative functions reductive groups unipotent groups

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A FINITENESS CRITERION FOR NON-LINEAR INPUT-OUTPUT DIFFERENTIAL-SYSTEMS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1983年第5期21卷 721-728页

作者： FLIESS, M KUPKA, I UNIV SCI & MED GRENOBLE INST FOURIERCNRSMATH PURE LABF-38402 ST MARTIN HERESFRANCE

We solve the following problem which has been dealt with in several publications: when does a system have the same input–output behaviour as a system depending linearly on the state variables? Precise definitions of this question are supplied in the text. A particular case of the problem is the equivalence of a system with a bilinear one. At the end, we illustrate our methods and the scope of our results by a brief study of several examples.

关键词： nonlinear systems state-affine systems bilinear systems representative functions noncommutative generating power series

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