We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan, the least-squares estimate is given by a co...
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We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan, the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.
We show that the weighted additive DEA score (WA) for the additive DEA model is simultaneously the (dual) support function for a translation of the DEA technology and the gauge of the dual polar set of the translated ...
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We show that the weighted additive DEA score (WA) for the additive DEA model is simultaneously the (dual) support function for a translation of the DEA technology and the gauge of the dual polar set of the translated technology. Those results are used: to show that WA and the indicator function for the translated technology form a dual conjugate pair;to show that WA and the translated technology's gauge function form a dual polar pair;to develop a simple but exact link between WA and the profit function for the DEA technology;to show that WA and the directional distance function form a dual polar pair;and to develop an exact decomposition of profit inefficiency that extends the Cooper, Pastor, Aparicio, and Borras (2011) and Aparicio et al. (2016) decomposition.& COPY;2022 Elsevier B.V. All rights reserved.
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications i...
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Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. The quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. The convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the practical applicability of the algorithm on the pseudospectral functions.
This paper provides a study of multiobjective fractional variational programs involving support functions. It then explains the concept of higher-order K-eta convex. The paper's motivation is to study the duality ...
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This paper provides a study of multiobjective fractional variational programs involving support functions. It then explains the concept of higher-order K-eta convex. The paper's motivation is to study the duality results for the value of primal and dual programs. The numerical example of functional is discussed, which is higher-order K-eta convex but not first-order K-eta convex. A real-world example is considered to verify the results of the weak duality theorem.
In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate funct...
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In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.
In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machine...
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In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new calculus results on intersection rules for normal cones to convex sets and on infimal convolutions of support functions.
This paper is devoted to the numerical study of geometrical shape optimization problems in fluid mechanics, which consist in minimizing some criterion volume cost functionals on a family of admissible doubly connected...
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This paper is devoted to the numerical study of geometrical shape optimization problems in fluid mechanics, which consist in minimizing some criterion volume cost functionals on a family of admissible doubly connected domains, constrained by steady-state Stokes boundary value problems. We establish the existence of the shape derivative of the considered cost functionals, by means of Minkowski deformation, using the shape derivative formulas, recently established in Boulkhemair and Chakib (2014). This allows us to express the shape derivative by means of the support function and to avoid the tedious computations required when one use the gradient optimization process based on the classical shape derivative, involving the vector fields, notably when one opt for the finite element discretization. So, based on the established shape derivative formulas, we propose a shape optimization numerical process for solving these problems, using the gradient descent algorithm performed by the finite element discretization, for approximating the auxiliary boundary value Stokes problems. Finally, in order to show the validity and the effectiveness of the proposed approach, we present some numerical tests obtained by solving some shape optimization problems of minimizing different cost functionals on various configurations of domains, constrained by steady-state stokes boundary value problems with different boundary conditions. These numerical simulations include some comparison results showing that the proposed approach is more efficient than the gradient approach based on the classical shape derivative, in terms of the accuracy of the solution and central processing unit (CPU) time execution.
This work deals with the simulation of fuzzy random variables, which can be used to model various realistic situations, where uncertainty is not only present in form of randomness but also in form of imprecision, desc...
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This work deals with the simulation of fuzzy random variables, which can be used to model various realistic situations, where uncertainty is not only present in form of randomness but also in form of imprecision, described by means of fuzzy sets. Utilizing the common arithmetics in the space of all fuzzy sets only induces a conical structure. As a consequence, it is difficult to directly apply the usual simulation techniques for functional data. In order to overcome this difficulty two different approaches based on the concept of support functions are presented. The first one makes use of techniques for simulating Hilbert space-valued random elements and afterwards projects on the cone of all fuzzy sets. It is shown by empirical results that the practicability of this approach is limited. The second approach imitates the representation of every element of a separable Hilbert space in terms of an orthonormal basis directly on the space of fuzzy sets. In this way, a new approximation of fuzzy sets useful to approximate and simulate fuzzy random variables is developed. This second approach is adequate to model various realistic situations. (c) 2008 Elsevier Inc. All rights reserved.
We derive three partial differential equations describing the attainable set dynamics from the local integral funnel equation. They can be considered as new partial differential equations for optimal control. The Bell...
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We derive three partial differential equations describing the attainable set dynamics from the local integral funnel equation. They can be considered as new partial differential equations for optimal control. The Bellman equation is a special case of one of them. Three examples are given.
作者：
Boulkhemair, A.Univ Nantes
CNRS UMR6629 Lab Math Jean Leray 2 Rue HoussiniereBP 92208 F-44322 Nantes France
We extend a formula for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains, using the so-called support functions and gauge functions to express it. This i...
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We extend a formula for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains, using the so-called support functions and gauge functions to express it. This is a priori a formula in shape optimization theory. However, the result also happens to be an extension of a well-known formula from the Brunn-Minkowski theory of convex bodies.
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