Search results for "Solution set"
showing 10 items of 13 documents
Closure properties for integral problems driven by regulated functions via convergence results
2018
Abstract In this paper we give necessary and sufficient conditions for the convergence of Kurzweil–Stieltjes integrals with respect to regulated functions, using the notion of asymptotical equiintegrability. One thus generalizes several well-known convergence theorems. As applications, we provide existence and closure results for integral problems driven by regulated functions, both in single- and set-valued cases. In the particular setting of bounded variation functions driving the equations, we get features of the solution set of measure integrals problems.
Witness computation for solving geometric constraint systems
2014
International audience; In geometric constraint solving, the constraints are represented with an equation system F(U, X) = 0, where X denotes the unknowns and U denotes a set of parameters. The target solution for X is noted XT. A witness is a couple (U_W, X_W) such that F(U_W, X_W) = 0. The witness is not the target solution, but they share the same combinatorial features, even when the witness and the target lie on two distinct connected components of the solution set of F(U, X) = 0. Thus a witness enables the qualitative study of the system: the detection of over- and under-constrained systems, the decomposition into irreducible subsystems, the computation of subsystems boundaries. This …
Termination of a set of rules modulo a set of equations
2006
The problem of termination of a set R of rules modulo a set E of equations, called E-termination problem, arises when trying to complete the set of rules in order to get a Church-Rosser property for the rules modulo the equations. We first show here that termination of the rewriting relation and E-termination are the same whenever the used rewriting relation is E-commuting, a property inspired from Peterson and Stickel’s E-compatibility property. More precisely, their results can be obtained by requiring termination of the rewriting relation instead of E-termination if E-commutation is used instead of E-compatibility. When the rewriting relation is not E-commuting, we show how to reduce E-t…
Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints
1994
In this paper, we establish some general metric regularity results for multivalued functions on Banach spaces. Then, we apply them to derive second-order necessary optimality conditions for the problem of minimizing a functionf on the solution set of an inclusion 0?F(x) withx?C, whenF has a closed convex second-order derivative.
The price of multiobjective robustness : Analyzing solution sets to uncertain multiobjective problems
2021
Defining and finding robust efficient solutions to uncertain multiobjective optimization problems has been an issue of growing interest recently. Different concepts have been published defining what a “robust efficient” solution is. Each of these concepts leads to a different set of solutions, but it is difficult to visualize and understand the differences between these sets. In this paper we develop an approach for comparing such sets of robust efficient solutions, namely we analyze their outcomes under the nominal scenario and in the worst case using the upper set-less order from set-valued optimization. Analyzing the set of nominal efficient solutions, the set of minmax robust efficient …
A non dominated ranking Multi Objective Genetic Algorithm and electre method for unequal area facility layout problems
2013
The unequal area facility layout problem (UA-FLP) comprises a class of extremely difficult and widely applicable optimization problems arising in diverse areas and meeting the requirements for real-world applications. Genetic Algorithms (GAs) have recently proven their effectiveness in finding (sub) optimal solutions to many NP-hard problems such as UA-FLP. A main issue in such approach is related to the genetic encoding and to the evolutionary mechanism implemented, which must allow the efficient exploration of a wide solution space, preserving the feasibility of the solutions and ensuring the convergence towards the optimum. In addition, in realistic situations where several design issues…
Solving the pentahedron problem
2015
Nowadays, all geometric modelers provide some tools for specifying geometric constraints. The 3D pentahedron problem is an example of a 3D Geometric Constraint Solving Problem (GCSP), composed of six vertices, nine edges, five faces (two triangles and three quadrilaterals), and defined by the lengths of its edges and the planarity of its quadrilateral faces. This problem seems to be the simplest non-trivial problem, as the methods used to solve the Stewart platform or octahedron problem fail to solve it. The naive algebraic formulation of the pentahedron yields an under-constrained system of twelve equations in eighteen unknowns. Even if the use of placement rules transforms the pentahedron…
Nonlinear vector Duffing inclusions with no growth restriction on the orientor field
2019
We consider nonlinear multivalued Dirichlet Duffing systems. We do not impose any growth condition on the multivalued perturbation. Using tools from the theory of nonlinear operators of monotone type, we prove existence theorems for the convex and the nonconvex problems. Also we show the existence of extremal trajectories and show that such solutions are $C_0^1(T,\mathbb{R}^N)$-dense in the solution set of the convex problem (strong relaxation theorem).
On Regulated Solutions of Impulsive Differential Equations with Variable Times
2020
In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.
Two-Sided Estimates of the Solution Set for the Reaction–Diffusion Problem with Uncertain Data
2009
We consider linear reaction–diffusion problems with mixed Dirichlet–Neumann–Robin conditions. The diffusion matrix, reaction coefficient, and the coefficient in the Robin boundary condition are defined with an uncertainty which allow bounded variations around some given mean values. A solution to such a problem cannot be exactly determined (it is a function in the set of “possible solutions” formed by generalized solutions related to possible data). The problem is to find parameters of this set. In this paper, we show that computable lower and upper bounds of the diameter (or radius) of the set can be expressed throughout problem data and parameters that regulate the indeterminacy range. Ou…