1,357 research outputs found
Characterizing the universal rigidity of generic frameworks
A framework is a graph and a map from its vertices to E^d (for some d). A
framework is universally rigid if any framework in any dimension with the same
graph and edge lengths is a Euclidean image of it. We show that a generic
universally rigid framework has a positive semi-definite stress matrix of
maximal rank. Connelly showed that the existence of such a positive
semi-definite stress matrix is sufficient for universal rigidity, so this
provides a characterization of universal rigidity for generic frameworks. We
also extend our argument to give a new result on the genericity of strict
complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
Optimal Control of Nonconvex Differential Inclusions
The paper concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization, and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings, and nonsmooth functions), which naturally appear, e.g., in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this paper
Variational Analysis of Evolution Inclusions
The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities. with generally nonsmooth functions. We develop a variational analysis of such roblems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W^1,2-convergence of optimal solutions under consistent perturbations of endpoint constraints) and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler-Lagrange and Weierstrass-Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority /finite codimension/normal compactness assumptions. The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces
Convex Optimization of Launch Vehicle Ascent Trajectory with Heat-Flux and Splash-Down Constraints
This paper presents a convex programming approach to the optimization of a
multistage launch vehicle ascent trajectory, from the liftoff to the payload
injection into the target orbit, taking into account multiple nonconvex
constraints, such as the maximum heat flux after fairing jettisoning and the
splash-down of the burned-out stages. Lossless and successive convexification
are employed to convert the problem into a sequence of convex subproblems.
Virtual controls and buffer zones are included to ensure the recursive
feasibility of the process and a state-of-the-art method for updating the
reference solution is implemented to filter out undesired phenomena that may
hinder convergence. A hp pseudospectral discretization scheme is used to
accurately capture the complex ascent and return dynamics with a limited
computational effort. The convergence properties, computational efficiency, and
robustness of the algorithm are discussed on the basis of numerical results.
The ascent of the VEGA launch vehicle toward a polar orbit is used as case
study to discuss the interaction between the heat flux and splash-down
constraints. Finally, a sensitivity analysis of the launch vehicle carrying
capacity to different splash-down locations is presented.Comment: 2020 AAS/AIAA Astrodynamics Specialist Virtual Lake Tahoe Conferenc
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