119,087 research outputs found
Complete Solutions to General Box-Constrained Global Optimization Problems
This paper presents a global optimization method for solving general nonlinear programming
problems subjected to box constraints. Regardless of convexity or nonconvexity, by introducing a
differential flow on the dual feasible space, a set of complete solutions to the original problem is obtained,
and criteria for global optimality and existence of solutions are given. Our theorems improve and
generalize recent known results in the canonical duality theory. Applications to a class of constrained
optimal control problems are discussed. Particularly, an analytical form of the optimal control is
expressed. Some examples are included to illustrate this new approach
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Novel Transcription Techniques for Multiple-Spacecraft, Multiple-Target Global Trajectory Optimization
With ever increasingly capable tools available to interplanetary mission designers, newer challenging classes of missions become accessible. Among these new classes of missions are Distributed Spacecraft Missions: designs where multiple spacecraft cooperate to achieve coordinated science objectives. Several applications being explored include, but are not limited to: coordinated launch, coordinated rendezvous, mega-constellation design, precision formation flying (PFF), very long baseline interferometry (VLBI), and distributed aperture space telescopes. These mission architectures promise to widen our gaze on the scientific phenomena in our solar system and beyond. However, current operational state-of-the art global trajectory optimization platforms lack the core capabilities to pose these complex new classes missions. These capabilities include: the ability to model multiple individual trajectory optimization problems as one single coupled trajectory optimization problem, the imposition of coordination constraints and cost functions, and the ability to traverse massive search spaces. In this dissertation, we present a fully automated multi-agent multi-objective technique for solving multi-spacecraft multi-target trajectory optimization problems using a hybrid optimal control approach. We apply this technique to two benchmark problems: an Ice Giants Dual Manifest mission design, and a variable fleet size Very Long Baseline Interferometry mission design. The techniques developed in this work create a general transcription for Multiple Traveling Salesmen Problems applied to global spacecraft trajectory optimization
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems
In this paper we propose a distributed dual gradient algorithm for minimizing
linearly constrained separable convex problems and analyze its rate of
convergence. In particular, we prove that under the assumption of strong
convexity and Lipshitz continuity of the gradient of the primal objective
function we have a global error bound type property for the dual problem. Using
this error bound property we devise a fully distributed dual gradient scheme,
i.e. a gradient scheme based on a weighted step size, for which we derive
global linear rate of convergence for both dual and primal suboptimality and
for primal feasibility violation. Many real applications, e.g. distributed
model predictive control, network utility maximization or optimal power flow,
can be posed as linearly constrained separable convex problems for which dual
gradient type methods from literature have sublinear convergence rate. In the
present paper we prove for the first time that in fact we can achieve linear
convergence rate for such algorithms when they are used for solving these
applications. Numerical simulations are also provided to confirm our theory.Comment: 14 pages, 4 figures, submitted to Automatica Journal, February 2014.
arXiv admin note: substantial text overlap with arXiv:1401.4398. We revised
the paper, adding more simulations and checking for typo
Nonnegative Matrix Inequalities and their Application to Nonconvex Power Control Optimization
Maximizing the sum rates in a multiuser Gaussian channel by power control is a nonconvex NP-hard problem that finds engineering application in code division multiple access (CDMA) wireless communication network. In this paper, we extend and apply several fundamental nonnegative matrix inequalities initiated by Friedland and Karlin in a 1975 paper to solve this nonconvex power control optimization problem. Leveraging tools such as the Perron–Frobenius theorem in nonnegative matrix theory, we (1) show that this problem in the power domain can be reformulated as an equivalent convex maximization problem over a closed unbounded convex set in the logarithmic signal-to-interference-noise ratio domain, (2) propose two relaxation techniques that utilize the reformulation problem structure and convexification by Lagrange dual relaxation to compute progressively tight bounds, and (3) propose a global optimization algorithm with ϵ-suboptimality to compute the optimal power control allocation. A byproduct of our analysis is the application of Friedland–Karlin inequalities to inverse problems in nonnegative matrix theory
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