1,316 research outputs found
A Convergent Approximation of the Pareto Optimal Set for Finite Horizon Multiobjective Optimal Control Problems (MOC) Using Viability Theory
The objective of this paper is to provide a convergent numerical
approximation of the Pareto optimal set for finite-horizon multiobjective
optimal control problems for which the objective space is not necessarily
convex. Our approach is based on Viability Theory. We first introduce the
set-valued return function V and show that the epigraph of V is equal to the
viability kernel of a properly chosen closed set for a properly chosen
dynamics. We then introduce an approximate set-valued return function with
finite set-values as the solution of a multiobjective dynamic programming
equation. The epigraph of this approximate set-valued return function is shown
to be equal to the finite discrete viability kernel resulting from the
convergent numerical approximation of the viability kernel proposed in [4, 5].
As a result, the epigraph of the approximate set-valued return function
converges towards the epigraph of V. The approximate set-valued return function
finally provides the proposed numerical approximation of the Pareto optimal set
for every initial time and state. Several numerical examples are provided
An efficient method for multiobjective optimal control and optimal control subject to integral constraints
We introduce a new and efficient numerical method for multicriterion optimal
control and single criterion optimal control under integral constraints. The
approach is based on extending the state space to include information on a
"budget" remaining to satisfy each constraint; the augmented
Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our
approach hinges on the causality in that PDE, i.e., the monotonicity of
characteristic curves in one of the newly added dimensions. A semi-Lagrangian
"marching" method is used to approximate the discontinuous viscosity solution
efficiently. We compare this to a recently introduced "weighted sum" based
algorithm for the same problem. We illustrate our method using examples from
flight path planning and robotic navigation in the presence of friendly and
adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures.
Since the previous version: typos fixed, formatting improved, one mistake in
bibliography correcte
Hamilton-Jacobi scaling limits of Pareto peeling in 2D
Pareto hull peeling is a discrete algorithm, generalizing convex hull
peeling, for sorting points in Euclidean space. We prove that Pareto peeling of
a random point set in two dimensions has a scaling limit described by a
first-order Hamilton-Jacobi equation and give an explicit formula for the
limiting Hamiltonian, which is both non-coercive and non-convex. This contrasts
with convex peeling, which converges to curvature flow. The proof involves
direct geometric manipulations in the same spirit as Calder (2016).Comment: 50 pages, 18 figures; v2 improves exposition and extends main theorem
to cover any norm in R^
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