9 research outputs found
Technical Report: Distributed Asynchronous Large-Scale Mixed-Integer Linear Programming via Saddle Point Computation
We solve large-scale mixed-integer linear programs (MILPs) via distributed
asynchronous saddle point computation. This is motivated by the MILPs being
able to model problems in multi-agent autonomy, e.g., task assignment problems
and trajectory planning with collision avoidance constraints in multi-robot
systems. To solve a MILP, we relax it with a nonlinear program approximation
whose accuracy tightens as the number of agents increases relative to the
number of coupled constraints. Next, we form an equivalent Lagrangian saddle
point problem, and then regularize the Lagrangian in both the primal and dual
spaces to create a regularized Lagrangian that is
strongly-convex-strongly-concave. We then develop a parallelized algorithm to
compute saddle points of the regularized Lagrangian. This algorithm partitions
problems into blocks, which are either scalars or sub-vectors of the primal or
dual decision variables, and it is shown to tolerate asynchrony in the
computations and communications of primal and dual variables. Suboptimality
bounds and convergence rates are presented for convergence to a saddle point.
The suboptimality bound includes (i) the regularization error induced by
regularizing the Lagrangian and (ii) the suboptimality gap between solutions to
the original MILP and its relaxed form. Simulation results illustrate these
theoretical developments in practice, and show that relaxation and
regularization together have only a mild impact on the quality of solution
obtained.Comment: 14 pages, 2 figure
Towards Totally Asynchronous Primal-Dual Convex Optimization in Blocks
We present a parallelized primal-dual algorithm for solving constrained
convex optimization problems. The algorithm is "block-based," in that vectors
of primal and dual variables are partitioned into blocks, each of which is
updated only by a single processor. We consider four possible forms of
asynchrony: in updates to primal variables, updates to dual variables,
communications of primal variables, and communications of dual variables. We
explicitly construct a family of counterexamples to rule out permitting
asynchronous communication of dual variables, though the other forms of
asynchrony are permitted, all without requiring bounds on delays. A first-order
update law is developed and shown to be robust to asynchrony. We then derive
convergence rates to a Lagrangian saddle point in terms of the operations
agents execute, without specifying any timing or pattern with which they must
be executed. These convergence rates contain a synchronous algorithm as a
special case and are used to quantify an "asynchrony penalty." Numerical
results illustrate these developments
Totally Asynchronous Primal-Dual Convex Optimization in Blocks
We present a parallelized primal-dual algorithm for solving constrained
convex optimization problems. The algorithm is "block-based," in that vectors
of primal and dual variables are partitioned into blocks, each of which is
updated only by a single processor. We consider four possible forms of
asynchrony: in updates to primal variables, updates to dual variables,
communications of primal variables, and communications of dual variables. We
construct a family of explicit counterexamples to show the need to eliminate
asynchronous communication of dual variables, though the other forms of
asynchrony are permitted, all without requiring bounds on delays. A first-order
primal-dual update law is developed and shown to be robust to asynchrony. We
then derive convergence rates to a Lagrangian saddle point in terms of the
operations agents execute, without specifying any timing or pattern with which
they must be executed. These convergence rates include an "asynchrony penalty"
that we quantify and present ways to mitigate. Numerical results illustrate
these developments.Comment: arXiv admin note: text overlap with arXiv:2004.0514