Optimization Problems on Graphs over Convex Regions

Tobia Marcucci, Jack Umenberger, Pablo Parrilo, and Russ Tedrake

Follow live at https://slides.com/russtedrake/brc2020/live

(or later at https://slides.com/russtedrake/brc2020)

Main Idea

Mathematical certification of dynamic UAVs

Background

Funnel libraries

Sequential composition

... at runtime

Collision-free planning with dynamic constraints

IRIS (Fast approximate convex segmentation)

• Iteration between (large-scale) quadratic program and (relatively compact) semi-definite program (SDP)
• Scales to high dimensions, millions of obstacles
• ... enough to work on raw sensor data

• Guaranteed collision-free along dynamic trajectories
• Complete/globally optimal within convex decomposition

Motivation (summarized)

At the start of this project

• Beginnings of mathematical certificates for planning aggressive maneuvers in complex environments
  • Can we scale to more complex (multivehicle?) scenarios
  • Planning with contact dynamics still too hard

• Still can't certify with perception in the loop

Last review: Warm-starting Mixed-Integer MPC

• Solve nearly identical optimization problems on every timestep.
• B&B traditionally uses warm-start inside one solve tree.  Can we reuse previous B&B solution to warm-start the next timestep?

• Challenges:
  • Even given the optimal solution, proving optimality is NP-hard
  • If our best guess is not integer feasible, we throw it away.
• Approach:
  • Dual solutions provide tight lower bounds for next time-step
  • Disturbances/model-errors handled naturally.
• Robust feasibility + fast runtime

Shortest Path Problems over Convex Regions

Deep dive

from Choset, Howie M., et al. Principles of robot motion: theory, algorithms, and implementation. MIT press, 2005.

Main Idea

• Let \(G := (V, E)\) be a directed graph with vertices \(V\) and edges \(E\).

• For each vertex \(i \in V\), we have a compact convex set \(X_i \subset \R^d\) and a point \(x_i\) contained in it.

• The length of an edge \((i,j) \in E\) is determined by the location of the points \(x_i\) and \(x_j\) via a nonnegative closed convex function \[ \ell(x_i, x_j) : \R^{2d} \rightarrow \R \cup \{\infty\}. \]

• We assume each edge \((i,j) \in E\) to connect distinct vertices \(i \neq j\), but allow finite self-transitions by making copies of a region.

• A path \(P\) from a source \(s \in V\) to a target \(t \in V - \{s\}\) is a sequence of distinct vertices \((i_k)_{k=0}^K\) such that \(i_0=s\), \(i_K=t\),  and \((i_k, i_{k+1})\in E\) for all \(k=0, \ldots, K-1\).

• Find the shortest path \[ \min_P \min_{(x_i)_{i \in P}} \sum_{(i,j) \in P} \ell(x_i,x_j).\]

• Interesting variants with additional constraints on \(x_i, x_j\).

Mixed-integer convex formulation

Introduce \(\varphi_{ij} \in \{0,1\}\) per edge \((i,j) \in E\).

\(\varphi_{ij} = 1\) if the edge \((i,j)\) in shortest path, otherwise \(\varphi_{ij} = 0\).

Use convex hull reformulation + perspective functions to make this convex.

Remarks:

• When sets \( X_i \) are points, reduces to standard LP formulation of the shortest path (known to be tight).
• There are instances of this problem that are NP-hard.

Mixed-integer convex formulation

Example: "Footstep planning" with \(x_{n+1}=Ax_n + Bu_n\)

MICP solution

Example: "Footstep planning" with \(x_{n+1}=Ax_n + Bu_n\)

Convex relaxation from previous approaches

Example: "Footstep planning" with \(x_{n+1}=Ax_n + Bu_n\)

Convex relaxation from our approach

Scaling

Euclidean shortest path

Finding the shortest path from A to B while avoiding polygonal obstacles (“Euclidean shortest path”):

• Solvable in polytime in 2d (with a visibility graph)
• NP-hard from 3 dimensions on
• For the 3d case there exists an approximation algorithm which gives you eps-optimality in poly time
• Nothing is known for \(d \ge 4\)

Our approach:

• Provides polynomial-time algorithm for \(d \ge 4\) that is often tight.
• Solves a more general class of problems (e.g. can add dynamic constraints).
• WIP: Certificate of optimal solutions with high probability given a suitable rounding strategy.

Should be suitable for many graph-based optimization problems

Going forward...

Example: Bipartite matching (with convex regions)

Summary

Project Goal: Mathematical certificates (potentially probabilistic) for multi-vehicle planning/control in complex environments

Today:

• New strong mixed-integer convex formulation for shortest path problems over convex regions
  • reduces to shortest path as regions become points
  • NP-hard; but strong formulation \(\Rightarrow\) efficient B&B
  • Convex relaxations are often tight!  \(\Rightarrow\) Rounding strategies
• Should generalize to many graph optimizations / applications