We show that contact-rich motion planning is also sparsity-rich when viewed as polynomial optimization (POP). We can exploit not only the correlative and term sparsity patterns that are general to all POPs, but also specialized sparsity patterns from the robot kinematic structure and the separability of contact modes. Such sparsity enables the design of high-order but sparse semidefinite programming (SDPs) relaxations--building upon Lasserre's moment and sums of squares hierarchy--that (i) can be solved in seconds by off-the-shelf SDP solvers, and (ii) compute near globally optimal solutions to the nonconvex contact-rich planning problems with small certified suboptimality. Through extensive experiments both in simulation (Push Bot, Push Box, Push Box with Obstacles, and Planar Hand) and real world (Push T), we demonstrate the power of using convex SDP relaxations to generate global contact-rich motion plans. As a contribution of independent interest, we release the Sparse Polynomial Optimization Toolbox (SPOT)--implemented in C++ with interfaces to both Python and Matlab--that automates sparsity exploitation for robotics and beyond.

The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of finite-dimensional convex problems. At each step of the associated hierarchy, one needs to solve a fixed size semidefinite program, which can be in turn solved with efficient numerical tools. On the practical side however, there is no-free lunch and such optimization methods usually encompass severe scalability issues. Fortunately, for many applications, we can look at the problem in the eyes and exploit the inherent data structure arising from the cost and constraints describing the problem, for instance sparsity or symmetries. This book presents several research efforts to tackle this scientific challenge with important computational implications, and provides the development of alternative optimization schemes that scale well in terms of computational complexity, at least in some identified class of problems. The presented algorithmic framework in this book mainly exploits the sparsity structure of the input data to solve large-scale polynomial optimization problems. We present sparsity-exploiting hierarchies of relaxations, for either unconstrained or constrained problems. By contrast with the dense hierarchies, they provide faster approximation of the solution in practice but also come with the same theoretical convergence guarantees. Our framework is not restricted to static polynomial optimization, and we expose hierarchies of approximations for values of interest arising from the analysis of dynamical systems. We also present various extensions to problems involving noncommuting variables, e.g., matrices of arbitrary size or quantum physic operators.

A highly influential ingredient of many techniques designed to exploit sparsity in numerical optimization is the so-called chordal extension of a graph representation of the optimization problem. The definitive relation between chordal extension and the performance of the optimization algorithm that uses the extension is not a mathematically understood task. For this reason, we follow the current research trend of looking at Combinatorial Optimization tasks by using a Machine Learning lens, and we devise a framework for learning elimination rules yielding high-quality chordal extensions. As a first building block of the learning framework, we propose an on-policy imitation learning scheme that mimics the elimination ordering provided by the (classical) minimum degree rule. The results show that our on-policy imitation learning approach is effective in learning the minimum degree policy and, consequently, produces graphs with desirable fill-in characteristics.