Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for handling collision detection between polyhedral shapes. We frame the signed distance between two polyhedral bodies as the optimal value of a convex optimization, and consider constraining the signed distance in a bilevel optimization problem. To avoid relying on specialized bilevel solvers, our method exploits the fact that the signed distance is the minimal point of a convex region related to the two bodies. Our method enumerates the values obtained at all extreme points of this region and lists them as constraints in the higher-level problem. We compare our formulation to existing methods in terms of reliability and speed when solved using the same mixed complementarity problem solver. We demonstrate that our approach more reliably solves difficult collision detection problems with multiple obstacles than other methods, and is faster than existing methods in some cases.

We study large-scale, nonsmooth, nonconconvex optimization problems. In particular, we focus on nonconvex problems with \emph{composite} objectives. This class of problems includes the extensively studied convex, composite objective problems as a special case. To tackle composite nonconvex problems, we introduce a powerful new framework based on asymptotically \emph{nonvanishing} errors, avoiding the common convenient assumption of eventually vanishing errors. Within our framework we derive both batch and incremental nonconvex proximal splitting algorithms.

We study large-scale, nonsmooth, nonconconvex optimization problems. In particular, we focus on nonconvex problems with \emph{composite} objectives. This class of problems includes the extensively studied convex, composite objective problems as a special case. To tackle composite nonconvex problems, we introduce a powerful new framework based on asymptotically \emph{nonvanishing} errors, avoiding the common convenient assumption of eventually vanishing errors. Within our framework we derive both batch and incremental nonconvex proximal splitting algorithms.

In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter. In this paper, we propose a continuation algorithm that is applicable to a large class of nonsmooth regularized risk minimization problems, can be flexibly used with a number of existing solvers for the underlying smoothed subproblem, and with convergence results on the whole algorithm rather than just one of its subproblems. In particular, when accelerated solvers are used, the proposed algorithm achieves the fastest known rates of $O(1/T^2)$ on strongly convex problems, and $O(1/T)$ on general convex problems. Experiments on nonsmooth classification and regression tasks demonstrate that the proposed algorithm outperforms the state-of-the-art.