BC-ADMM: An Efficient Non-convex Constrained Optimizer with Robotic Applications

TL;DR

BC-ADMM addresses non-convex constrained optimization in robotics by introducing a bi-convex relaxation within ADMM, enabling large-step, parallelizable subproblems. The core idea is to relax non-convex constraints into a bi-convex form with a slack $z$ and a surrogate with Lipschitz gradient, enabling convergence analysis and an $O(\epsilon^{-2})$ rate under suitable conditions. The authors validate on multi-agent navigation, UAV trajectory optimization, deformable object simulation, and soft robot grasping, showing faster wall-clock time than gradient descent and Newton in large-scale problems, with attention to scalability and parameter sensitivity. The work delivers a practical, anytime-feasible optimizer for robotics, with potential impact on real-time planning and simulation, while noting limitations such as centralized feasibility checks and reliance on the bi-convex relaxation.

Abstract

Non-convex constrained optimizations are ubiquitous in robotic applications such as multi-agent navigation, UAV trajectory optimization, and soft robot simulation. For this problem class, conventional optimizers suffer from small step sizes and slow convergence. We propose BC-ADMM, a variant of Alternating Direction Method of Multiplier (ADMM), that can solve a class of non-convex constrained optimizations with biconvex constraint relaxation. Our algorithm allows larger step sizes by breaking the problem into small-scale sub-problems that can be easily solved in parallel. We show that our method has both theoretical convergence speed guarantees and practical convergence guarantees in the asymptotic sense. Through numerical experiments in a row of four robotic applications, we show that BC-ADMM has faster convergence than conventional gradient descent and Newton's method in terms of wall clock time.

Figures (23)

• Figure 1: We optimize the position of a circular robot with a radius $r$ under collision constraint, which is located at $x_1\in\mathbb{R}^2$. The robot needs to reach the goal position $x_1^\star$, while avoiding the triangular obstacle spanned by its three vertices $x_2,x_3,x_4$. We can formulate this problem as an optimization with the collision constraint $\text{inf}_{x'\in\text{CH}(x_2,x_3,x_4)}\|x_1-x'\|\geq r$. Here $\text{CH}$ denotes the closed convex set spanned by a set of vertices.
• Figure 2: We illustrate two cases of the non-smooth collision constraint function. Left: When $x_1\notin\text{CH}^\circ(x_2,x_3,x_4)$, the function is locally differentiable with a gradient (red arrow) satisfying $\left\|{\partial{\left[\text{inf}_{x'\in\text{CH}(x_2,x_3,x_4)}\|x_1-x'\|-r\right]}}/{\partial{x_1}}\right\|=1$. Right: When $x_1\in\text{CH}^\circ(x_2,x_3,x_4)$, then gradient vanishes.
• Figure 3: We optimize the position of three circular robots with a uniform radius $r$ under collision constraints, where their position variables are denoted as $x_{1,2,3}\in\mathbb{R}^2$ concatenated into a decision variable $x\in\mathbb{R}^6$. The objective function $f:\mathbb{R}^6\to\mathbb{R}^+$ is designed for the three robots to reach their distinctive goals $x_i^\star$, which is defined as $\sum_{i=1}^3\|x_i-x_i^\star\|^2$. Each collision constraint is modeled as an extended-real log-barrier function $\tilde{P}\in\mathbb{R}^4\to\mathbb{R}^+\cup\{\infty\}$ defined as $\tilde{P}_r(x_i,x_j)=-\log_\epsilon(\|x_i-x_j\|-2r)$, with $\log_\epsilon$ being a locally supported variant of log-barreir function. Altogether, we need three barrier functions $\tilde{P}_r(x_1,x_2)$, $\tilde{P}_r(x_2,x_3)$, and $\tilde{P}_r(x_3,x_1)$, of which the corresponding constraint graph has a loop. Unfortunately, existing ADMM algorithm does not have convergence guarantee for this problem.
• Figure 4: We can reformulate the log-barrier function $\tilde{P}_r(x_i,x_j)$ in fig:fail into an augmented function $P_r(x_i,x_j,n_{ij},d_{ij})$. This is done by introducing separating planes between the pair of robots $x_i$ and $x_j$ and require that the two robots lie on different sides of the separating plane, with the normals and offsets of these separating planes (red) denoted as $n_{ij}$ and $d_{ij}$, respectively. We cancatenate all these variables into $z=\left({n_{12}}^T,{d_{12}}^T,{n_{23}}^T,{d_{23}}^T,{n_{31}}^T,{d_{31}}^T\right)^T$. By proper design, we can make the function $P_r$ convex in $\left({x_i},{x_j}\right)$ and in $\left({n_{ij}},{d_{ij}}\right)$ but not both. We refer readers to sec:collision-potential for more details on the design of the function $P_r$.
• Figure 5: An illustration of our technique of analysis. We replace the extended real function $g$ (gray) with a real function $G$ with Lipschitz continuous gradient (red). With $g$ replaced by $G$, we derive a modified BC-ADMM algorithm denoted as BC-ADMM$_G$. The solution sequence generated by BC-ADMM$_G$ (gray dots and arrows), is restricted to the compact set where $g=G$, which establishes the convergence of BC-ADMM.

Theorems & Definitions (23)

• Remark 2
• Remark 4
• Theorem 5
• Theorem 6
• Corollary 7
• Corollary 8
• Lemma 10
• Lemma 11
• Lemma 12
• Lemma 13