The convergence of many numerical optimization techniques is highly dependent on the initial guess given to the solver. To address this issue, we propose a novel approach that utilizes tensor methods to initialize existing optimization solvers near global optima. Our method does not require access to a database of good solutions. We first transform the cost function, which depends on both task parameters and optimization variables, into a probability density function. Unlike existing approaches, the joint probability distribution of the task parameters and optimization variables is approximated using the Tensor Train model, which enables efficient conditioning and sampling. We treat the task parameters as random variables, and for a given task, we generate samples for decision variables from the conditional distribution to initialize the optimization solver. Our method can produce multiple solutions (when they exist) faster than existing methods. We first evaluate the approach on benchmark functions for numerical optimization that are hard to solve using gradient-based optimization solvers with a naive initialization. The results show that the proposed method can generate samples close to global optima and from multiple modes. We then demonstrate the generality and relevance of our framework to robotics by applying it to inverse kinematics with obstacles and motion planning problems with a 7-DoF manipulator.

翻译：许多数值优化技术的收敛性高度依赖于求解器获得的初始猜测。针对这一问题，我们提出了一种新颖方法，利用张量方法在全局最优解附近初始化现有优化求解器。该方法无需访问优质解数据库。我们首先将依赖任务参数和优化变量的代价函数转化为概率密度函数。与现有方法不同，我们采用张量列模型逼近任务参数与优化变量的联合概率分布，从而实现了高效的条件化与采样。我们将任务参数视为随机变量，对于给定任务，从条件分布中生成决策变量样本以初始化优化求解器。我们的方法能够比现有方法更快地生成多个解（当它们存在时）。首先在难以通过梯度优化求解器配合朴素初始化求解的数值优化基准函数上评估该方法，结果表明该方法能生成靠近全局最优解且源自多个模态的样本。随后通过对7自由度机械臂的带障碍逆运动学与运动规划问题的应用，展示了该框架在机器人学中的通用性与相关性。