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. The joint probability distribution of the task parameters and optimization variables is approximated using the Tensor Train model which enables efficient conditioning and sampling. Unlike existing methods, 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 for a given task from different modes when they exist. 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的初始猜测。针对这一问题，我们提出了一种利用张量方法将现有优化求解器初始化至全局最优解附近的新方法。该方法无需访问优质解数据库。我们首先将同时依赖于任务参数和优化变量的代价函数转化为概率密度函数。采用张量列车模型近似任务参数与优化变量的联合概率分布，从而支持高效的条件概率计算与采样。与现有方法不同，我们将任务参数视为随机变量，针对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。当存在多模态解时，该方法可为给定任务生成来自不同模态的多个解。我们首先在数值优化基准函数上评估该方法——这类函数在朴素初始化下难以通过基于梯度的优化器求解。结果表明，所提方法能生成接近全局最优解且来自多模态的样本。随后，我们通过将其应用于带障碍物的逆向运动学和七自由度机械臂运动规划问题，验证了该框架在机器人领域的通用性和实用性。