论文标题
在全球优化中引起的带有拉格朗日的消失的艾科纳尔方程
An Eikonal equation with vanishing Lagrangian arising in Global Optimization
论文作者
论文摘要
我们展示了连续函数$ f $的全局不受约束优化与弱KAM理论的连接，用于在Ergodic Control中出现的Eikonal型方程。关键的汉密尔顿 - 雅各比方程的解决方案$ v $是通过较小的折扣近似以及相关进化方程的长时间限制而构建的。然后，$ v $表示目标是控制问题的价值函数，目标轨迹的最佳轨迹是由描述$ v $的梯度下降的差分包含驱动的。事实证明，使用控制理论和职业措施的工具，这种轨迹被证明融合到了$ f $的最小值。我们还证明，在某些情况下，在有限的时间内达到了最小值。
We show a connection between global unconstrained optimization of a continuous function $f$ and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution $v$ of the critical Hamilton-Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then $v$ is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of $v$. Such trajectories are proved to converge to the set of minima of $f$, using tools in control theory and occupational measures. We prove also that in some cases the set of minima is reached in finite time.
