凸规划精确罚方法中的路径跟踪
Path Following in the Exact Penalty Method of Convex Programming
AI总结：
本文提出一种路径跟踪策略，通过连续变化罚常数从无约束解追踪到约束解，适用于精确罚方法，并应用于多种凸规划问题。
摘要：
经典罚方法求解一系列无约束问题，这些问题的约束满足程度越来越严格。当罚常数趋于无穷大时，恢复约束解。在精确罚方法中，平方罚被绝对值罚替代，且对于有限罚常数值即可恢复解。实践中，罚函数中的折点和罚常数未知大小阻碍了精确罚方法在非线性规划中的广泛应用。本文研究了一种与精确罚方法一致的路径跟踪策略。我们不是在一个固定的罚常数下进行优化，而是将解作为罚常数的连续函数进行追踪。因此，路径跟踪从无约束解开始，随着罚常数增加沿解路径移动。在此过程中，解路径会碰到、沿着滑动并离开各种约束。对于二次规划，解路径是分段线性的，并在约束之间大幅跳跃。对于一般凸规划，解路径是分段光滑的，路径跟踪通过逐段数值求解常微分方程来实现。我们在a) 凸集投影、b) 非负最小二乘、c) 二次约束二次规划、d) 几何规划和e) 半定规划中的多种应用展示了路径跟踪的机制和潜力。最后，图像去噪的示例展示了路径跟踪在逆问题正则化估计中的相关性。在正则化估计中，我们随着罚常数从大值减小而追踪解路径。
英文摘要：
Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to $\infty$, one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and the solution is recovered for a finite value of the penalty constant. In practice, the kinks in the penalty and the unknown magnitude of the penalty constant prevent wide application of the exact penalty method in nonlinear programming. In this article, we examine a strategy of path following consistent with the exact penalty method. Instead of performing optimization at a single penalty constant, we trace the solution as a continuous function of the penalty constant. Thus, path following starts at the unconstrained solution and follows the solution path as the penalty constant increases. In the process, the solution path hits, slides along, and exits from the various constraints. For quadratic programming, the solution path is piecewise linear and takes large jumps from constraint to constraint. For a general convex program, the solution path is piecewise smooth, and path following operates by numerically solving an ordinary differential equation segment by segment. Our diverse applications to a) projection onto a convex set, b) nonnegative least squares, c) quadratically constrained quadratic programming, d) geometric programming, and e) semidefinite programming illustrate the mechanics and potential of path following. The final detour to image denoising demonstrates the relevance of path following to regularized estimation in inverse problems. In regularized estimation, one follows the solution path as the penalty constant decreases from a large value.
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