Solving large-scale systems of nonlinear equations/inequalities is a fundamental problem in computing and optimization. In this paper, we propose a generic successive projection (SP) framework for this problem. The SP sequentially projects the current iterate onto the constraint set corresponding to each nonlinear (in)equality. It extends von Neumann's alternating projection for finding a point in the intersection of two linear subspaces, Bregman's method for finding a common point of convex sets and the Kaczmarz method for solving systems of linear equations to the more general case of multiple nonlinear and nonconvex sets. The existing convergence analyses on randomized Kaczmarz are merely applicable to linear case. There are no theoretical convergence results of the SP for solving nonlinear equations. This paper presents the first proof that the SP locally converges to a solution of nonlinear equations/inequalities at a linear rate. Our work establishes the convergence theory of the SP for the case of multiple nonlinear and nonconvex sets. Besides cyclic and randomized projections, we devise two new greedy projection approaches that significantly accelerate the convergence. Furthermore, the theoretical bounds of the convergence rates are derived. We reveal that the convergence rates are related to the Hoffman constants of the Jacobian matrix of the nonlinear functions at the solution. Applying the SP to solve the graph realization problem, which attracts much attention in theoretical computer science, is discussed.
翻译:解决非线性方程式/ 不平等的大规模系统是计算和优化的根本问题。 在本文中, 我们提出一个通用的连续预测框架( SP) 。 SP 依次将当前循环投射到每个非线性( 内) 等值的制约下。 扩展 von Neumann 的轮流预测, 以找到两个线性子空间的交叉点, 即 Bregman 找到一个共点共点的方法 和 Kaczmarz 方法, 用来解决线性方程式系统, 与多非线性和非线性计算机组合的更普通案例 。 随机的卡茨马尔兹现有趋同分析仅适用于线性案例 。 SP 没有解决非线性等值的理论趋同结果 。 本文首次证明, SP 本地与非线性方程式/ 等离线性方程式/ 等离子的解决方法相交汇点。 我们的工作确定了SP 的共解理论, 在多非线性和非康韦克斯 组中, 的解性双轨性化和随机化的加式的卡兹马尔基化趋同性趋同率的趋同率。 我们设计了两种新的内基化办法, 。 加速地显示的趋同式的递合的递合率 。