We propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract Hilbert spaces under the Guignard constraint qualification. The method is equivalent to performing projected backward Euler timestepping on a projected gradient/antigradient flow of the augmented Lagrangian. The projected backward Euler equations can be interpreted as the necessary optimality conditions of a primal-dual proximal regularization of the original problem. The regularized problems are always feasible, satisfy a strong constraint qualification guaranteeing uniqueness of Lagrange multipliers, yield unique primal solutions provided that the stepsize is sufficiently small, and can be solved by a continuation in the stepsize. We show that equilibria of the projected gradient/antigradient flow and critical points of the optimization problem are identical, provide sufficient conditions for the existence of global flow solutions, and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient/antigradient flow, practically eradicating convergence to saddle points and maxima. The sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework. We demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth Newton approach for the regularized subproblems. In contrast to the published article, this version contains a correction that the associate editor considers as too insignificant to justify publication in the journal.

翻译：本文提出了一种序列同伦方法，用于求解在Guignard约束规范下于抽象希尔伯特空间中表述的数学规划问题。该方法等价于在增广拉格朗日函数的投影梯度/反梯度流上执行投影后向欧拉时间步进。投影后向欧拉方程可解释为原始问题经原始-对偶邻近正则化后的必要最优性条件。正则化问题始终可行，满足保证拉格朗日乘子唯一性的强约束规范，且在步长足够小时能产生唯一的原始解，并可通过步长延拓法求解。我们证明了投影梯度/反梯度流的平衡点与优化问题的临界点相同，给出了全局流解存在的充分条件，并证明了具有下降曲线分支的临界点不能是该投影梯度/反梯度流的渐近稳定平衡点，从而在实践上消除了收敛到鞍点和极大值的可能性。该序列同伦方法可用于将任何能在同伦框架中使用的局部收敛优化方法全局化。我们通过一类高度非线性且病态的控制约束椭圆型最优控制问题，结合正则化子问题的半光滑牛顿法，验证了该方法的有效性。与已发表版本相比，本版本包含一项修正，但副主编认为其重要性不足以在期刊上正式发表。