Stable sequential Kuhn-Tucker theorem in iterative form or a regularized Uzawa algorithm in a regular nonlinear programming problem
The paper of M. I. Sumin is a valuable contribution to the analytic theory and numerical mathematics of Mathematical and Continuous Optimization and of Optimal Control Theory which have become a core technology in various disciplines, especially, in Sciences and Engineering, in Economics, Finance and OR. It pays special attention to the noisy or uncertain nature of initial data and to a constructive use of regularization techniques. In fact, the author investigates a parametric nonlinear programming problem in a metric space with an operator equality constraint in a Hilbert space presuming that its lower semicontinuous value function at a chosen individual parameter value has particular properties of subdifferentiability in the sense of nonlinear and nonsmooth analysis. That subdifferentiability can be interpreted as the existence of a proximal subgradient or a Fréchet subdifferential. This means that an individual problem has a corresponding generalized Kuhn-Tucker vector. Given that assumption, a stable subsequential Kuhn-Tucker Theorem in nonsmooth iterative form is demonstrated and discussed in terms on minimizing sequences, essentially, through the dual regularization method. This result provides necessary and sufficient conditions for a stable construction of a minimizing approximate solution in Warga's sense for the considered problem, whose initial data can be specified approximately. A core difference of the theorem proved from its classical analogue of the same name consists in the fact that the former one takes into account the possible sensitivity of the problem under perturbations of the initial data and, therefore, permits the inherited instability of classical optimality conditions. That result can be treated as a regularized extension of the well-known Uzawa algorithm to nonlinear programming problems. Eventually, the theorem is applied to the ``easiest'' nonlinear optimal control problem, which is a time optimal control problem. This excellent article is well-structured, mathematically deep, demonstrated and exemplified carefully and well-written. The first part and the following seven sections of this detailed paper are as follows: Introduction, 1. Formulation of the Problem, 2. Subdifferentials of Lower Semicontinuous Functions, Augmented Lagrangians and Dual Problems, 3. Dual Regularization for the Solution of the Nonlinear Mathematical Programming Problem, 4. Stable Sequential Kuhn-Tucker Theorem in Nonlinear Programming, 5. Dual Iterative Regularization for Solving the Nonlinear Mathematical Programming Problem with a Stopping Rule, 6. Stable Sequential Kuhn-Tucker Theorem in Iterative Form in Nonlinear Programming with a Stopping Rule, and 7. Stable Sequential Iterative Pontryagin Maximum Principle in the Time-Optimal Control Problem. Further analytic and algorithmic refinements and extensions, strong results and numerical algorithms may be expected in the future, inspired and prepared by this research paper, within the scientific community. For example, we may think of Optimal Control of Partial and Stochastic Differential Equations and of Stochastic Hybrid Systems, representing regime switches, and possibly including delay and further forms of Memory. Those future progresses could support important achievements in science and technology, finance and economics, and in Operational Research (OR), eventually leading to improvements for our modern industries and economies, and for the living conditions of our people.
Lagrange principle Kuhn-Tucker Theorem in nondifferential form
scientific article; zbMATH DE number 6486283
scientific article