Global Optima is not Limit Computable

非凸全局优化作为最优稳定化问题：收敛速率

Non-Convex Global Optimization as an Optimal Stabilization Problem: Convergence Rates

本文将非凸全局优化问题转化为折扣无限时间最优控制问题，建立非凸、连续且可能非光滑目标函数的显式指数收敛速率理论，无需凸性、可微性或Lojasiewicz型条件。

An adaptive Newton algorithm for optimal control problems with application to optimal electrode design

In this work we present an adaptive Newton-type method to solve nonlinear constrained optimization problems in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first order optimality conditions by the Newton-type method. This strategy allows to balance the two errors and to derive effective stopping criteria for the Newton-itera…

On convergence to the global optima

We show that there is no general algorithm which finds a sequence of points with finite-precision converging computably to the global optima of any continuous function.

Certifying Global Optimality of AC-OPF Solutions via sparse polynomial optimization

We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy -- a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor's relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarchy scales well with the problem size and is indeed useful in certifying global optimality of solutions for large-scale real world problems, e.g., the AC-OPF problem. In particular, we are able to certify 1% global optimality for a challenging AC-OP…

Global minima for semilinear optimal control problems

We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. We provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. Numerical examples with unique global solutions are presented.

Complexity and mission computability of adaptive computing systems

There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic polynomial time complexity class called mission class, as many polynomial problems are not computable in mission time. By focusing on such subclass of languages in the context for successful military applications, we also discuss their computational and communicational constraints. We investigate feasible (non)linear models that will minimize energy and maximize …