Robustness of Convergence Proofs in Numerical Methods in Unconstrained Optimization

Numerical methods to solve unconstrained optimization problems may be viewed as control systems. An important principle in dynamic control system theory is that control policies should be prescribed in a feedback manner rather than in an open loop manner. This is to ensure that the outcomes are not sensitive to small errors in the state variables. A standard proof in numerical methods in unconstrained optimization like the Zoutendijk method is, from the control theory point of view, an open loop type of analysis as it studies what happens along a total trajectory for various initial state variables. In this chapter, an example is constructed to show that the eventual outcome and convergence to a global minimum point or otherwise can be very sensitive to initial values of the state variable. Convergence of a numerical method in unconstrained optimization can also be established by using the Lyapunov function theorem. The Lyapunov function convergence theorem provides feedback type analysis and thus the outcomes are robust to small numerical errors in the initial states. It requires that the level sets of the objective function are properly nested everywhere in order to have global convergence. This means the level sets of the objective function must be topologically equivalent to concentric spherical surfaces.