Lecture 10: Systems of equations; optimization in one variable (part III)

Function optimization: Positive definite Hessian indicates a local minimum; indefinite Hessian, a saddle point. Convex functions have a unique minimum. Newton's method (f'/f'') finds critical points; successive parabolic interpolation offers a secant-like approach. Bisection method analogy for minimization will be discussed. This segment contrasts Newton's method and the Secant method for finding minima of functions. It explains how Newton's method, adapted for minimization, uses the first and second derivatives of the function, while the Secant method (or its analogue, successive parabolic interpolation) avoids the need for the second derivative, using only function values at previous iterations to approximate the minimum. The trade-offs between these methods in terms of computational cost and convergence are discussed. This segment explains the relationship between the Hessian matrix (second derivative) of a multivariable function at a critical point and the nature of that critical point (local minimum, maximum, or saddle point). It clarifies that a positive definite Hessian indicates a local minimum, a negative definite Hessian indicates a local maximum, and an indefinite Hessian indicates a saddle point. The case of a non-invertible Hessian is also discussed, highlighting its implications for optimization.