Optimization Part II G.Anuradha

Review of previous lecture- Steepest Descent Choose the next step so that the function decreases: For small changes in x we can approximate F(x): where If we want the function to decrease: We can maximize the decrease by choosing:

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Necessary and sufficient conditions for a function with single variable

Functions with two variables Sufficient conditions Necessary conditions

Stationary Points

Effect of learning rate More the learning rate the trajectory becomes oscillatory. This will make the algorithm unstable The upper limit for learning rates can be set for quadratic functions

Stable Learning Rates (Quadratic) Stability is determined by the eigenvalues of this matrix. Eigenvalues of [I - aA]. (li - eigenvalue of A) Stability Requirement:

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Newton’s Method Take the gradient of this second-order approximation and set it equal to zero to find the stationary point:

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This is used for finding line minimization methods and their stopping criteria Initial bracketing Line searches Newton’s method Secant method Sectioning method

Initial Bracketing Helps in finding the range which contains the relative minimum Bracketing some assumed minimum in the starting interval is required Two schemes are used for this purpose

Sectioning methods