Titlebook: Optimization; Kenneth Lange Textbook 2013Latest edition Springer Science+Business Media New York 2013 Convexity.Differentiation.Gauge Inte

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Differentiation,ues surrounding differentiation were settled long ago. For multivariate differentiation, there are still some subtleties and snares. We adopt a definition of differentiability that avoids most of the pitfalls and makes differentiation of vectors and matrices relatively painless. In later chapters, t

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Block Relaxation,st either minimization or maximization rather than generic optimization. Regardless of what one terms the strategy, in many problems it pays to update only a subset of the parameters at a time. Block relaxation divides the parameters into disjoint blocks and cycles through the blocks, updating only

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The MM Algorithm,guments and is particularly useful in high-dimensional problems such as image reconstruction [171]. This iterative method is called the MM algorithm. One of the virtues of this acronym is that it does double duty. In minimization problems, the first M of MM stands for majorize and the second M for m

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,Newton’s Method and Scoring,s defects, Newton’s method is the gold standard for speed of convergence and forms the basis of most modern optimization algorithms in low dimensions. Its many variants seek to retain its fast convergence while taming its defects. The variants all revolve around the core idea of locally approximatin

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Conjugate Gradient and Quasi-Newton,pecial features of the objective function . in overcoming the defects of Newton’s method. We now consider algorithms that apply to generic functions .. These algorithms also operate by locally approximating . by a strictly convex quadratic function. Indeed, the guiding philosophy behind many modern

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Analysis of Convergence,patterns separately. The local convergence rate of an algorithm provides a useful benchmark for comparing it to other algorithms. On this basis, Newton’s method wins hands down. However, the tradeoffs are subtle. Besides the sheer number of iterations until convergence, the computational complexity