By Wilhelm Forst
Optimization is a crucial box in its personal correct but additionally performs a critical function in several technologies, together with operations learn, administration technological know-how, economics, finance, and engineering.
Optimization — thought and Practice bargains a latest and well-balanced presentation of varied optimization options and their purposes. The book's transparent constitution, sound theoretical fundamentals complemented by means of insightful illustrations and instructive examples, makes it a fantastic introductory textbook and offers the reader with a finished starting place in a single of the main attention-grabbing and beneficial branches of mathematics.
Notable positive aspects include:
- Detailed causes of theoretic effects observed via helping algorithms and routines, usually supplemented by means of necessary tricks or MATLAB®/MAPLE® code fragments;
- an evaluate of the MATLAB® Optimization Toolbox and demonstrations of its makes use of with chosen examples;
- accessibility to readers with an information of multi-dimensional calculus, linear algebra, and easy numerical methods.
Written at an introductory point, this e-book is meant for complex undergraduates and graduate scholars, yet can also be used as a reference via lecturers and execs in arithmetic and the utilized sciences.
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Extra info for Optimization—Theory and Practice
2 Historical Overview In this book we almost exclusively deal with continuous ﬁnite-dimensional optimization problems. This historical overview, however, does not only consider the ﬁnite-dimensional, but also the inﬁnite-dimensional case (that is, Calculus of Variations and Optimal Control ), because their developments have for Historical Overview 21 large parts run in parallel and the theories are almost inextricably interweaved with each other. Optimization Problems in Antiquity The history of optimization begins, like so many other stories, with the ‘ancient’ Greeks.
One works with exact Hessians of the Lagrange function and uses the Lagrange multipliers of the preceding iteration step as its approximate values. If dk is a solution of the quadratic subproblem, then x(k+1) := x(k) + dk gives the new approximative value; note that at ﬁrst there is no ‘line search’. Similar to the classical Newton method one can prove local quadratic convergence for this method; in the literature it is called the Lagrange–Newton method. Beginning with Han (1976), Powell and Fletcher Wilson’s approach was taken up and developed further by numerous authors.
Besides Cauchy’s classical gradient method these were iterative methods which minimized with alternating coordinates and which worked cyclically like the Gauss–Seidel method or with relaxation control like the Gauss– Southwell method [Sou]. In the latter case the minimization was done with respect to the coordinate with the steepest descent. A Quasi-Newton method proposed by Davidon (1959) turned out to be revolutionary. It determined the descent directions dk = −Hk gk from the gradient gk using symmetric positive deﬁnite matrices Hk .