Practical methods of optimization, Volume 1
Fully describes optimization methods that are currently most valuable in solving real-life problems. Since optimization has applications in almost every branch of science and technology, the text emphasizes their practical aspects in conjunction with the heuristics useful in making them perform more reliably and efficiently. To this end, it presents comparative numerical studies to give readers a feel for possibile applications and to illustrate the problems in assessing evidence. Also provides theoretical background which provides insights into how methods are derived. This edition offers revised coverage of basic theory and standard techniques, with updated discussions of line search methods, Newton and quasi-Newton methods, and conjugate direction methods, as well as a comprehensive treatment of restricted step or trust region methods not commonly found in the literature. Also includes recent developments in hybrid methods for nonlinear least squares; an extended discussion of linear programming, with new methods for stable updating of LU factors; and a completely new section on network programming. Chapters include computer subroutines, worked examples, and study questions.
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Structure of Methods
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active constraints active set method algorithm applied approximation arise assumption basic BFGS method bound Broyden calculated column computed Consider constraint problem convex function convex set defined descent direction described in Section dual elimination equality constraint equations equivalent exact line searches exact penalty function example exists factors feasible direction feasible point feasible region Figure Fletcher follows Gauss-Newton method given gives global convergence hence Hessian matrix implies inequality constraints integer iteration Lagrange multipliers Lagrangian Lemma line search linear constraints LP problem Newton's method node non-singular non-smooth nonlinear programming objective function obtained optimization order sufficient conditions orthogonal positive definite possible primal problem minimize Proof quadratic function quasi-Newton method Question reduced result satisfies second order conditions second order sufficient sequence simplex method solve SQP method steepest descent subproblem Taylor series termination Theorem transformation trust region unconstrained updating variables vector zero ZTGZ