Optimization by Vector Space MethodsEngineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book. |
Contents
INTRODUCTION | 1 |
The Main Principles | 8 |
NORMED LINEAR SPACES | 22 |
HILBERT SPACE | 46 |
APPROXIMATION | 55 |
OTHER MINIMUM NORM PROBLEMS | 64 |
LEASTSQUARES ESTIMATION | 78 |
DUAL SPACES | 103 |
ADJOINTS | 150 |
OPTIMIZATION IN HILBERT SPACE | 160 |
OPTIMIZATION OF FUNCTIONALS | 169 |
GLOBAL THEORY OF CONSTRAINED OPTIMIZATION | 213 |
LOCAL THEORY OF CONSTRAINED OPTIMIZATION | 239 |
Inequality Constraints KuhnTucker Theorem | 247 |
OPTIMAL CONTROL THEORY | 254 |
References | 269 |
EXTENSION FORM OF THE HAHNBANACH | 110 |
GEOMETRIC FORM OF THE HAHNBANACH | 129 |
LINEAR OPERATORS AND ADJOINTS | 143 |
METHODS FOR SOLVING CONSTRAINED | 297 |
| 321 | |
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Common terms and phrases
adjoint applied arbitrary assume Banach space bounded linear functional Cauchy sequence chapter components conjugate functional consider constraints contains continuous functions convergence convex functional convex set defined definition denoted derivatives dimensional element equivalent Example finite finite-dimensional follows Fréchet differentiable functional ƒ G(xo Gateaux differential geometric given gradient Hahn-Banach theorem hence Hilbert space hyperplane inequality inf f(x inner product interior point Lagrange multiplier Lemma linear combination linear operator linear variety linear vector space linearly independent mapping matrix minimize f(x minimum norm problems n-dimensional Newton's method nonlinear nonzero normed linear space normed space optimization problems orthogonal polynomial pre-Hilbert space projection theorem Proof Proposition random variables real numbers result satisfying scalar Section Show solution solved space H sphere subset subspace Suppose t₁ t₂ theory unique vector space x₁ Xn+1 y₁ zero