## 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. |

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### Contents

INTRODUCTION | 1 |

LINEAR SPACES | 11 |

HILBERT SPACE | 46 |

APPROXIMATION | 55 |

OTHER MINIMUM NORM PROBLEMS | 64 |

LEASTSQUARES ESTIMATION | 78 |

DUAL SPACES | 103 |

EXTENSION FORM OF THE HAHNBANACH | 110 |

LINEAR OPERATORS AND ADJOINTS | 143 |

ADJOINTS | 150 |

OPTIMIZATION OF FUNCTIONALS | 169 |

GLOBAL THEORY OF CONSTRAINED OPTIMIZATION | 213 |

LOCAL THEORY OF CONSTRAINED OPTIMIZATION | 239 |

OPTIMAL CONTROL THEORY | 254 |

I0 ITERATIVE METHODS OF OPTIMIZATION | 271 |

GEOMETRIC FORM OF THE HAHNBANACH | 129 |

### Common terms and phrases

adjoint applied arbitrary assume Banach space bounded linear functional Cauchy sequence chapter closed subspace components conjugate functional consider constraints contains continuous functions control problem convergence convex functional convex set corresponding defined definition denoted derivatives dimensional dual space element equal equivalent Example exists finite finite-dimensional follows Frechet differentiable Gateaux differential geometric given gradient Hahn-Banach theorem hence Hilbert space inequality inner product interior point inverse Lagrange multiplier Lemma linear combination linear operator linear variety linear vector space linearly independent Lp spaces mapping matrix maximize minimum norm problems Newton's method nonlinear nonzero normal equations normed linear space normed space obtain optimal control optimization problems orthogonal orthonormal point x0 polynomial pre-Hilbert space projection theorem Proof Proposition random variables random vector real numbers real-valued result satisfying scalar Section Show solution solved space H spave sphere subset Suppose technique theory transformation unique vector space zero