What is the difference between global and partial optimization

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Sign in with your library card Please enter your library card number. Show Summary Details Title Pages Dedication Preface Acknowledgements 1 Introduction 2 Optimal shape design 3 Partial differential equations for fluids 4 Some numerical methods for fluids 5 Sensitivity evaluation and automatic differentiation 6 Parameterization and implementation issues 7 Local and global optimization 8 Incomplete sensitivities 9 Consistent approximations and approximate gradients 10 Numerical results on shape optimization 11 Control of unsteady flows 12 From airplane design to microfluidics 13 Topological optimization for fluids 14 Conclusions and prospectives Index.

Local and global optimization Local and global optimization Chapter: p. If you think you should have access to this title, please contact your librarian. Title Pages Dedication Preface Acknowledgements 1 Introduction 2 Optimal shape design 3 Partial differential equations for fluids 4 Some numerical methods for fluids 5 Sensitivity evaluation and automatic differentiation 6 Parameterization and implementation issues 7 Local and global optimization 8 Incomplete sensitivities 9 Consistent approximations and approximate gradients 10 Numerical results on shape optimization 11 Control of unsteady flows 12 From airplane design to microfluidics 13 Topological optimization for fluids 14 Conclusions and prospectives Index.

All rights reserved. In one dimension, if X is a real-valued variable with the current value of v , the next value should be. Figure 4. It starts at a position marked as 1. The derivative is a big positive value, so it takes a step to the left to position 2. Here the derivative is negative, and closer to zero, so it takes a smaller step to the right to position 3. At position 3, the derivative is negative and closer to zero, so it takes a smaller step to the right.

As it approaches the local minimum value, the slope becomes closer to zero and it takes smaller steps. For multidimensional optimization, when there are many variables, gradient descent takes a step in each dimension proportional to the partial derivative of that dimension.

The new value for X i is. Applying it to the point v 1 , … , v n gives. If the partial derivative of h can be computed analytically, it is usually good to do so. Gradient descent is used for parameter learning , in which there may be thousands or even millions of real-valued parameters to be optimized. There are many variants of this algorithm. For example, instead of using a constant step size, the algorithm could do a binary search to determine a locally optimal step size.

For smooth functions, where there is a minimum, if the step size is small enough, gradient descent will converge to a local minimum.

If the step size is too big, it is possible that the algorithm will diverge. If the step size is too small the algorithm will be very slow. If there is a unique local minimum, gradient descent, with a small enough step size, will converge to that global minimum. When there are multiple local minima, not all of which are global minima, it may need to search to find a global minimum, for example by doing a random restart or a random walk.

These are not guarantee to find a global minimum unless the whole search space is exhausted, but are often as good as we can get.