Mathematics of Optimization, Smooth and Nonsmooth Case

Derivative-free optimization

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The book is intended for people graduates, researchers, but also undergraduates with a good mathematical background involved in the study of static optimization problems in finite-dimensional spaces. It contains a lot of material, from basic tools of convex analysis to optimality conditions for smooth optimization problems, for non smooth optimization problems and for vector optimization problems. The development of the subjects are self-contained and the bibliographical references are usually treated in different books only a few books on optimization theory deal also with vector problems , so the book can be a starting point for further readings in a more specialized literature.

Kundrecensioner

Assuming only a good even if not advanced knowledge of mathematical analysis and linear algebra, this book presents various aspects of the mathematical theory in optimization problems. The treatment is performed in finite-dimensional spaces and with no regard to algorithmic questions.

Convert currency. Add to Basket. Book Description Elsevier Science, Condition: New. More information about this seller Contact this seller. Language: English. Brand new Book. After two chapters concerning, respectively, introductory subjects and basic tools and concepts of convex analysis, the book treats extensively mathematical programming problems in the smmoth case, in the nonsmooth case and finally vector optimization problems. Seller Inventory EOD Book Description Elsevier Science , Book Description Elsevier, Condition: NEW.

For all enquiries, please contact Herb Tandree Philosophy Books directly - customer service is our primary goal. Book Description Elsevier Science and Technology, New Book. Delivered from our UK warehouse in 4 to 14 business days. They can compute it numerically, but will perform better if you can pass them the gradient:. Newton methods use a local quadratic approximation to compute the jump direction.

For this purpose, they rely on the 2 first derivative of the function: the gradient and the Hessian. Note that, as the quadratic approximation is exact, the Newton method is blazing fast. Here we are optimizing a Gaussian, which is always below its quadratic approximation. As a result, the Newton method overshoots and leads to oscillations.

In scipy, you can use the Newton method by setting method to Newton-CG in scipy. Here, CG refers to the fact that an internal inversion of the Hessian is performed by conjugate gradient. Noisy optimization problem.

Smooth vs non-smooth. Curve fitting. Convex function. Finding a minimum in a flat neighborhood. Optimization with constraints. Constraint optimization: visualizing the geometry.

Plotting the comparison of optimizers. Alternating optimization. Gradient descent. Gallery generated by Sphinx-Gallery.

Here BFGS does better than Newton, as its empirical estimate of the curvature is better than that given by the Hessian. L-BFGS keeps a low-rank version.

An active-set algorithm for solving large-scale nonsmooth optimization models with box constraints

The Nelder-Mead algorithms is a generalization of dichotomy approaches to high-dimensional spaces. The algorithm works by refining a simplex , the generalization of intervals and triangles to high-dimensional spaces, to bracket the minimum. Strong points : it is robust to noise, as it does not rely on computing gradients.

Thus it can work on functions that are not locally smooth such as experimental data points, as long as they display a large-scale bell-shape behavior. However it is slower than gradient-based methods on smooth, non-noisy functions. Using the Nelder-Mead solver in scipy. If your problem does not admit a unique local minimum which can be hard to test unless the function is convex , and you do not have prior information to initialize the optimization close to the solution, you may need a global optimizer.

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The parameters are specified with ranges given to numpy. By default, 20 steps are taken in each direction:. All methods are exposed as the method argument of scipy. Computing gradients, and even more Hessians, is very tedious but worth the effort. Symbolic computation with Sympy may come in handy.

General Information

A very common source of optimization not converging well is human error in the computation of the gradient. You can use scipy. It returns the norm of the different between the gradient given, and a gradient computed numerically:. See also scipy. This function admits a minimum in 0, 0. Starting from an initialization at 1, 1 , try to get within 1e-8 of this minimum point. Least square problems, minimizing the norm of a vector function, have a specific structure that can be used in the Levenberg—Marquardt algorithm implemented in scipy.

What if we compute the norm ourselves and use a good generic optimizer BFGS :. If the function is linear, this is a linear-algebra problem, and should be solved with scipy. Least square problems occur often when fitting a non-linear to data. While it is possible to construct our optimization problem ourselves, scipy provides a helper function for this purpose: scipy. Box bounds correspond to limiting each of the individual parameters of the optimization.

Note that some problems that are not originally written as box bounds can be rewritten as such via change of variables. Both scipy. Equality and inequality constraints specified as functions: and. References to Chapter V. References to Chapter VI.

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