A derivative-free solver for unconstrained minimization

## Project description

directsearch is a package for solving unconstrained minimization, without requiring derivatives of the objective. It is particularly useful when evaluations of the objective function are expensive and/or noisy.

It implements a family of direct search methods. For general references on these methods, please consult:

1. A R Conn, K Scheinberg, and L N Vicente. Introduction to derivative-free optimization. SIAM, 2009.

2. C Audet, and W. Hare. Derivative-Free and Blackbox Optimization. Springer, 2017.

3. T G Kolda, R M Lewis, and V Torczon. Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods. SIAM Review, 45(3), 2003, 385-482.

This package extends general direct search methods to use randomized methods for improved practical performance and scalability.

## Citation

If you use this package, please cite:

L Roberts, and C W Royer. Direct search based on probabilistic descent in reduced spaces, In preparation, (2022).

## Installation

$pip install [--user] directsearch To instead install from source run: $ git clone git@github.com:lindonroberts/directsearch.git
$cd directsearch$ pip install -e .

The -e option to pip allows you to modify the source code and for your Python installation to recognize this.

## Usage

This package can solve unconstrained nonlinear optimization problems of the form: min_{x in R^n} f(x). The simplest usage of directsearch is

soln = directsearch.solve(f, x0)

where

• f is a callable objective function, taking in a numpy.ndarray the same shape as x0 and returing a single float.

• x0 is a one-dimensional numpy.ndarray (i.e. len(x0.shape)==1), the starting point for the algorithm. It should be the best available guess of the minimizer.

The output is an object with fields:

• soln.x: the approximate minimizer, the best x value found (a numpy.ndarray the same shape as x0).

• soln.f: the minimum value equal to f(soln.x).

• soln.nf: the number of evaluations of f required by the solve routine.

• soln.flag: an integer indicating the reason for termination.

• soln.msg: a string with a human-readable termination message.

The possible values of soln.flag are:

• soln.EXIT_MAXFUN_REACHED: termination on maximum number of objective evaluations.

• soln.EXIT_ALPHA_MIN_REACHED: termination on small step size (success).

You can print information about the solution using print(soln). The examples directory has several scripts showing the usage of directsearch.

Interfaces to solver instances

There are many configurable options for the solver in directsearch and several ways to call specific direct search algorithm implementations. The full set of available functions is:

• directsearch.solve() applies a direct-search method to a given optimization problem. It is the most flexible available routine.

• directsearch.solve_directsearch() applies regular direct-search techniques without sketching [1,2,3].

• directsearch.solve_probabilistic_directsearch() applies direct search based on probabilistic descent without sketching [4].

• directsearch.solve_subspace_directsearch() applies direct-search schemes based on polling directions in random subspaces [5].

• directsearch.solve_stp() applies the stochastic three points method, a particular direct-search technique [6].

See usage.txt for full details on how to call these functions. The most commonly used optional inputs (to all functions) are:

• maxevals: the maximum number of allowed evaluations of f during the solve.

• verbose: a bool for whether or not to print progress information.

• print_freq: an int indicating how frequently to print progress information (1 is at every iteration).

Choosing a solver instance

As a rule of thumb, if len(x0) is not too large (e.g. less than 50), then solve_directsearch() or solve_probabilistic_directsearch() are suitable choices. Of these, generally solve_probabilistic_directsearch() will solve with fewer evaluations of f, but solve_directsearch() is a deterministic algorithm. If len(x0) is larger, then directsearch.solve_subspace_directsearch() may be a better option. Note that solve_directsearch() is the only deterministic algorithm (i.e. reproducible without setting the numpy random seed).

References

1. A R Conn, K Scheinberg, and L N Vicente. Introduction to derivative-free optimization. SIAM, 2009.

2. C Audet, and W. Hare. Derivative-Free and Blackbox Optimization. Springer, 2017.

3. T G Kolda, R M Lewis, and V Torczon. Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods. SIAM Review, 45(3), 2003, 385-482.

4. S Gratton, C W Royer, L N Vicente, and Z Zhang. Direct Search Based on Probabilistic Descent. SIAM J. Optimization, 25(3), 2015, 1515-1541.

5. L Roberts, and C W Royer. Direct search based on probabilistic descent in reduced spaces, In preparation, (2022).

6. E H Bergou, E Gorbunov, and P Richtarik. Stochastic Three Points Method for Unconstrained Smooth Minimization. SIAM J. Optimization, 30(4), 2020, 2726-2749.

## Bugs

Please report any bugs using GitHub’s issue tracker.

This algorithm is released under the GNU GPL license.

## Project details

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