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THE SLIDE RULE OF SILICON DESIGN

Free Analog Circuit Simulation

Circuit Optimization with SPICE OPUS

Introduction

A new command named optimize has been added. It realises an optimisation loop in Spice Opus. When we run particular optimisation method on a given circuit with optimize command, a general optimisation loop is performed and it is always the same. Its algorithm is as follows:

```do
if number of iterations > max break
change parameter values
verify explicit constraints and correct parameter values if necessary
execute all analysis commands (analysis 0, analysis 1, ..., analysis N)
number of iterations = number of iterations + 1
if not implicit constraint 0 or not implicit constraint 1 or ...
or not implicit constraint M continue
calculate the cost function
while termination criteria not satisfied
```

Number of iterations is checked in every iteration. This way the user has some kind of control over the optimisation algorithm and an infinitive optimisation loop (in case of no convergence) is avoided. The optimisation method, which was chosen, is hidden in the way of changing parameter values. If one parameter violate its explicit constraint, its value is corrected to the explicit border. More analyses lines can be defined with optimize analysis commands. They are performed in order and produce necessary results for verifying implicit constraints and evaluating the cost function. If at least one implicit constraint is violated, than algorithm continues with next iteration. Otherwise the cost function is calculated and terminating criteria decides whether the loop will continue or stop.

The optimize command is placed on top of Nutmeg by its function. In fact it just prepares commands in Nutmeg language and executes them in an appropriate order, what leads to optimising process. But on the other hand optimize command is written as an additional Nutmeg command and can be used as one.

The optimize command also generates special group of data (plot). It is named optimizex, where x repesents a plot number. It contains a vector called parameter with final parameter values, a vector called cost with values of the cost function in each iteration, and vectors with parameter values in each iteration.

When the optimize command finishes and finds some optimal parameter values, which are better than initial values, the initial parameter values are changed to optimal ones. That way we can proceed with the next optimize command (maybe with some other method) on the same circuit from the best known point as initial guess.

If the initial_guess option is set (equal or greater than 1), the above is not true. In this case the initial parameter values are set to the values where parameter space was not searched yet. This way many dimensional parameter spaces can be searched by successive optimisation processes. The value of initial_guess option gives the weight factor of the point with the greatest value of the cost function. The weight factor of the point with the smallest value is always 1.

Optimization methods

One can choose among several optimisation methods which are coded in the optimize command:

• steepest descent
• Newton's method
• Davidon-Fletcher-Powell's method
• random search
• grid search method
• search along coordinate axes
• Powell's method
• Hooke-Jeeves's method
• constrained simplex method
• simple genetic algorithm
• simulated annealing algorithm
• parallel simulated annealing with differential evolution
• checking cost function in axes directions through current point
• evaluating cost function across whole parameter space

Further description of a particular method can be found in the literature. Here are brief descriptions of optimisation methods and their parameters which can be manually adjusted.

Steepest Descent (steepest_descent)

If the expression for the i-th component of the gradient of the cost function is not given, then it is calculated numerically. All parameters, that are to be optimised, have to be given before this method is chosen. The parameters of the method are:

Newton's Method (newton)

Gradient and Hesse matrix of the cost function are calculated numerically. The parameters of the method are:

 r constant used in determining constraints where the minimum lies in the desired direction. It must be between 1.5 and 2. method method used for searching in the desired direction. Possible values are quadratic, golden and fibonacci. epsilon satisfying relative distance (in percentage) between two points when searching in the desired direction. number_of_iterations maximum number of iterations when searching in the desired direction. transformation type of transformation from explicitly constrained to unconstrained space. Possible values are no, sin and arcctg.

Davidon-Fletcher-Powell's method (davidon_fletcher_powell)

If the expression for the i-th component of the gradient of the cost function is not given, then it is calculated numerically. All parameters, that are to be optimised, have to be given before this method is chosen. The parameters of the method are:

 r constant used in determining constraints where the minimum lies in the desired direction. It must be between 1.5 and 2. method method used for searching in the desired direction. Possible values are quadratic, golden and fibonacci. epsilon satisfying relative distance (in percentage) between two points when searching in the desired direction. number_of_iterations maximum number of iterations when searching in the desired direction. modification modification of the Davidon-Fletcher-Powell's method. Possible values are no, modified, first and second. transformation type of transformation from explicitly constrained to unconstrained space. Possible values are no, sin and arcctg. gradient0 expression of the first component of the gradient gradient1 expression of the second component of the gradient gradient2 expression of the third component of the gradient etc.

Random Search (monte_carlo)

Parameter distribution is regarded only if direction = no. On the other hand parameters r, epsilon and number_of_iterations are taken into account when direction != no. The parameters of the method are:

 direction the direction of search is determined randomly. Possible values are no, quadratic, golden and fibonacci. distribution the distribution of the random parameter values. Possible values are linear and normal. r constant used in determining constraints where the minimum lies in the randomly determined direction. It must be between 1.5 and 2. epsilon satisfying relative distance (in percentage) between two points when searching in the randomly determined direction. number_of_iterations maximum number of iterations when searching in the randomly determined direction.

Grid Search Method (grid_search)

The parameters of the method are:

 level grid level used. Possible values are 2 and 3. alpha the magnifying factor. epsilon satisfying relative grid size (in percentage).

Search Along Coordinate Axes (axis_search)

The parameters of the method are:

 r constant used in determining constraints where the minimum lies in the desired direction. It must be between 1.5 and 2. method method used for searching in the desired direction. Possible values are quadratic, golden and fibonacci. epsilon satisfying relative distance (in percentage) between two points when searching in the desired direction. number_of_iterations maximum number of iterations when searching in the desired direction.

Powell's Method (powell)

The parameters of the method are:

 r constant used in determining constraints where the minimum lies in the desired direction. It must be between 1.5 and 2. method method used for searching in the desired direction. Possible values are quadratic, golden and fibonacci. epsilon satisfying relative distance (in percentage) between two points when searching in the desired direction. number_of_iterations maximum number of iterations when searching in the desired direction. transformation type of transformation from explicitly constrained to unconstrained space. Possible values are no, sin and arcctg.

Hooke-Jeeves's Method (hooke_jeeves)

The parameters of the method are:

 alpha the magnifying factor. epsilon satisfying relative step (in percentage).

Constrained Simplex Method (complex)

The parameters of the method are:

 k number of points in constrained simplex. The default value is 2 times number of parameters. It should be greater than number of parameters. alpha the mirroring factor. size satisfying average relative distance (in percentage) of the points to the centre of the simplex. oscillation_detection detect and prevent an oscillation behavior around the lowest point of the constrained simplex. nmirror number of points mirrored in each iteration. width portion of explicitly constrained parameter space, where initial simplex will be chosen around the initial point.

Simple Genetic Algorithm (genetic)

The parameters of the method are:

 popsize population size. Must be an even number. lchrom chromosome length (number of bytes per dimension). Maximum length is 4 bytes per dimension. maxgen number of generations. pcross crossover probability. pmutation mutation probability. scaling cost function scaling factor. Must be greater than 1. coding parameter values coding. Possible values are binary and gray. elitism the best individual in each generation is copied to the next one. Possible values are yes and no. The default is no.

Simulated Annealing Algorithm (simulated_annealing)

The parameters of the method are:

 init_num_of_pts number of points used for determining initial temperature. tmin absolute final temperature value. range_min maximum reduction of parameter range. Moves are constrained by parameter range around current point. Parameter range is reduced during the optimisation run. range_min defines maximum reduction of parameter range, which always has to be greater than range_min * (upper_limit - lower_limit). moves number of moves at initial temperature stage. alpha cooling factor.

Parallel Simulated Annealing with Differential Evolution Algorithm (psade)

Several simulated annealing processes in parallel interacting through differential evolution. The parameters of the method are:

 np number of points in population. tmin minimum absolute temperature value. rmin minimum relative range for generating random moves. sfreq save frequency (save entire population after every sfreq iterations). minCF stopping cost function value. pL local step probability.

Checking cost function in axes directions through current point (cost_profile)

The parameter of the method is:

 resolution number of points in each axis direction.

Evaluating cost function across whole parameter space (parameter_space)

The parameters of the method are:

 outfile output filename, where the results are stored. npts0 number of points in the grid along the first parameter. npts1 number of points in the grid along the second parameter. npts2 number of points in the grid along the third parameter etc.

The command syntax

optimize [command]

 command = analysis [number delete | number expression] | cost [expression] | implicit [number delete | number expression] | method [definition] | parameter [number delete | number [name] [low value] [high value] [initial value] [increment value] [mean value] [deviation value] [abstol value] [reltol value] [lin | log]] | options [initial_guess value] [number_of_iterations value] [stop_cost value] [centering off | on] [discrete_space off | on] [setparams] | reset [analysis] [cost] [implicit] [method] [parameter] [options] [population]