optimbase

Purpose

The goal of this component is to provide a building block for optimization methods. The goal is to provide a building block for a large class of specialized optimization methods. This component manages

• the number of variables,

• the minimum and maximum bounds,

• the number of non linear inequality constraints,

• the cost function,

• the logging system,

• various termination criteria,

• etc...

This toolbox is designed with Oriented Object ideas in mind.

Features

The following is a list of features the Optimbase toolbox currently provides :

• Manage cost function

  • optionnal additionnal argument

  • direct communication of the task to perform : cost function or inequality constraints

• Manage various termination criteria, including

  • maximum number of iterations,

  • tolerance on function value (relative or absolute),

  • tolerance on x (relative or absolute),

  • maximum number of evaluations of cost function,

• Manage the history of the convergence, including

  • history of function values,

  • history of optimum point.

• Provide query features for

  • the status of the optimization process,

  • the number of iterations,

  • the number of function evaluations,

  • function value at initial point,

  • function value at optimal point,

  • the optimum parameters,

  • etc...

Description

This set of commands allows to manage an abstract optimization method. The goal is to provide a building block for a large class of specialized optimization methods. This component manages the number of variables, the minimum and maximum bounds, the number of non linear inequality constraints, the logging system, various termination criteria, the cost function, etc...

The optimization problem to solve is the following

min f(x)
l_i <= x_i <= h_i, i = 1,n
g_i(x) >= 0, i = 1,nbineq

where

n

number of variables

nbineq

number of inequality constraints

Functions

The following functions are available.

newobj = optimbase_new ()

Creates a new optimization object.

newobj

The new object.

this = optimbase_destroy (this)

Destroy the given object.

this

The current object.

this = optimbase_configure (this,key,value)

Configure the current object with the given value for the given key.

this

The current object.

key

the key to configure. The following keys are available.

-verbose

a 1-by-1 matrix of doubles, positive, integer value, set to 1 to enable verbose logging (default verbose = 0).

-verbosetermination

a 1-by-1 matrix of doubles, positive, integer value, set to 1 to enable verbose termination logging (default verbosetermination = 0).

-x0

a n-by-1 matrix of doubles, where n is the number of variables, the initial guess.

There is no default value, i.e. the user must provide x0.

-maxfunevals

a 1-by-1 matrix of doubles, positive, integer value, the maximum number of function evaluations (default maxfunevals = 100). If this criteria is triggered, the status of the optimization is set to "maxfuneval".

-maxiter

a 1-by-1 matrix of doubles, positive, integer value, the maximum number of iterations (default maxiter = 100). If this criteria is triggered, the status of the optimization is set to "maxiter".

-tolfunabsolute

a 1-by-1 matrix of doubles, positive, the absolute tolerance for the function value (default tolfunabsolute = 0).

-tolfunrelative

a 1-by-1 matrix of doubles, positive, the relative tolerance for the function value (default tolfunrelative = %eps).

-tolfunmethod

a 1-by-1 matrix of booleans, set to %t to enable termination with tolerance on function value (default tolfunmethod = %f).

If this criteria is triggered, the status of the optimization is set to "tolf".

-tolxabsolute

a 1-by-1 matrix of doubles, positive, the absolute tolerance on x (default tolxabsolute = 0).

-tolxrelative

a 1-by-1 matrix of doubles, positive, the relative tolerance on x (default tolxrelative = sqrt(%eps)).

-tolxmethod

a 1-by-1 matrix of booleans, set to %t to enable the tolerance on x in the termination criteria (default tolxmethod = %t).

If this criteria is triggered, the status of the optimization is set to "tolx".

-function

a function or a list, the objective function. This function computes the value of the cost and the non linear constraints, if any.

There is no default value, i.e. the user must provide f.

See below for the details of the communication between the optimization system and the cost function.

-outputcommand

a function or a list, a function which is called back for output.

See below for the details of the communication between the optimization system and the output command function.

-numberofvariables

a 1-by-1 matrix of doubles, positive, integer value, the number of variables to optimize (default numberofvariables = 0).

-storehistory

a 1-by-1 matrix of booleans, set to %t to enable the history storing (default storehistory = %f).

-boundsmin

a n-by-1 matrix of doubles, the minimum bounds for the parameters where n is the number of variables (default boundsmin = [], i.e. there are no minimum bounds).

-boundsmax

a n-by-1 matrix of doubles, the maximum bounds for the parameters where n is the number of variables (default boundsmax = [], i.e. there are no maximum bounds).

-nbineqconst

a 1-by-1 matrix of doubles, positive, integer value, the number of inequality constraints (default nbineqconst = 0).

-logfile

the name of the log file

-withderivatives

a 1-by-1 matrix of booleans, set to %t if the algorithm uses derivatives (default withderivatives = %f).

value

the value.

value = optimbase_cget (this,key)

Get the value for the given key. If the key is unknown, generates an error.

this

The current object.

key

the name of the key to quiery. The list of available keys is the same as for the optimbase_configure function.

this = optimbase_checkbounds ( this )

Check if the bounds are consistent and produces an error message if not.

this

The current object.

opt = optimbase_checkx0 ( this )

Check if the initial guess is consistent with the bounds and the non linear inequality constraints, if any and produces an error message if not.

this

The current object.

optimbase_function

Call the cost function and return the required results.

If a cost function additionnal argument is defined in current object, pass it to the function as the last argument.

The following calling sequences are available (see below for more details).

[ this , f , index ] = optimbase_function ( this , x , index )
[ this , f , c , index ] = optimbase_function ( this , x , index )
[ this , f , g , index ] = optimbase_function ( this , x , index )
[ this , f , g , c , gc , index ] = optimbase_function ( this , x , index )

this

The current object.

x

the current point, as a column vector

index

what value to compute.

See below in the section "Cost function" for details on this index.

f

the value of the cost function

g

the gradient of the cost function

c

the nonlinear, positive, inequality constraints

gc

the gradient of the nonlinear, positive, inequality constraints

this = optimbase_set ( this , key , value )

Set the value for the given key. If the key is unknown, generates an error.

this

The current object.

key

the key to set

The following keys are available :

-funevals

the number of function evaluations

-iterations

the number of iterations

-xopt

the x optimum

-fopt

the optimum cost function value

-historyxopt

an array, with nbiter values, containing the history of x during the iterations.

This array is available after optimization if the history storing was enabled with the -storehistory option.

-historyfopt

an array, with nbiter values, containing the history of the function value during the iterations.

This array is available after optimization if the history storing was enabled with the -storehistory option.

-fx0

the function value for the initial guess

-status

a string containing the status of the optimization

value

the value to set

value = optimbase_get (this,key)

Get the value for the given key. If the key is unknown, generates an error. This command corresponds with options which are not available directly to the optimbase_configure function, but are computed internally.

this

The current object.

key

the name of the key to quiery.

The list of available keys is the same as the optimbase_set function.

[ this , hasbounds ] = optimbase_hasbounds ( this )

Returns %T if current problem has bounds.

this

The current object.

[ this , hascons ] = optimbase_hasconstraints ( this )

Returns %T if current problem has bounds constraints, linear constraints or non linear constraints.

this

The current object.

[ this , hasnlcons ] = optimbase_hasnlcons ( this )

Returns %T if current problem has non linear constraints.

this

The current object.

this = optimbase_histset ( this , iter , key , value )

Set the history value at given iteration for the given key. If the key is unknown, generates an error.

this

The current object.

iter

the iteration number to get

key

the name of the key to quiery.

The list of available keys is the following : "-xopt", "-fopt".

value

the value to set

value = optimbase_histget ( this , iter , key )

Returns the history value at the given iteration number for the given key. If the key is unknown, generates an error.

this

The current object.

iter

the iteration number to get

key

the name of the key to quiery.

The list of available keys is the same as the optimbase_histset function.

this = optimbase_incriter ( this )

Increments the number of iterations.

this

The current object.

[ this , isfeasible ] = optimbase_isfeasible ( this , x )

Returns 1 if the given point satisfies bounds constraints and inequality constraints. Returns 0 if the given point is not in the bounds. Returns -1 if the given point does not satisfies inequality constraints.

this

The current object.

x

the current point

this = optimbase_log (this,msg)

If verbose logging is enabled, prints the given message in the console. If verbose logging is disabled, does nothing. If the -lofgile option has been set, writes the message into the file instead of writing to the console. If the console is too slow, writing into a file can be a solution, since it is very fast.

this

The current object.

msg

the message to print

stop = optimbase_outputcmd ( this , state , data )

Calls back user's output command. See below for details, in the "The output function" section.

this

The current object.

state

a 1-by-1 matrix of strings, the current state of the algorithm

data

a data structure with type T_OPTDATA. This is typically the output of the optimbase_outstruct function, with potentially additionnal fields.

data = optimbase_outstruct ( this )

Returns a data structure with type T_OPTDATA. This data structure contains basic optimization fields. The output argument data is designed to be the input of the optimbase_outputcmd function which, in turn, calls back the output function. This data structure may be enriched by children (specialized) optimization methods.

this

The current object.

[ this , p ] = optimbase_proj2bnds ( this , x )

Returns a point, which is the projection of the given point into the bounds.

this

The current object.

x

the current point

this = optimbase_stoplog ( this , msg )

Prints the given stopping rule message if verbose termination is enabled. If verbose termination is disabled, does nothing.

this

The current object.

msg

the message to print

[this , terminate , status] = optimbase_terminate (this , previousfopt , currentfopt , previousxopt , currentxopt )

Returns 1 if the algorithm terminates. Returns 0 if the algorithm must continue. If the -verbosetermination option is enabled, messages are printed detailing the termination intermediate steps. The optimbase_terminate function takes into account the number of iterations, the number of evaluations of the cost function, the tolerance on x and the tolerance on f. See below in the section "Termination" for more details.

this

The current object.

previousfopt

the previous value of the cost function

currentfopt

the current value of the cost function

previousxopt

the previous x optimum

currentxopt

the current x optimum

terminate

%t if the algorithm must terminate, %f if the algorithm must continue

status

if terminate = %t, the detailed status of the termination, as a string. If terminate = %f, the status is "continue".

The following status are available :

"maxiter"

the maximum number of iterations, provided by the -maxiter option, is reached.

"maxfuneval"

the maximum number of function evaluations, provided by the -maxfunevals option, is reached

"tolf"

the tolerance on the function value is reached. This status is associated with the -tolfunmethod, -tolfunabsolute and -tolfunrelative options.

"tolx"

the tolerance on x is reached. This status is associated with the -tolxmethod, -tolxabsolute and -tolxrelative options.

this = optimbase_checkcostfun ( this )

Check that the cost function is correctly connected. Generate an error if there is one. Takes into account for the cost function at the initial guess x0 only. Checks that all values of the index argument are valid. If there are nonlinear constraints, check that the matrix has the correct shape.

This function requires at least one call to the cost function to make the necessary checks.

this

The current object.

[ this , isfeasible ] = optimbase_isinbounds ( this , x )

Returns isfeasible = %t if the given point satisfies bounds constraints. Returns isfeasible = %f if the given point is not in the bounds.

this

The current object.

isfeasible

a boolean

[ this , isfeasible ] = optimbase_isinnonlinconst ( this , x )

Returns isfeasible = %t if the given point satisfies the nonlinear constraints. Returns isfeasible = %f if the given point does not satisfy the nonlinear constraints.

this

The current object.

isfeasible

a boolean

The cost function

The -function option allows to configure the cost function. The cost function is used, depending on the context, to compute the cost, the nonlinear inequality positive constraints, the gradient of the function and the gradient of the nonlinear inequality constraints.

The cost function can also be used to produce outputs and to terminate an optimization algorithm.

In the following, the variables are

• f: scalar, the objective,

• g: row matrix, the gradient of the objective,

• c: row matrix, the constraints,

• gc: matrix, the gradient of the constraints.

Each calling sequence of the optimbase_function function corresponds to a specific calling sequence of the user-provided cost function.

• If the -withderivatives is false and there is no nonlinear constraint, the calling sequence is

  [ this , f , index ] = optimbase_function ( this , x , index )

  which corresponds to the cost functions

  [ f , index ] = costf ( x , index )

• If the -withderivatives is false and there are nonlinear constraints, the calling sequence is

  [ this , f , c , index ] = optimbase_function ( this , x , index )

  which corresponds to the cost functions

  [ f , c , index ] = costf ( x , index )

• If the -withderivatives is true and there is no nonlinear constraint, the calling sequence is

  [ this , f , g , index ] = optimbase_function ( this , x , index )

  which corresponds to the cost functions

  [ f , g , index ] = costf ( x , index )

• If the -withderivatives is true and there is are nonlinear constraints, the calling sequence is

  [ this , f , g , c , gc , index ] = optimbase_function ( this , x , index )

  which corresponds to the cost functions

  [ f , g , c , gc , index ] = costf ( x , index )

Each calling sequence corresponds to a particular class of algorithms, including for example

• unconstrained, derivative-free algorithms,

• nonlinearily constrained, derivative-free algorithms,

• unconstrained, derivative-based algorithms,

• nonlinearilyconstrained, derivative-based algorithms,

• etc...

The current component is designed in order to be able to handle many situations.

The index input parameter has the following meaning.

• index = 1: nothing is to be computed, the user may display messages, for example

• index = 2: compute f

• index = 3: compute g

• index = 4: compute f and g

• index = 5: compute c

• index = 6: compute f and c

• index = 7: compute f, g, c and gc

The index output parameter has the following meaning.

• index > 0: everything is fine,

• index = 0: the optimization must stop,

• index < 0: one function could not be avaluated.

It might happen that the function requires additionnal arguments to be evaluated. In this case, we can use the following feature. The argument fun can also be the list (f,a1,a2,...). In this case f, the first element in the list, must be a function and must have the header:

[ f , index ] = f ( x , index , a1 , a2 , ... )
[ f , c , index ] = f ( x , index , a1 , a2 , ... )
[ f , g , index ] = f ( x , index , a1 , a2 , ... )
[ f , g , c , gc , index ] = f ( x , index , a1 , a2 , ... )

The output function

The option -outputcommand allows to configure a command which is called back at the start of the optimization, at each iteration and at the end of the optimization.

The output function must have the following header

stop = outputcmd(state, data)

where

state

a 1-by-1 matrix of strings, the current state of the algorithm. Available values are "init", "iter", "done".

data

a data structure with type T_OPTDATA containing at least the following fields

x

the current optimum

fval

the current function value

iteration

the current iteration index

funccount

the number of function evaluations

stop

a 1-by-1 matrix of booleans, stop is true if the optimization algorithm must be stopped, stop is false if the optimization algorithm must continue.

It might happen that the output function requires additionnal arguments to be evaluated. In this case, we can use the following feature. The argument outputcmd can also be the list (outf,a1,a2,...). In this case outf, the first element in the list, must be a function and must have the header:

stop = outf ( state, data, a1, a2, ... )

Termination

The current component takes into account for several generic termination criterias. Specialized termination criterias should be implemented in specialized optimization algorithms, by calling the optimbase_termination function and adding external criterias, rather than by modification of this function.

The optimbase_terminate function uses a set of rules to compute if the termination occurs, which leads to an optimization status which is equal to one of the following : "continue", "maxiter", "maxfunevals", "tolf", "tolx". The set of rules is the following.

• By default, the status is "continue" and the terminate flag is 0.

• The number of iterations is examined and compared to the -maxiter option : if the following condition

  iterations >= maxiter

  is true, then the status is set to "maxiter" and terminate is set to %t.

• The number of function evaluations and compared to the -maxfunevals option is examined : if the following condition

  funevals >= maxfunevals

  is true, then the status is set to "maxfuneval" and terminate is set to %t.

• The tolerance on function value is examined depending on the value of the -tolfunmethod.

  "disabled"

  then the tolerance on f is just skipped.

  "enabled"

  if the following condition

  abs(currentfopt) < tolfunrelative * abs(previousfopt) + tolfunabsolute

  is true, then the status is set to "tolf" and terminate is set to %t.

  The relative termination criteria on the function value works well if the function value at optimum is near zero. In that case, the function value at initial guess fx0 may be used as previousfopt.

  The absolute termination criteria on the function value works if the user has an accurate idea of the optimum function value.

• The tolerance on x is examined depending on the value of the -tolxmethod.

  %f

  then the tolerance on x is just skipped.

  %t

  if the following condition

  norm(currentxopt - previousxopt) < tolxrelative * norm(currentxopt) + tolxabsolute

  is true, then the status is set to "tolx" and terminate is set to %t.

  The relative termination criteria on x works well if x at optimum is different from zero. In that case, the condition measures the distance between two iterates.

  The absolute termination criteria on x works if the user has an accurate idea of the scale of the optimum x. If the optimum x is near 0, the relative tolerance will not work and the absolute tolerance is more appropriate.

Example : Setting up an optimization

In the following example, one searches the minimum of the 2D Rosenbrock function. One begins by defining the function "rosenbrock" which computes the Rosenbrock function. The traditionnal initial guess [-1.2 1.0] is used. The initial simplex is computed along the axes with a length equal to 0.1. The Nelder-Mead algorithm with variable simplex size is used. The verbose mode is enabled so that messages are generated during the algorithm. After the optimization is performed, the optimum is retrieved with quiery features.

function [f, index]=rosenbrock(x, index)
f = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
endfunction

opt = optimbase_new();
opt = optimbase_configure(opt,"-numberofvariables",2);
nbvar = optimbase_cget(opt,"-numberofvariables");
opt = optimbase_configure(opt,"-function",rosenbrock);
[ opt , f , index ] = optimbase_function ( opt , [0.0 0.0] , 1 );
expectedf = 1
disp(f)
opt = optimbase_destroy(opt);

Example : Passing extra parameters

In the following example, we consider a function which has two additionnal parameters a and b. In this case, we can configure the "-function" option as a list, where the first element is the function and the two extra arguments are located at the end of the list.

function [f, index]=rosenbrock2(x, index, a, b)
f = a*(x(2)-x(1)^2)^2 + (b-x(1))^2;
endfunction

opt = optimbase_new();
opt = optimbase_configure(opt,"-numberofvariables",2);
nbvar = optimbase_cget(opt,"-numberofvariables");
a = 100;
b = 1;
opt = optimbase_configure(opt,"-function",list(rosenbrock2,a,b));
[ opt , f , index ] = optimbase_function ( opt , [0.0 0.0] , 1 );
expectedf = 1
disp(f)
opt = optimbase_destroy(opt);

TODO

• manage equality constraints

• manage linear constraints

• manage quadratic objective

• manage linear objective

• manage linear inequality constraints

• manage non linear equality constraints

• manage linear equality constraints