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Adaptive Response Surface Method (ARSM)

Adaptive Response Surface Method (ARSM)

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Adaptive Response Surface Method (ARSM)

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Adaptive response surface method works by internally building response surfaces and adaptively updating them as new evaluations are available.  The first response surface it builds is a linear regression polynomial, then it finds the optimum on this surface and validates it with the exact simulation. If the output response values from the response surface and the exact simulation are not close; ARSM updates the surface with the new evaluation and finds the optimum in this updated surface. ARSM repeats this loop until it meets one of the convergence criteria.

 

Usability Characteristics

•ARSM is the default method. However, if the number of input variables is large, or if a global optima is required, then it is suggested that you use GRSM instead.
•For Revisions A-multi and B-multi, ARSM can take advantage of parallel execution. The number of runs in the iterative stages after N+1 evaluations (N is the number of variables) can be controlled by the setting parameter "Points per iteration".
•It is an efficient optimization method because it utilizes response surfaces. It is recommend to use ARSM directly on a solver and not on a Fit.
•In the case of a failed run, it is possible to ignore a failed analysis or terminate an optimization. When omitting failed runs, the optimizer will back up half of a step between the failed run and the previous design.
•ARSM terminates when one of the following conditions are met:
-One of the convergence criteria is satisfied.
-The maximum number of allowable analysis (MAXDES) is reached.
-An analysis fails and the "Terminate Optimization" option is the default (IGFAIL).
•Supports input variable constraints.

 

The flowchart below illustrates the different phases of the ARSM process.

arsm_flowchart

 

Settings

In the Specifications step, you can change the settings of ARSM from the following tabs:

hmtoggle_arrow1Settings

In the Settings tab, you can access the settings listed below. Please note that for most applications the default settings work optimally, and you may only need to change the Maximum Iterations and On Failed Analysis.

Setting

Default Value

Range

Description

Maximum Iterations

(MAXDES)

25

> 0

Maximum number of analyses (only for ARSM number of analysis is equal to number of iterations) allowed.

Absolute convergence

(ABSOBJ)

0.001

>0.0

Absolute convergence parameter.  Determines an absolute convergence tolerance, which is constant and equal to ABSOBJ, times the initial objective function value.  The design has converged when there are two consecutive designs for which the absolute change in the objective function is less than this tolerance.  There also must not be any constraint whose allowable violation is exceeded in the last design.

A larger value allows for faster convergence, but worse results could be achieved.

The above description can be formulated as:

arsm2

Where f is the objective value; arsm_f0is the objective value of the initial design; k is the current iteration number; mslm_epsilon is the absolute convergence parameter; arsm_cmax is the maximum constraint violation; arsm_gmazx is the allowable constraint violation.

Relative convergence

(RELOBJ)

1.0

> 0.0

Relative percentage convergence parameter. The design has converged if the relative (percent) change in the objective function is less than this value for two consecutive designs. There also must not be any constraint whose allowable violation is exceeded in the last design.

A larger value allows for faster convergence, but worse results could be achieved.

The above description can be formulated as:

arsm3

Where, f is the objective value; k is the current iteration number; mslm_epsilon is the relative convergence parameter; arsm_cmax is the maximum constraint violation; arsm_gmazx is the allowable constraint violation.

Constraint violation tol.

(GMAX)

0.5

> 0.0

Global maximum allowable percentage constraint violation.  Constraints must not be violated by more than this value in the converged design.

Input variable convergence

(DVCONV)

0.001

> 0.0

Input variable convergence parameter.  Design has converged when there are two consecutive designs for which the change in each input variable is less than both (1) DVCONV times the difference between its bounds, and (2) DVCONV times the absolute value of its initial value (simply DVCONV if its initial value is zero).  There also must not be any constraint whose allowable violation is exceeded in the last design.

A larger value allows for faster convergence, but worse results could be achieved.

The above description can be formulated as:

arsm4

Where, x is input variable; x0 is the initial design; XL, XU are lower bound and upper bound of input variables respectively; k is the current iteration number; n is the number of input variables; y2 is the input variable convergence parameter.

On failed analysis

(IGFAIL)

Termination

Termination or Ignore Analysis

0

ARSM terminates with an error message when an analysis run fails (default).

1

ARSM ignores the failed analysis run, reduces the preceding step size by 50%, and attempts the analysis again.

 
hmtoggle_arrow1More

In the More tab, you can access the setting listed below. Please note that for most applications, the default settings work optimally.

Setting

Default

Range

Description

Initial linear move

(MPERT)

By DV Initial

By DV Initial or By DV Bounds

By DV initial.

Initial move = PERT * MOVE * abs(INI)

(Default when initial value of input variable is non-zero)

An exception is that initial move will be set to minimum move if it is less than minimum move.

Minimum move = MINMOVE * (UB-LB) if (UB-LB) is less than 1.

Minimum move = MINMOVE if (UB-LB) is not less than 1 and absolute value of INI is less than 1.

Minimum move = MINMOVE * min((UB-LB),abs(INI)) if (UB-LB) is not less than 1 and absolute value of INI is not less than 1.

By DV bounds.

Initial move = PERT * MOVE * (UB-LB)

(Default when initial value of input variable is zero)

INI  Initial input variable value

LB, UB  Lower and upper bounds on input variable

Move limit fraction

(MOVE)

0.15

0.0 < MOVE < 1.0

Move limit fraction.

Smaller value allow more steadily convergence (smaller fluctuation of the output response values), but more computational effort could be consumed.  The value will be adaptively updated during optimization process.

Initial DV perturbation

(PERT)

1.1

≠ 0.0

Initial input variable perturbation value.

Larger value results in wider spread of the initial N designs (n is the number of input variables; the ndesigns together with the start design can determine a linear response surface).  ARSM will search the design space more widely.

Constraint screening (%)

(CONRET)

50.0

real value

Constraint Screening.

> 0.0

Constraint is retained (not screened out) if it is violated or within the given percentage of its critical value (bound).

< 0.0

As many constraints are retained as memory permits.

Minimal Move Factor

(MINMOVE)

0.1

0.0 < MINMOVE < MOVE

Minimal move factor. It is to avoid too small of the step size. It is used in the initial sampling step (See the usage of MINMOVE in MPERT) and also in the preceding move limit strategy.

Response surface

(TPARSM)

SORS

SORS,

SRSM

SORS= Second order response surface (SORS) is used.

SRSM= Scalable response surface method (SRSM) is used.

SRSM is used, the limit on MAXDES >= N+2 should be deleted, where N is number of input variables.

When there are a lot of input variables and the computational effort is limited, SRSM is a good choice.

Solver

(OPTARSM)

SQP

MFD,

SQP,

Hybrid

Method ARSM uses to solve the response surface based optimization problem.

It is recommended to use 2 when there are a lot of discrete variables.

Points per Iteration

(PARALLE)

1

> 0

Controls the number of points used in an iteration after the first iteration. The number of points used per iteration can result in different iteration histories.

Sample points

(NPTSRS)

0

>=0

0

Automatically determined; in SRSM, NPTSRS is set to n. n is the number of input variables.

>0

Use the user defined value.

NPTSRS is useful only if TPARSM = 1.

Use SVD

(SVDRSM)

False

False or

True

This parameter can be useful in case of soft convergence. In case of soft convergence:

•if Use SVD is false,  ARSM is terminated.
•if Use SVD is true, Singular Value Decomposition is activated to re-build the response surfaces and optimization process will be continued.

Revision

(VERARSM)

A-multi

A-multi and B-multi

This parameter is used to help when there is a convergence difficulty. By default, "A-multi" is selected meaning the legacy algorithm.

Note:A-multi and B-multi are new versions of A and B that support multi-execution. The classification of iteration points is different between A and A-multi (and B and B-multi).

Constraint threshold

(EPSCON)

1.0e-4

> 0.0

This parameter is used for constraint value calculation.  In general, constraint value is normalized to its bound value.  One exception is that constraint value is not normalized if its absolute bound value is less than this parameter.  Recommended range is 1.0e-6 ~ 1.0.

Use Inclusion Matrix

(INCLUSI)

No

No, With Initial, Without Initial
•No ignores the Inclusion matrix
•With Initial runs the initial point. The best point of the inclusion or the initial point is used as the starting point.
•Without Initial does not run the initial point. The best point of the inclusion is used as the starting point.