Minimzation function error: "Convergence FAILURE: CANNOT DECREASE OBJECTIVE FUNCTION"

Question

I am working on a project in JMP 16, where I use the “Minimize”-function. When running the function I often encounter the error “Convergence FAILURE: CANNOT DECREASE OBJECTIVE”. Looking at the final results of the script, they are as expected. It therefore seems that the error has little influence on the final results. However, to ensure the validity of the script, I would still like to know the reasoning behind the error. I have had difficulties finding any documentation about this specific error, and therefore hoped someone in the JMP community could help identify the cause.

Using the “<< Details( Both )” option for the minimize function, below image of the log was obtained:

Best regards,

Louis

Craige_Hales · Answer

You could make a graph, like this:

expression=expr(Log10( (ChiSquare Quantile( alpha, df ) / df - Vratio) ^ 2 ));

alpha = 0.025;
Vratio =  0.39135;
df=2;
Minimize( expression, { df( 2, 1000 ) }, << Use Numeric Deriv( "true" ) );

show(df);

New Window( "Example",
	Graph Box(xscale(-1,26),yscale(-10,0),
		Pen Color( "red" );
		Y Function( expression, df );
	)
);

there is a sharp drop at the df found by the minimize function.

you can zoom in on the sharp drop and see it is going to be computationally difficult to get a perfect answer:

9 significant digits, and more to go.

minimize found df = 13.3517718809044;

I'd guess that the minimize algorithm hits a point on the curve that finds the next point on the far side of the true minimum, and might even be in a cycle bouncing across the minimum. With a different starting value, it might get close enough to decide to stop. Close enough is also a parameter you can pass to minimize (tolerance).

Jeff_Perkinson · Answer

Looking at the Profiler for the function (see below, and the attached data table), it does look like Minimize() is giving a good answer.

Perhaps @chris_gotwalt1 can give an explanation of the error message you're receiving from the call to Minimize().

Community 358762 Minimize function.jmp

Mark_Bailey · Answer

I used this script based on your answers.

Names Default to Here( 1 );

alpha = 0.025;
variance ratio = 0.94917 ;
df = 2;

expression = Expr( Log10( (Chi Square Quantile( alpha, df ) / df - variance ratio) ^ 2 ) );

Minimize( expression, { df( 2, 1000 ) }, << Use Numeric Deriv( "true" ) );

I get the following using the two variance ratios that you gave to me:

//:*/
Names Default to Here( 1 );

alpha = 0.025;
variance ratio = 0.39135;
df = 2;

expression = Expr( Log10( (Chi Square Quantile( alpha, df ) / df - variance ratio) ^ 2 ) );

Minimize( expression, { df( 2, 1000 ) }, << Use Numeric Deriv( "true" ) );
/*:

Optimization failed: Failed: Cannot Decrease Objective Function

-19.6490758040614
//:*/
Names Default to Here( 1 );

alpha = 0.025;
variance ratio = 0.94917 ;
df = 2;

expression = Expr( Log10( (Chi Square Quantile( alpha, df ) / df - variance ratio) ^ 2 ) );

Minimize( expression, { df( 2, 1000 ) }, << Use Numeric Deriv( "true" ) );
/*:

-2.91402948948765