This seems too easy, (which usually means I didn't understand the question but...):

#! perl -slw
use strict;

sub minpos {
    my( $v, $s, $e ) = @_;
    my( $n, $min, $o ) = ( 0, 1e30, 0 );
    for my $p ( $s .. $e ) {
        $n += vec( $v, $p, 1 ) ? 1 : -1;
        ( $min, $o ) = ( $n, $p ) if $n < $min;
    }
    return $o;
}

my $vec = pack 'b*', '100101000100';
my( $s, $e ) = ( 1 , 7 );

printf "The minima between %d - %d is at %d\n",
    $s, $e, minpos( $vec, $s, $e );

__END__

C:\test>junk5
The minima between 1 - 7 is at 7
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