MINIF

Returns the minimum value in a range of values that match a certain condition.

'rgeCriteria - The range of cells containing the criteria you want to check.
'sCriteria - The criteria value you want to match.
'rgeMinRange - The range of corresponding values you want the minimum of.

Public Function MINIF(ByVal rgeCriteria As Range, _ 
                      ByVal sCriteria As String, _ 
                      ByVal rgeMinRange As Range) As Single 
          
Dim iconditioncolno As Integer 
Dim inumberscolno As Integer 
Dim lrowno As Long 
Dim sngmin As Single 
Dim vcellvalue As Variant 

   iconditioncolno = rgeCriteria.Column 
   inumberscolno = rgeMinRange.Column 
   For lrowno = 1 To rgeCriteria.Rows.Count 
      vcellvalue = rgeCriteria.Parent.Cells(rgeCriteria.Row + lrowno - 1, inumberscolno).Value 
      If rgeCriteria.Parent.Cells(rgeCriteria.Row + lrowno - 1, iconditioncolno).Value = sCriteria And _ 
         IsNumeric(vcellvalue) = True Then 
         If sngmin = 0 Then sngmin = vcellvalue 
         If vcellvalue < sngmin Then sngmin = vcellvalue 
      End If 
      
      If sngmin <> 0 And IsEmpty(vcellvalue) = True Then Exit For 
   Next lrowno 
    
   For lrowno = 1 To rgeCriteria.Rows.Count 
      vcellvalue = rgeCriteria.Parent.Cells(rgeCriteria.Row + lrowno - 1, inumberscolno).Value 
      If rgeCriteria.Parent.Cells(rgeCriteria.Row + lrowno - 1, iconditioncolno).Value = sCriteria And _ 
         IsNumeric(vcellvalue) = True Then 
         If rgeCriteria.Parent.Cells(rgeCriteria.Row + lrowno - 1, inumberscolno).Value < sngmin Then 
            sngmin = rgeCriteria.Parent.Cells(rgeCriteria.Row + lrowno - 1, inumberscolno).Value 
         End If 
      End If 
      
      If sngmin <> 0 And IsEmpty(vcellvalue) = True Then Exit For 
   Next lrowno 
    
   MINIF = sngmin 
End Function 

MINIFS was added as a built-in function in 2016 Jan.
There is an Array Formula equivalent if you do not want to use the user defined function.
This includes hidden rows.
This user defined function is similar to the built-in SUMIF function.

Array Formula

To find out more about Array Formulas please refer to the Array Formulas section.
To find out more about the individual functions please refer to the MIN function and IF function pages.
This example illustrates how to use an array formula to obtain the minimum value in a column based on a condition.

The formula in row 16 returns the smallest total in the "Sales" department.
The formula in row 17 returns the smallest total in the "IT" department.
The formula [C3:C14="Sales"] returns an array of 1's and 0's indicating if the value in column "C" satisfies the condition.
Every number 1 in this array represents the value True in the IF statement and therefore has its corresponding value returned from column "D".
Every number 0 in this array represents the value False in the IF statement and therefore an empty string is returned in this case. The MIN function will ignore any string values.
The use of the empty string is to ensure that the formula works with both positive and negative numbers.
This array of numbers is then passed to the MIN() function to obtain the largest number.
Remember that you must press (Ctrl + Shift + Enter) to enter the formulas.

Important

You cannot use column references (such as "A:A" or "D:D") in your array formulas.
Any cell ranges passed to an array formula must have the same number of rows. This is to ensure that all the temporary arrays used are the same length.