Summary of monotonic optimization of dynamic programming--min/max

General form:

$max \{min (AX+BY+C,DF (x) +eg (y) +f) \}, where f (x) and g (Y) are monotonic functions. $

Or

$min \{max (AX+BY+C,DF (x) +eg (y) +f) \}, where f (x) and g (Y) are monotonic functions. $

(discussed in the first form)

(1) DF (x) increases with the increase of Ax, EG (y) increases with the by.

Both Ax and by take the maximum value.

(2) DF (x) increases with the increase of Ax and the EG (Y) decreases with the increase of by.

Ax must take the maximum value, and Ax and DF (x) become constants.

This becomes:

$H (y) =max\{min (By+c,eg (y) +f) \}$

H (y) is a single-peak function.

(3) DF (x) decreases with the increase of Ax, EG (y) increases with the by.

Similar to (2).

(4) DF (x) decreases with the increase of Ax, EG (y) decreases with the increase of by.

From small to large enumeration ax, when Ax becomes larger, DF (x) becomes smaller:

$max \{min (AX↑+BY+C,DF (x) ↓+eg (y) +f) \}$

If by also becomes larger, then eg (y) becomes smaller:

$max \{min (AX↑+BY↑+C,DF (x) ↓+eg (y) ↓+f) \}$

As we take the min, there is no egg to use.

So by only getting smaller.

$max \{min (AX↑+BY↓+C,DF (x) ↓+eg (y) ↑+f) \}$

So with our small to large enumeration Ax,ay monotonically decrements.

When you enumerate ax, ax and DF (x) become constants.

This becomes:

$H (y) =max\{min (By+c,eg (y) +f) \}$

H (y) is a single-peak function.

Summary of monotonic optimization of dynamic programming--min/max