# Finding the maximum of a piecewise function

I have the following piecewise function of the variable $$e_f$$:

$$g(a,b,c,w,F,e_h,e_f)=\begin{cases} \frac{(c-a e_f) (e_f (4e_f w-a)+c)}{8 b e_f^2} & \left(e_f=e_h\land e_f>\frac{c}{a}\right)\lor e_f\geq \frac{c}{a-2 \sqrt{b} \sqrt{F}} \\ 0 & \text{otherwise} \end{cases}$$

where all the parameters $$a$$, $$b$$, $$c$$, $$w$$, $$F$$, $$e_h$$ and $$e_f$$ are strictly positive ($$\gt 0$$).

g[a_, b_, c_, w_, F_, eh_, ef_] := Piecewise[{{
((c - a ef) (c + ef (-a + 4 ef w)))/(8 b ef^2),
(ef == eh && ef > c/a) || ef >= c/(a - 2 Sqrt[b] Sqrt[F])
}}, 0]


For given numerical values of $$a$$, $$b$$, $$c$$, $$w$$, $$F$$, and for a given $$e_h$$, I would like to find the value of $$e_f$$ that maximises $$g$$. I tried to use the FindMaximum function, but this seems to miss the point where $$e_f=e_h$$ where the function may be defined and maximised. For example:

FindMaximum[g[10, 1, 1, 5, 10, 0.24, ef], {ef, 0.2}] returns {0., {ef -> 0.2}} and FindMaximum[g[10, 1, 1, 5, 10, 0.24, ef], {ef, 0.3}] returns {0.698102, {ef -> 0.272076}} which is the maximum on the continuous part for $$e_f\geq \frac{c}{a-2 \sqrt{b} \sqrt{F}}$$. So in both cases, the point $$e_f=0.24$$ where the global maximum $$g(10, 1, 1, 5, 10, 0.24, 0.24)=0.753472$$ is missed.

Ultimately, I would like to plot the argmax of $$g(e_f)$$ as a function of $$e_h$$ for given values of the other parameters. What's the best way to do this?

• In this case, you could add the piecewise conditions as constraints on the optimisation Commented Jul 1, 2020 at 6:00
• Maximize[g[10, 1, 1, 5, 10, 0.24, ef], ef ] (*{0.698102, {ef -> 0.272076}}*) gets a little bit closer to the expected maximum. Commented Jul 1, 2020 at 6:13
• Maximize[g[10, 1, 1, 5, 10, 24/100, ef] performs {217/288, {ef -> 6/25}}. Commented Jul 1, 2020 at 8:10
• @Bill: yes, it was a typo. Sorry about that and thanks for spotting it! I corrected it now. Commented Jul 1, 2020 at 13:37

FullSimplify[g[10, 1, 1, 5, 10, 0.24, ef] // N]


I believe these general min/maximize functions use search strategies that start with some initial points, and Mathematica doesn't expect that the max point is located at the isolated point $$e_f=0.24$$. Therefore, you may need to treat this specially.

# Method 1

If[g[10, 1, 1, 5, 10, 0.24, 0.24] > #1,
0.24, #2[[1, 2]]
] & @@ NMaximize[
FullSimplify[g[10, 1, 1, 5, 10, 0.24, ef]], ef]

0.24

Plot[
If[g[10, 1, 1, 5, 10, eh, eh] > #1,
eh, #2[[1, 2]]
] & @@ NMaximize[
FullSimplify[g[10, 1, 1, 5, 10, eh, ef]], ef],
{eh, 0, 0.5}, PlotRange -> {0, Automatic}]


# Method 2

Put the special data together with the result of NMaximize in same format, and then take the largest data according to the first element (value). This is more general.

MaximalBy[
{
{g[10, 1, 1, 5, 10, 0.24, 0.24], {ef -> 0.24}},
NMaximize[FullSimplify@g[10, 1, 1, 5, 10, 0.24, ef], ef]
},
First
][[1, 2, 1, 2]]

0.24

Plot[
MaximalBy[
{
{g[10, 1, 1, 5, 10, eh, eh], {ef -> eh}},
NMaximize[
FullSimplify@g[10, 1, 1, 5, 10, eh, ef], ef]
}, First
][[1, 2, 1, 2]],
{eh, 0, 0.5},
PlotRange -> {0, Automatic},
AxesLabel -> {"\!$$\*SubscriptBox[\(e$$, $$h$$]\)",
\!$$\*UnderscriptBox[\("\<arg max\>"$$,
SubscriptBox[$$e$$, $$f$$]]\) g[Subscript[e, h], Subscript[e, f]]}
]


• Thank you! I see the difficulty of finding the isolated point. As I am not very advanced with Mathematica, just to understand what your code does: the #2 and #1 refer to the output of the NMaximize, right? #2[[1, 2]] is the value of the maximum? And #1 is the argmax? So for any eh in (0,0.6), if g[10, 1, 1, 5, 10, eh, eh] is greater than the maximum found through the NMaximize, the plot draws eh, otherwise it draws the argmax found through the NMaximize, is that correct? Commented Jul 1, 2020 at 16:11
• @Lednacek: Oops! The #1 and #2[[1,2]] were misplaced in my code. The plot is incorrect. I'll modify it later. As for what you say, they're right. Commented Jul 1, 2020 at 16:32
• @Lednacek: I've updated my answer. Commented Jul 1, 2020 at 17:17
• Thank you SO much! This is great! I have two more questions regarding your answer: (1) In the plot, where the curve drops vertically, I believe this should be a discontinuity (a jump). Why does it appear as a continuous (vertical) line? (2) Is there any way I can store the relationship plotted (the argmax of g(ef) as a function of eh) and use it for some later calculations? In particular, I need to calculate the intersection of the curve plotted with another function that links ef and eh. Many thanks again! Commented Jul 1, 2020 at 21:31
• @Lednacek: (1) I think it's because we're using functions that are non-differentiable to Mathematica like If and NMaximize, so the discontinuities are not detected. I can't think of a good solution but you can take a look at a related question: 10501. (2) Related question: 199037. Commented Jul 2, 2020 at 4:49

First consider the valid region of the parameters eh,ef

cond[a_?NumericQ, b_?NumericQ, c_?NumericQ, w_?NumericQ,F_?NumericQ ] := (ef == eh && ef > c/a) ||ef >= c/(a - 2 Sqrt[b] Sqrt[F])
RegionPlot[ cond[10, 1, 1, 5, 10] , {ef, .2, .3} , {eh, 0.23, .28},PlotPoints -> {100, {eh == ef}}, FrameLabel -> Automatic,Prolog -> {Red, Point[{.24, .24}]}]


The plot shows that the point ef==eh==.24 you expect the maximum isn't allowed!

NMaximize evaluates the maximum

Maximize[g[10, 1, 1, 5, 10, 0.24, ef], ef ]  (*{0.698102, {ef -> 0.272076}}*)


Obviously Mathematica didn't find the complete valid region. But Maximize is able to solve the problem if you add the constraints  ef > 0, eh > 0 and maximize in two dimensions {ef,eh}:

Maximize[{g [10, 1, 1, 5, 10, eh, ef], ef > 0, eh > 0}, {ef, eh}] // N
(*{0.753847, {ef -> 0.242362, eh -> 0.242362}}*)


If you are looking for a maximum for given parameters a, b, c, w, F, eh define a region depending on these parameters

reg[a_, b_, c_, w_, F_, eh_] =ImplicitRegion[(ef == eh && ef > c/a) ||ef >= c/(a - 2 Sqrt[b] Sqrt[F]), ef ]


and maximize

NMaximize[ g [10, 1, 1, 5, 10, .24, ef]  , Element[{ef}, reg [10, 1, 1, 5, 10, .24]]]
(*{0.753472, {ef -> 0.24}}*)

• How about g[10, 1, 1, 5, 10, 24/100, 24/100] which produces 217/288? See my comment to the question. Commented Jul 1, 2020 at 13:27
• @user64494 Yes, but the point eh=ef=.24 lies outside the valid region! Commented Jul 1, 2020 at 13:31
• I am sorry but I don't see what is this "valid region"? The function g is defined for any positive ef. It is given by the expression in the first line when ef>c/(a - 2 Sqrt[b] Sqrt[F]) OR when ef>c/a AND ef=eh. (Otherwise, it is zero.) So eh=ef=.24>1/10 is a valid point and it corresponds indeed to the maximum that I would like to be able to find, but don't know how. Commented Jul 1, 2020 at 13:55
• Yes, that's the region plotted in my answer! Commented Jul 1, 2020 at 14:17
• @Lednacek Perhaps my final addendum shows the way to solve your problem. Commented Jul 2, 2020 at 11:07