Let $T>0$, $H>0$, $D$, and $No>0$ are known parameter.

Let $\{p_1,p_2,\cdots, p_T\}$ are the optimization variables.

Let $\{w_1,w_2,\cdots, w_T\}$ are known parameters.

Now, I want to perform the following optimization

minimize $\sum_{t=1}^Tp_t$

subject to $\sum_{t=1}^Tw_t\log_2\left(1+\frac{Hp_t}{w_t No}\right)=Do$

I want to express $P_t$ as a function of other parameters.

I want to do it as follows

W = Array[w, T];
P = Array[p, T];

Expr = Total[W Log2[1 + H P/(W No)]];

Assuming[H > 0 && No > 0 && T > 0 && Do > 0, Simplify[Minimize[{Total[P], Expr == Do}, P]]]

How to denote $W$ and $P$ so that I can perform the optimization and express $P_t$ as a function of other system parameters?

  • 1
    $\begingroup$ N and D are already defined in Mathematica, you shouldn't use them for variables. More constructively, if you put in your known values for H, N, T, D, and the w_i, the minimisation you've written should work. $\endgroup$ – AnotherShruggingPhysicist Oct 13 '17 at 9:14
  • $\begingroup$ @AnotherShruggingPhysicist, Thank you very much. But I need to express $p_t$ in a symbolic form. $\endgroup$ – George Farnandez Oct 13 '17 at 9:31
  • $\begingroup$ Since this is a frequent issue: Do you just want $p_t$ in symbolic form or do you really need it in symbolic form? $\endgroup$ – Henrik Schumacher Oct 13 '17 at 17:13
  • $\begingroup$ @HenrikSchumacher, I really need it in symbolic form. Can you help me out? $\endgroup$ – George Farnandez Oct 16 '17 at 2:45

If you are really interested in a symbolic solution, I would propose to investigate the KKT-conditions for this optimization problem. Here a quick script that generates them for $T=5$ variables. (\[Lambda] is a Lagrange multiplier.)

T = 5;
W = Table[w[[i]], {i, 1, T}];
P = Table[p[[i]], {i, 1, T}];
F = P \[Function] Evaluate[Total[P]];
DF = P \[Function] Evaluate[D[F[P], {P, 1}]];
G = p \[Function] Evaluate[Total[W + Log[1 + c P/W]]];
DG = p \[Function] Evaluate[D[G[P], {P, 1}]];
eq = Join[DF[P] + \[Lambda] DG[P], {G[P] - c0}];
Thread[eq == 0]

enter image description here

These can be simplified a bit; moreover, I eleminated \[Lambda]:

eq2 = Join[Table[w[[i]] + c p[[i]] + c \[Lambda], {i, 1, T}], eq[[{-1}]]];
lambdasol = Solve[Total[eq2[[1 ;; T]]] == 0, \[Lambda]][[1]];
p1sol = Solve[eq2[[-1]] == 0, p[[1]]][[1]] /. ConditionalExpression[bla_, blubb_] :> bla;
eq3 = T eq2[[1 ;; T-1]] /. lambdasol /. p1sol // Expand;
Thread[eq3 == 0]

enter image description here

Multiplying everything by with the nasty denominator would produce T-1 polynomial equations of order T in the T-1variables p[[2]],...,p[[T]]. Good luck with solving them!

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.