Maximise $p_1p_2p_3p_4p_5$ subject to constraints

With the help of the slack variables $\epsilon_i$ and calling

$$ L(p,\mu,\epsilon,\lambda)= p_1 p_2 p_3 p_4 p_5+\mu _5 \left(p_1-p_2-\epsilon _5^2\right)+\mu _4 \left(p_2-p_3-\epsilon _4^2\right)+\mu _3 \left(p_3-p_4-\epsilon _3^2\right)+\mu _2 \left(p_4-p_5-\epsilon _2^2\right)+\mu _1 \left(p_5-\epsilon _1^2\right)+\lambda \left(p_1 x_1+p_2 \left(x_2-x_1\right)+p_3 \left(x_3-x_2\right)+p_4 \left(x_4-x_3\right)+p_5 \left(x_5-x_4\right)-1\right) $$

and solving the sationary conditions

$$ \nabla L = 0 $$

we obtain a set of solutions jointly with a set of conditions $\epsilon_i^2\ge 0$ to qualify those solutions. To be feasible the solution requires that $\epsilon_i^2\ge 0$ Also when $\epsilon_i^2 = 0$ it means that the corresponding restriction is active.

Due to the length of the symbolic response, we leave a script in MATHEMATICA that summarizes the results. There are sixteen non trivial solutions in res0 with structure $\{p_i,\epsilon_i^2,p_1p_2p_3p_4p_5\}$

n = 5;
X = Table[Subscript[x, k], {k, 1, n}];
P = Table[Subscript[p, k], {k, 1, n}];
EE = Table[Subscript[epsilon, k], {k, 1, n}];
M = Table[Subscript[mu, k], {k, 1, n}];
vars = Join[Join[Join[P, M], EE], {lambda}]
prod = Product[P[[k]], {k, 1, n}]
L = prod
L += lambda (Sum[P[[k]] (X[[k]] - X[[k - 1]]), {k, 2, n}] + P[[1]] X[[1]] - 1)
L += Sum[M[[k]] (P[[n - k + 1]] - P[[n - k + 2]] + EE[[k]]^2), {k, 2, n}] + M[[1]] (P[[n]] - EE[[1]]^2)
grad = Grad[L, vars]
equs = Thread[grad == 0];
E2 = EE^2;
results = Join[Join[P, E2], {prod}];
sols = Quiet[Solve[equs, vars]];
results0 = results /. sols // FullSimplify;
For[i = 1; res = {}, i <= Length[results0], i++,
  If[NumberQ[results0[[i]][[2 n + 1]]] == False, AppendTo[res,results0[[i]]]]
]
res0 = Union[res];
res0 // MatrixForm

Results for $n = 3$

$$ \left[ \begin{array}{ccccccc} p_1&p_2&p_3&\epsilon_1^2&\epsilon_2^2&\epsilon_3^2&p_1p_2p_3\\ \frac{1}{3 x_1} & \frac{1}{3 \left(x_2-x_1\right)} & \frac{1}{3 \left(x_3-x_2\right)} & \frac{1}{3 \left(x_3-x_2\right)} & \frac{1}{3 x_2-3 x_3}+\frac{1}{3 \left(x_2-x_1\right)} & \frac{1}{3} \left(\frac{1}{x_1-x_2}+\frac{1}{x_1}\right) & \frac{1}{27 \left(x_1^2-x_1 x_2\right) \left(x_2-x_3\right)} \\ \frac{1}{3 x_1} & \frac{2}{3 \left(x_3-x_1\right)} & \frac{2}{3 \left(x_3-x_1\right)} & \frac{2}{3 \left(x_3-x_1\right)} & 0 & \frac{1}{3} \left(\frac{2}{x_1-x_3}+\frac{1}{x_1}\right) & \frac{4}{27 x_1 \left(x_1-x_3\right){}^2} \\ \frac{2}{3 x_2} & \frac{2}{3 x_2} & \frac{1}{3 \left(x_3-x_2\right)} & \frac{1}{3 \left(x_3-x_2\right)} & \frac{1}{3 x_2-3 x_3}+\frac{2}{3 x_2} & 0 & \frac{4}{27 x_2^2 \left(x_3-x_2\right)} \\ \frac{1}{x_3} & \frac{1}{x_3} & \frac{1}{x_3} & \frac{1}{x_3} & 0 & 0 & \frac{1}{x_3^3} \\ \end{array} \right] $$

Results for $n = 4$

$$ \left[ \begin{array}{ccccccccc} p_1 & p_2 & p_3 & p_4 & \epsilon_1^2&\epsilon_2^2&\epsilon_3^2&\epsilon_4^2&p_1p_2p_3p_4\\ \frac{1}{4 x_1} & \frac{1}{4 \left(x_2-x_1\right)} & \frac{1}{4 \left(x_3-x_2\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 x_3-4 x_4}+\frac{1}{4 \left(x_3-x_2\right)} & \frac{1}{4 x_2-4 x_3}+\frac{1}{4 \left(x_2-x_1\right)} & \frac{1}{4} \left(\frac{1}{x_1-x_2}+\frac{1}{x_1}\right) & -\frac{1}{256 \left(x_1^2-x_1 x_2\right) \left(x_2-x_3\right) \left(x_3-x_4\right)} \\ \frac{1}{4 x_1} & \frac{1}{4 \left(x_2-x_1\right)} & \frac{1}{2 \left(x_4-x_2\right)} & \frac{1}{2 \left(x_4-x_2\right)} & \frac{1}{2 \left(x_4-x_2\right)} & 0 & \frac{1}{2 x_2-2 x_4}+\frac{1}{4 \left(x_2-x_1\right)} & \frac{1}{4} \left(\frac{1}{x_1-x_2}+\frac{1}{x_1}\right) & -\frac{1}{64 x_1 \left(x_1-x_2\right) \left(x_2-x_4\right){}^2} \\ \frac{1}{4 x_1} & \frac{1}{2 \left(x_3-x_1\right)} & \frac{1}{2 \left(x_3-x_1\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 x_3-4 x_4}+\frac{1}{2 \left(x_3-x_1\right)} & 0 & \frac{1}{4} \left(\frac{2}{x_1-x_3}+\frac{1}{x_1}\right) & \frac{1}{64 x_1 \left(x_1-x_3\right){}^2 \left(x_4-x_3\right)} \\ \frac{1}{4 x_1} & \frac{3}{4 \left(x_4-x_1\right)} & \frac{3}{4 \left(x_4-x_1\right)} & \frac{3}{4 \left(x_4-x_1\right)} & \frac{3}{4 \left(x_4-x_1\right)} & 0 & 0 & \frac{1}{4} \left(\frac{3}{x_1-x_4}+\frac{1}{x_1}\right) & \frac{27}{256 x_1 \left(x_4-x_1\right){}^3} \\ \frac{1}{2 x_2} & \frac{1}{2 x_2} & \frac{1}{4 \left(x_3-x_2\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 x_3-4 x_4}+\frac{1}{4 \left(x_3-x_2\right)} & \frac{1}{4 x_2-4 x_3}+\frac{1}{2 x_2} & 0 & \frac{1}{64 x_2^2 \left(x_2-x_3\right) \left(x_3-x_4\right)} \\ \frac{1}{2 x_2} & \frac{1}{2 x_2} & \frac{1}{2 \left(x_4-x_2\right)} & \frac{1}{2 \left(x_4-x_2\right)} & \frac{1}{2 \left(x_4-x_2\right)} & 0 & \frac{1}{2} \left(\frac{1}{x_2-x_4}+\frac{1}{x_2}\right) & 0 & \frac{1}{16 x_2^2 \left(x_2-x_4\right){}^2} \\ \frac{3}{4 x_3} & \frac{3}{4 x_3} & \frac{3}{4 x_3} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 \left(x_4-x_3\right)} & \frac{1}{4 x_3-4 x_4}+\frac{3}{4 x_3} & 0 & 0 & \frac{27}{256 x_3^3 \left(x_4-x_3\right)} \\ \frac{1}{x_4} & \frac{1}{x_4} & \frac{1}{x_4} & \frac{1}{x_4} & \frac{1}{x_4} & 0 & 0 & 0 & \frac{1}{x_4^4} \\\end{array} \right] $$

and for $n = 5$

$$ \left[ \begin{array}{ccccccccccc} p_1&p_2&p_3&p_4&p_5&\epsilon_1^2&\epsilon_2^2&\epsilon_3^2&\epsilon_4^2&\epsilon_5^2&p_1p_2p_3p_4p_5\\ \frac{1}{5 x_1} & -\frac{1}{5 \left(x_1-x_2\right)} & -\frac{1}{5 \left(x_2-x_3\right)} & -\frac{1}{5 \left(x_3-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_3-2 x_4+x_5}{5 \left(x_3-x_4\right) \left(x_4-x_5\right)} & \frac{x_2-2 x_3+x_4}{5 \left(x_2-x_3\right) \left(x_3-x_4\right)} & \frac{x_1-2 x_2+x_3}{5 \left(x_1-x_2\right) \left(x_2-x_3\right)} & \frac{2 x_1-x_2}{5 x_1 \left(x_1-x_2\right)} & \frac{1}{3125 x_1 \left(x_1-x_2\right) \left(x_2-x_3\right) \left(x_3-x_4\right) \left(x_4-x_5\right)} \\ \frac{1}{5 x_1} & -\frac{1}{5 \left(x_1-x_2\right)} & -\frac{1}{5 \left(x_2-x_3\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & 0 & \frac{2 x_2-3 x_3+x_5}{5 \left(x_2-x_3\right) \left(x_3-x_5\right)} & \frac{x_1-2 x_2+x_3}{5 \left(x_1-x_2\right) \left(x_2-x_3\right)} & \frac{2 x_1-x_2}{5 x_1 \left(x_1-x_2\right)} & \frac{4}{3125 x_1 \left(x_1-x_2\right) \left(x_2-x_3\right) \left(x_3-x_5\right){}^2} \\ \frac{1}{5 x_1} & -\frac{1}{5 \left(x_1-x_2\right)} & -\frac{2}{5 \left(x_2-x_4\right)} & -\frac{2}{5 \left(x_2-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_2-3 x_4+2 x_5}{5 \left(x_2-x_4\right) \left(x_4-x_5\right)} & 0 & \frac{2 x_1-3 x_2+x_4}{5 \left(x_1-x_2\right) \left(x_2-x_4\right)} & \frac{2 x_1-x_2}{5 x_1 \left(x_1-x_2\right)} & \frac{4}{3125 x_1 \left(x_1-x_2\right) \left(x_2-x_4\right){}^2 \left(x_4-x_5\right)} \\ \frac{1}{5 x_1} & -\frac{1}{5 \left(x_1-x_2\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & 0 & 0 & \frac{3 x_1-4 x_2+x_5}{5 \left(x_1-x_2\right) \left(x_2-x_5\right)} & \frac{2 x_1-x_2}{5 x_1 \left(x_1-x_2\right)} & \frac{27}{3125 x_1 \left(x_1-x_2\right) \left(x_2-x_5\right){}^3} \\ \frac{1}{5 x_1} & -\frac{2}{5 \left(x_1-x_3\right)} & -\frac{2}{5 \left(x_1-x_3\right)} & -\frac{1}{5 \left(x_3-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_3-2 x_4+x_5}{5 \left(x_3-x_4\right) \left(x_4-x_5\right)} & \frac{x_1-3 x_3+2 x_4}{5 \left(x_1-x_3\right) \left(x_3-x_4\right)} & 0 & \frac{3 x_1-x_3}{5 x_1 \left(x_1-x_3\right)} & \frac{4}{3125 x_1 \left(x_1-x_3\right){}^2 \left(x_3-x_4\right) \left(x_4-x_5\right)} \\ \frac{1}{5 x_1} & -\frac{2}{5 \left(x_1-x_3\right)} & -\frac{2}{5 \left(x_1-x_3\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & 0 & \frac{2 \left(x_1-2 x_3+x_5\right)}{5 \left(x_1-x_3\right) \left(x_3-x_5\right)} & 0 & \frac{3 x_1-x_3}{5 x_1 \left(x_1-x_3\right)} & \frac{16}{3125 x_1 \left(x_1-x_3\right){}^2 \left(x_3-x_5\right){}^2} \\ \frac{1}{5 x_1} & -\frac{3}{5 \left(x_1-x_4\right)} & -\frac{3}{5 \left(x_1-x_4\right)} & -\frac{3}{5 \left(x_1-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_1-4 x_4+3 x_5}{5 \left(x_1-x_4\right) \left(x_4-x_5\right)} & 0 & 0 & \frac{4 x_1-x_4}{5 x_1 \left(x_1-x_4\right)} & \frac{27}{3125 x_1 \left(x_1-x_4\right){}^3 \left(x_4-x_5\right)} \\ \frac{1}{5 x_1} & -\frac{4}{5 \left(x_1-x_5\right)} & -\frac{4}{5 \left(x_1-x_5\right)} & -\frac{4}{5 \left(x_1-x_5\right)} & -\frac{4}{5 \left(x_1-x_5\right)} & -\frac{4}{5 \left(x_1-x_5\right)} & 0 & 0 & 0 & \frac{5 x_1-x_5}{5 x_1 \left(x_1-x_5\right)} & \frac{256}{3125 x_1 \left(x_1-x_5\right){}^4} \\ \frac{2}{5 x_2} & \frac{2}{5 x_2} & -\frac{1}{5 \left(x_2-x_3\right)} & -\frac{1}{5 \left(x_3-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_3-2 x_4+x_5}{5 \left(x_3-x_4\right) \left(x_4-x_5\right)} & \frac{x_2-2 x_3+x_4}{5 \left(x_2-x_3\right) \left(x_3-x_4\right)} & \frac{3 x_2-2 x_3}{5 x_2 \left(x_2-x_3\right)} & 0 & -\frac{4}{3125 x_2^2 \left(x_2-x_3\right) \left(x_3-x_4\right) \left(x_4-x_5\right)} \\ \frac{2}{5 x_2} & \frac{2}{5 x_2} & -\frac{1}{5 \left(x_2-x_3\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & 0 & \frac{2 x_2-3 x_3+x_5}{5 \left(x_2-x_3\right) \left(x_3-x_5\right)} & \frac{3 x_2-2 x_3}{5 x_2 \left(x_2-x_3\right)} & 0 & -\frac{16}{3125 x_2^2 \left(x_2-x_3\right) \left(x_3-x_5\right){}^2} \\ \frac{2}{5 x_2} & \frac{2}{5 x_2} & -\frac{2}{5 \left(x_2-x_4\right)} & -\frac{2}{5 \left(x_2-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_2-3 x_4+2 x_5}{5 \left(x_2-x_4\right) \left(x_4-x_5\right)} & 0 & \frac{2 \left(2 x_2-x_4\right)}{5 x_2 \left(x_2-x_4\right)} & 0 & -\frac{16}{3125 x_2^2 \left(x_2-x_4\right){}^2 \left(x_4-x_5\right)} \\ \frac{2}{5 x_2} & \frac{2}{5 x_2} & -\frac{3}{5 \left(x_2-x_5\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & -\frac{3}{5 \left(x_2-x_5\right)} & 0 & 0 & \frac{5 x_2-2 x_5}{5 x_2 \left(x_2-x_5\right)} & 0 & -\frac{108}{3125 x_2^2 \left(x_2-x_5\right){}^3} \\ \frac{3}{5 x_3} & \frac{3}{5 x_3} & \frac{3}{5 x_3} & -\frac{1}{5 \left(x_3-x_4\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{x_3-2 x_4+x_5}{5 \left(x_3-x_4\right) \left(x_4-x_5\right)} & \frac{4 x_3-3 x_4}{5 x_3 \left(x_3-x_4\right)} & 0 & 0 & \frac{27}{3125 x_3^3 \left(x_3-x_4\right) \left(x_4-x_5\right)} \\ \frac{3}{5 x_3} & \frac{3}{5 x_3} & \frac{3}{5 x_3} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & -\frac{2}{5 \left(x_3-x_5\right)} & 0 & \frac{5 x_3-3 x_5}{5 x_3 \left(x_3-x_5\right)} & 0 & 0 & \frac{108}{3125 x_3^3 \left(x_3-x_5\right){}^2} \\ \frac{4}{5 x_4} & \frac{4}{5 x_4} & \frac{4}{5 x_4} & \frac{4}{5 x_4} & -\frac{1}{5 \left(x_4-x_5\right)} & -\frac{1}{5 \left(x_4-x_5\right)} & \frac{5 x_4-4 x_5}{5 x_4 \left(x_4-x_5\right)} & 0 & 0 & 0 & -\frac{256}{3125 x_4^4 \left(x_4-x_5\right)} \\ \frac{1}{x_5} & \frac{1}{x_5} & \frac{1}{x_5} & \frac{1}{x_5} & \frac{1}{x_5} & \frac{1}{x_5} & 0 & 0 & 0 & 0 & \frac{1}{x_5^5} \\ \end{array} \right] $$

Well you could try with Am-Gm inequality:

$$ 5\sqrt[5]{p_1p_2...p_5x_1(x_2-x_1)...(x_5-x_4)}\leq 1$$

So $$p_1p_2...p_5 \leq {1\over 5^5x_1(x_2-x_1)...(x_5-x_4)}$$

and this value is achivable if $$ p_1 ={1\over 5x_1}$$

$$ p_2 ={1\over 5(x_2-x_1)}$$ $$ p_3 ={1\over 5(x_3-x_2)}$$ $$...$$

This answer addresses the case when $$ \frac{5}{x_5}\geq\max_{i=1,\dots,4}\big\{\frac{i}{x_i}\big\}. $$ In this case, the solution using the AM-GM inequality fails to satisfy the ordering constraint. There are still other cases to be considered, for example, $x=(1,\;3,\;4.75,\;8.75,\;13.75)$ which produces the solution $p_1=1/5$, $p_2=p_3=8/75$, $p_4=1/20$ and $p_5=1/25$ (computed this solution numerically).

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While the other answers are clever, I don't believe that they respect the ordering constraint $0\leq{p_5}\leq\dots\leq{p_1}$. My proposed solution is found using the KKT conditions. (Note: I assume $x_i>0$ for all $i$--the case when $x_i=0$ should following from a similar technique to what I use below).

Answer: $p_1=p_2=p_3=p_4=p_5=1/x_5$.

Solution: By taking the logarithm of the objective function, the given optimization problem can be formulated as a convex optimization problem in standard form: $$ \begin{array}{rl} \min & -\log(p_1)-\log(p_2)-\cdots-\log(p_5)\\ \text{s.t.} & p_1x_1+p_2(x_2-x_1)+p_3(x_3-x_2)+p_4(x_4-x_3)+p_5(x_5-x_4)-1=0\\ &p_2-p_1\leq0\\ &p_3-p_2\leq0\\ &p_4-p_3\leq0\\ &p_5-p_4\leq0\\ &\hspace{12pt}-p_5\leq0 \end{array} $$ Associate the multiplier $\lambda\in\mathbb{R}$ with the equality constraint, and $\mu_i\geq0$ with the remaining constraints $i=1,\dots,5$. The stationarity conditions for this optimization problem are \begin{align*} \frac{1}{p_1}&=\lambda{x_1}-\mu_1\\ \frac{1}{p_2}&=\lambda(x_2-x_1)+(\mu_1-\mu_2)\\ \frac{1}{p_3}&=\lambda(x_3-x_2)+(\mu_2-\mu_3)\\ \frac{1}{p_4}&=\lambda(x_4-x_3)+(\mu_3-\mu_4)\\ \frac{1}{p_5}&=\lambda(x_5-x_4)+(\mu_4-\mu_5)\\ \end{align*} A solution to the above system is \begin{align*} p_i&=\frac{1}{x_5}&&\text{for }i=1,\dots,5\\ \mu_i&=5x_i-i\cdot{x_5}&&\text{for }i=1,\dots,5\\ \lambda&=5 \end{align*} It's easy to verify that the other KKT conditions are satisfied (complementary slackness, etc.), and to check that the problem satisfies the Slater condition. Hence, we conclude that the proposed point is indeed globally optimal for this problem.