Find the max of $x_1 x_2 \cdots x_m$ when $x_1^2+x_2^2+ \cdots + x_m^2=1$

Find the max of a function $$f(x_1, x_2,...,x_m)=x_1 x_2 \cdots x_m$$ when $$x_1^2+x_2^2+ \cdots +x_m^2=1$$

I am looking for unique solutions. So far I've tried two different ways.

One of them was with Lagrange multiplier but there seem to be quite too many variables and I couldn't express $$x_1,...,x_m$$ separately.

The second one solution is with the Inequality of arithmetic and geometric means.

$$\frac{x_1^2+...+x_m^2}{m}\ge \sqrt[m]{x_1^2 \cdots x_m^2} \implies \frac{1}{m} \ge \sqrt[m]{x_1^2 \cdots x_m^2}=(x_1 \cdots x_m)^{\frac{2}{m}} \implies$$ $$x_1 \cdots x_m \le \frac{1}{m^{\frac{m}{2}}}$$ So we can assume that $$\max f(x_1, x_2,...,x_m)=\frac{1}{m^{\frac{m}{2}}} \quad \forall x_i=\frac{1}{\sqrt m}, \quad i=1,2,...,m.$$

I'd love to know if somebody solved this exercise with Langrange multiplier or in other way. Thanks in advance!

• Note: $x_i$ don't have to be positive. We can assume they are positive without loss of generality however. – Jakobian Feb 27 at 18:42

4 Answers

$$x_1x_2...x_m\neq\sqrt{x_1^2x_2^2...x_m^2}$$

The right way it's the following:

By AM-GM $$x_1x_2...x_m\leq\sqrt{x_1^2x_2^2...x_m^2}\leq\sqrt{\left(\frac{x_1^2+x_2^2+...+x_m^2}{m}\right)^m}=\frac{1}{m^{\frac{m}{2}}}$$ The equality occurs for $$x_1=x_2=...=x_m=\frac{1}{\sqrt{m}},$$ which says that we got a maximal value.

• That's basically the answer in OPs question. – Thomas Andrews Feb 27 at 18:50
• @Thomas Andrews See please better our solutions. They are different in some point. – Michael Rozenberg Feb 27 at 18:51

Let $$g(x) = ||x||_2^2 - 1 = 0$$ be the constraint. $$\nabla g(x) = 2x$$, $$\nabla f(x)= (f(x)/x_1,\dots,f(x)/x_m) = \lambda \nabla g(x) = 2\lambda x.$$ $$\frac{x_1\cdots x_m}{x_i} = 2 \lambda x_i \\ x_1\cdots x_m = 2 \lambda x_i^2$$ Observe the the LHS is symmetric by interchange of indices $$i$$ and $$j$$ in $$\{1,\dots,m\}$$, so $$x_i = \pm x_j$$. Sum the above inequality over $$i$$ to get $$m x_1 \cdots x_m = 2 \lambda$$. This should give $$|x_i| = \sqrt[m]{\dfrac{2|\lambda|}{m}}$$.

Substitute this back to $$g(x) = 0$$: $$m \cdot \sqrt[m]{\dfrac{4\lambda^2}{m^2}} = 1,$$ so $$\dfrac{2|\lambda|}{m} = \pm \dfrac{1}{\sqrt{m^m}},$$ giving a critical point $$x^* = \left(\dfrac{1}{\sqrt{m}},\dots,\dfrac{1}{\sqrt{m}}\right)$$ when $$\lambda^* = \dfrac{m^{1-m/2}}{2}$$, which is the same as the one found by AM-GM inequality in the question body.

Evaluating the objective at $$x^*$$, we find that $$f(x^*) = x_1^* x_2^* \dots x_m^* = \dfrac{1}{\sqrt{m^m}}$$.

To determine whether this is a local maximum, define the Lagrangian

\begin{aligned}{\mathcal {L}}(x,\lambda) &= f(x)-\lambda g(x)\\ \nabla{\cal L}(x,\lambda) &= \left({\frac {\partial {\mathcal {L}}}{\partial x}},{\frac {\partial {\mathcal {L}}}{\partial \lambda }}\right) = \left(\frac{x_1\cdots x_m}{x_1} - 2\lambda x_1, \dots, \frac{x_1\cdots x_m}{x_m} - 2\lambda x_m, -\sum_{i=1}^m x_i^2 \right) \end{aligned}

We compute the Hessian matrix of $${\cal L}$$ at $$x^*$$. For all $$i,j \in \{1,\dots, m\}$$ with $$i \ne j$$, \begin{alignat}{2} \frac{\partial^2{\mathcal {L}}}{\partial x_i^2} &= -2\lambda \quad & \frac{\partial^2{\mathcal {L}}}{\partial x_i \partial x_j} &= \frac{x_1\cdots x_m}{x_i x_j} \\ \frac{\partial^2{\mathcal {L}}}{\partial \lambda^2} &= 0 & \frac{\partial^2{\mathcal {L}}}{\partial \lambda \partial x_i} &= -2 x_i \end{alignat}

This gives a symmetric matrix

$$\begin{bmatrix} -2\lambda & \dfrac{x_1\cdots x_m}{x_1 x_2} & \dots & \dfrac{x_1\cdots x_m}{x_1 x_m} & -2 x_1 \\ \dfrac{x_1\cdots x_m}{x_1 x_2} & -2\lambda & \dots & \dfrac{x_1\cdots x_m}{x_2 x_m} & -2 x_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \dfrac{x_1\cdots x_m}{x_1 x_m} & \dfrac{x_1\cdots x_m}{x_2 x_m} & \dots & -2\lambda & -2 x_m \\ -2x_1 & -2x_2 & \dots & -2x_m & 0 \end{bmatrix}$$

Substituting $$(x^*,\lambda^*) = (m^{-1/2},\dots,m^{-1/2},m^{1-m/2}/2)$$, we have

$$H = \begin{bmatrix} -m^{1-m/2} & m^{1-m/2} & \dots & m^{1-m/2} & -2 m^{-1/2} \\ m^{1-m/2} & -m^{1-m/2} & \dots & m^{1-m/2} & -2 m^{-1/2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ m^{1-m/2} & m^{1-m/2} & \dots & -m^{1-m/2} & -2 m^{-1/2} \\ -2 m^{-1/2} & -2 m^{-1/2} & \dots & -2 m^{-1/2} & 0 \end{bmatrix}$$

For $$k = 1,\dots,m$$, let $$M_k$$ be the $$k$$-th leading principal minor of $$-H$$. Then the $$M_k$$'s are whose associated polynomial is $$f(x) = C \left( 1-\sum\limits_{\ell=1}^{k-1} x^\ell \right)$$, where $$C = m^{1-m/2} \in (0,1)$$ for $$m \ge 3$$.

Let $$\omega = e^{2\pi i /k}$$. A simple geometric sum would give

$$\sum_{\ell=0}^{k-1} (\omega^j)^\ell = \begin{cases} 0 &\text{ if } j = 1,\dots,k-1 \\ k &\text{ if } j = 0 \end{cases}$$

Then $$\det(M_k) = \prod\limits_{j = 0}^{k-1} f(\omega^j) = C^k \cdot [-(k-2)] \cdot 2^{k-1} < 0$$ for all $$k > 3$$.

• Are you sure? Why these give a maximal value? – Michael Rozenberg Feb 27 at 18:56
• @MichaelRozenberg Take one of the $x_i = 0$, say $(1,0,\dots,0)$, then that gives objective value zero, so the $x$ found by Lagrange multipliers is not a constrained minimum. – GNUSupporter 8964民主女神 地下教會 Feb 27 at 19:05
• Maybe we got a saddle point... Also, maybe a maximum occurs in another point. Your solution is very not full. – Michael Rozenberg Feb 27 at 19:08
• I don't understand how you get $\nabla f(x)=(1,1,\dots,1)$ Why isn't it $\left({f(x)\over x_1}, \dots, {f(x)\over x_m}\right)?$ – saulspatz Feb 27 at 19:45
• @saulspatz Nice catch! I'm rewriting my solution. – GNUSupporter 8964民主女神 地下教會 Feb 27 at 19:47

Either the function and the constraint are symmetrical wrt $$x_1=x_2= \cdots=x_m$$.

The function $$f(x_1,\cdots,x_m)=x_1 x_2 \cdots x_m=s$$ is one of the sheets of a 2-sheets hyperboloid, of revolution around the axis above, with minimum (apex) at $$x_1= \cdots =x_m=s^{1/m}$$.

The constraint is a sphere.

Thus ..

I would do it another way. The symmetry suggests that the maximum will occur when all the coordinates are equal, so I would try to show that if two coordinates say $$x_i$$ and $$x_j$$ are different, then we can increase the objective function by replacing both of them by some value $$c$$ and leaving the other coordinates the same. Since we have to maintain the constraint, this comes down to showing that the maximum of $$g(x,y)=xy$$ subject to $$x^2+y^2=k$$ is attained when $$x=y$$ This is easy with Lagrange multipliers.

Once we know that $$x_1=x_2=\cdots=x_m$$ we solve for the common value.