Orthogonal Projection onto the $ {L}_{1} $ Unit Ball

Solution 1:

$$ \DeclareMathOperator{\sign}{sign} $$

The Lagrangian of the problem can be written as:

$$ \begin{align} L \left( x, \lambda \right) & = \frac{1}{2} {\left\| x - y \right\|}^{2} + \lambda \left( {\left\| x \right\|}_{1} - 1 \right) && \text{} \\ & = \sum_{i = 1}^{n} \left( \frac{1}{2} { \left( {x}_{i} - {y}_{i} \right) }^{2} + \lambda \left| {x}_{i} \right| \right) - \lambda && \text{Component wise form} \end{align} $$

The Dual Function is given by:

$$ \begin{align} g \left( \lambda \right) = \inf_{x} L \left( x, \lambda \right) \end{align} $$

The above can be solved component wise for the term $ \left( \frac{1}{2} { \left( {x}_{i} - {y}_{i} \right) }^{2} + \lambda \left| {x}_{i} \right| \right) $ which is solved by the soft Thresholding Operator:

$$ \begin{align} {x}_{i}^{\ast} = \sign \left( {y}_{i} \right) { \left( \left| {y}_{i} \right| - \lambda \right) }_{+} \end{align} $$

Where $ {\left( t \right)}_{+} = \max \left( t, 0 \right) $.

Now, all needed is to find the optimal $ \lambda \geq 0 $ which is given by the root of the objective function (Which is the constrain of the KKT Sytsem):

$$ \begin{align} h \left( \lambda \right) & = \sum_{i = 1}^{n} \left| {x}_{i}^{\ast} \left( \lambda \right) \right| - 1 \\ & = \sum_{i = 1}^{n} { \left( \left| {y}_{i} \right| - \lambda \right) }_{+} - 1 \end{align} $$

The above is a Piece Wise linear function of $ \lambda $ and its Derivative given by:

$$ \begin{align} \frac{\mathrm{d} }{\mathrm{d} \lambda} h \left( \lambda \right) & = \frac{\mathrm{d} }{\mathrm{d} \lambda} \sum_{i = 1}^{n} { \left( \left| {y}_{i} \right| - \lambda \right) }_{+} \\ & = \sum_{i = 1}^{n} -{ \mathbf{1} }_{\left\{ \left| {y}_{i} \right| - \lambda > 0 \right\}} \end{align} $$

Hence it can be solved using Newton Iteration.

In a similar manner the projection onto the Simplex (See @Ashkan answer) can be calculated.
The Lagrangian in that case is given by:

$$ \begin{align} L \left( x, \mu \right) & = \frac{1}{2} {\left\| x - y \right\|}^{2} + \mu \left( \boldsymbol{1}^{T} x - 1 \right) && \text{} \\ \end{align} $$

The trick is to leave non negativity constrain implicit.
Hence the Dual Function is given by:

$$ \begin{align} g \left( \mu \right) & = \inf_{x \succeq 0} L \left( x, \mu \right) && \text{} \\ & = \inf_{x \succeq 0} \sum_{i = 1}^{n} \left( \frac{1}{2} { \left( {x}_{i} - {y}_{i} \right) }^{2} + \mu {x}_{i} \right) - \mu && \text{Component wise form} \end{align} $$

Again, taking advantage of the Component Wise form the solution is given:

$$ \begin{align} {x}_{i}^{\ast} = { \left( {y}_{i} - \mu \right) }_{+} \end{align} $$

Where the solution includes the non negativity constrain by Projecting onto $ {\mathbb{R}}_{+} $

Again, the solution is given by finding the $ \mu $ which holds the constrain (Pay attention, since the above was equality constrain, $ \mu $ can have any value and it is not limited to non negativity as $ \lambda $ above).

The objective function (From the KKT) is given by:

$$ \begin{align} h \left( \mu \right) = \sum_{i = 1}^{n} {x}_{i}^{\ast} - 1 & = \sum_{i = 1}^{n} { \left( {y}_{i} - \mu \right) }_{+} - 1 \end{align} $$

The above is a Piece Wise linear function of $ \mu $ and its Derivative given by:

$$ \begin{align} \frac{\mathrm{d} }{\mathrm{d} \mu} h \left( \mu \right) & = \frac{\mathrm{d} }{\mathrm{d} \mu} \sum_{i = 1}^{n} { \left( {y}_{i} - \mu \right) }_{+} \\ & = \sum_{i = 1}^{n} -{ \mathbf{1} }_{\left\{ {y}_{i} - \mu > 0 \right\}} \end{align} $$

Hence it can be solved using Newton Iteration.

I wrote MATLAB code which implements them both at Mathematics StackExchange Question 2327504 - GitHub.
There is a test which compares the result to a reference calculated by CVX.

Solution 2:

Hint: Because of the symmetric essences of the problem you may assume $x$ lies in first quadrant i.e, $x \ge 0$ and assume $x$ is out side of $\ell_1 $- Unit ball (other wise the answer is trivially $y=x$ ),Therefor under these assumption for sure we have $ 0 \leq y^{*} \leq x$ where $y^{*} $ is the unique optimal solution. To find $y^{*}$ you need to solve following Quadratic programming
\begin{aligned} & {\text{Min}} & & \sum_{i=1}^{n} (x_i -y_i)^2 \\ & \text{subject to} & & y \geq 0, \\ & & & \sum_{i=1}^{n} y_i =1 , \end{aligned}

Note that this is a smooth convex optimization problem with linear constraints, So it is easy to solve! To find a closed form solution set up $KKT$ systems.

Note that once you get solution from problem above, you can characterize all solutions for all cases depending on positions of $x$ in space. For example let $x = (-1, 2,0,0,3)$, you know the solution for above problem where $\bar{x}=(1,2,0,0,3),$ call it $\bar{y} =(y_1,y_2,..., y_n)$ then solution corresponding to $x$ is $y=(-y_1,y_2,...,y_n)$.