# Convex Iteration

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Convex iteration is method for constraining rank or cardinality in an otherwise convex optimization problem. | Convex iteration is method for constraining rank or cardinality in an otherwise convex optimization problem. | ||

<br> | <br> | ||

- | A rank or cardinality constraint is replaced by a | + | A rank or cardinality constraint is replaced by a linear regularization term added to the objective. |

<br> | <br> | ||

Then two convex problems are iterated until convergence where, ideally, solution to the original problem is found. | Then two convex problems are iterated until convergence where, ideally, solution to the original problem is found. | ||

==constraining cardinality of signed variable== | ==constraining cardinality of signed variable== | ||

- | + | Excerpt from Chapter 4 [http://meboo.convexoptimization.com/Meboo.html Semidefinite Programming]: | |

- | Consider a feasibility problem equivalent to the classical problem from linear algebra | + | <br>Consider a feasibility problem equivalent to the classical problem from linear algebra |

<math>A_{}x_{\!}=_{\!}b</math> , | <math>A_{}x_{\!}=_{\!}b</math> , | ||

- | but with an upper bound <math>k</math> on cardinality <math> | + | but with an upper bound <math>k</math> on cardinality <math>||x||_0</math> : |

- | for vector <math>b\!\in | + | for vector <math>b\!\in\mathcal{R}(A)</math> |

- | <math>\begin{array}{rl}\text | + | <math>\begin{array}{rl}{\text find}&_{}x\in_{}\mathbb{R}^n\\ |

\mbox{subject to}&A_{}x=b\\ | \mbox{subject to}&A_{}x=b\\ | ||

- | & | + | &||x||_0\leq_{}k |

- | \end{array} | + | \end{array}</math> (1) |

- | where <math> | + | where <math>||x||_0\leq k</math> means vector <math>x</math> has at most <math>k</math> nonzero entries; |

- | such a vector is presumed existent in the feasible set. <br>Convex iteration works with a nonnegative variable; | + | such a vector is presumed existent in the feasible set. |

- | absolute value <math>|x|</math> is therefore needed. | + | <br>Convex iteration works with a nonnegative variable; absolute value <math>|x|</math> is therefore needed. |

- | We propose that nonconvex problem (1) can be equivalently written | + | <br>We propose that nonconvex problem (1) can be equivalently written |

as a convex iteration: | as a convex iteration: | ||

for <math>\varepsilon</math> a relatively small positive constant, | for <math>\varepsilon</math> a relatively small positive constant, | ||

- | <math>\begin{array}{ | + | <math>\begin{array}{cl}\text{minimize}_{x_{}\in_{_{}}\mathbb{R}^n,~t_{}\in_{_{}}\mathbb{R}^n}&\langle t_{}\,,\,y+\varepsilon^{}\mathbf{1}\rangle\\ |

\mbox{subject to}&A_{}x=b\\ | \mbox{subject to}&A_{}x=b\\ | ||

&-t\preceq x\preceq_{_{}}t | &-t\preceq x\preceq_{_{}}t | ||

- | \end{array} | + | \end{array}</math> (2) |

is iterated with | is iterated with | ||

- | <math>\begin{array}{cl}\text{minimize}_{y_{}\in_{_{}}\mathbb{R} | + | <math>\begin{array}{cl}\text{minimize}_{y_{}\in_{_{}}\mathbb{R}^n}&\langle t^*,\,y+\varepsilon^{}\mathbf{1}\rangle\\ |

\mbox{subject to}&0\preceq y\preceq\mathbf{1}\\ | \mbox{subject to}&0\preceq y\preceq\mathbf{1}\\ | ||

&y^T\mathbf{1}=n-_{}k | &y^T\mathbf{1}=n-_{}k | ||

- | \end{array} | + | \end{array}</math> (3) |

to find ''direction vector'' <math>y</math>. | to find ''direction vector'' <math>y</math>. | ||

The cardinality constraint has been moved to the objective as a linear regularization. | The cardinality constraint has been moved to the objective as a linear regularization. | ||

- | The term <math>\langle t\,,_{_{}}\varepsilon^{}\mathbf{1}\rangle</math> in (2) is necessary to determine absolute value | + | <br>The term <math>\langle t\,,_{_{}}\varepsilon^{}\mathbf{1}\rangle</math> in (2) is necessary to determine absolute value |

- | <math>|x|_{\!}=_{}t^{ | + | <math>|x|_{\!}=_{}t^{*_{}}</math> because vector <math>y</math> can take zero values in its entries. |

By initializing direction vector <math>y</math> to <math>(1\!-_{}\!\varepsilon)\mathbf{1}_{}</math>, | By initializing direction vector <math>y</math> to <math>(1\!-_{}\!\varepsilon)\mathbf{1}_{}</math>, | ||

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<math>\begin{array}{ccc} | <math>\begin{array}{ccc} | ||

- | \begin{array}{ | + | \begin{array}{cl}{\text minimize}_{x\in\mathbb{R}^n,~t\in\mathbb{R}^n}&\langle t,\mathbf{1}\rangle\\ |

- | \mbox{subject to}& | + | \mbox{subject to}&Ax=b\\ |

- | &-t\preceq x\ | + | &-t\preceq x\preceq t |

\end{array} | \end{array} | ||

- | &\equiv& | + | &\equiv& |

- | \begin{array}{cl}\text | + | \begin{array}{cl}{\text minimize}_{x\in\mathbb{R}^n}&||x||_1\\ |

- | \mbox{subject to}& | + | \mbox{subject to}&Ax=b |

\end{array} | \end{array} | ||

- | \end{array} | + | \end{array}</math> (4) |

Subsequent iterations of problem (2) engaging cardinality term <math>\langle t_{}\,,\,y\rangle</math> | Subsequent iterations of problem (2) engaging cardinality term <math>\langle t_{}\,,\,y\rangle</math> | ||

- | can be interpreted as corrections to this 1-norm problem leading to a 0-norm solution. | + | can be interpreted as corrections to this 1-norm problem leading to a <math>0</math>-norm solution. |

<br> | <br> | ||

Iteration (2) (3) always converges to a locally optimal solution by virtue of | Iteration (2) (3) always converges to a locally optimal solution by virtue of | ||

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<br> | <br> | ||

There can be no proof of global optimality, defined by an optimal objective equal to 0. | There can be no proof of global optimality, defined by an optimal objective equal to 0. | ||

- | Local solutions are therefore detected by nonzero optimal objective. | + | <br>Local solutions are therefore detected by nonzero optimal objective. |

- | <br> | + | <br>Heuristics for reinitializing direction vector <math>y</math> can lead to a globally optimal solution. |

- | Heuristics for reinitializing direction vector <math>y</math> can lead to a globally optimal solution. | + | |

+ | ===equivalent=== | ||

+ | Several simple equivalents to linear programs (2) (3) are easily devised, | ||

+ | but their geometrical interpretation is not as apparent: ''e.g.'', equivalent in the limit <math>\,\varepsilon\!\rightarrow0^+\,</math> | ||

+ | |||

+ | <math>\begin{array}{cl}{\text minimize}_{x_{}\in_{_{}}\mathbb{R}^n,~t_{}\in_{_{}}\mathbb{R}^n}&\langle t\,,\,y\rangle\\ | ||

+ | \text{subject to}&_{}A_{}x=b\\ | ||

+ | &-t\preceq x\preceq_{_{}}t | ||

+ | \end{array}</math> | ||

+ | |||

+ | <math>\begin{array}{cl}{\text minimize}_{y_{}\in_{_{}}\mathbb{R}^n}&\langle |x^*|\,,\,y\rangle\\ | ||

+ | \text{subject to}&0\preceq y\preceq\mathbf{1}\\ | ||

+ | &y^{\rm T}\mathbf{1}=n-_{}k | ||

+ | \end{array}</math> | ||

---- | ---- | ||

For a coded numerical example, see [[Candes.m]] | For a coded numerical example, see [[Candes.m]] |

## Current revision

Convex iteration is method for constraining rank or cardinality in an otherwise convex optimization problem.

A rank or cardinality constraint is replaced by a linear regularization term added to the objective.

Then two convex problems are iterated until convergence where, ideally, solution to the original problem is found.

## constraining cardinality of signed variable

Excerpt from Chapter 4 Semidefinite Programming:

Consider a feasibility problem equivalent to the classical problem from linear algebra
,
but with an upper bound on cardinality :
for vector

(1)

where means vector has at most nonzero entries;
such a vector is presumed existent in the feasible set.

Convex iteration works with a nonnegative variable; absolute value is therefore needed.

We propose that nonconvex problem (1) can be equivalently written
as a convex iteration:
for a relatively small positive constant,

(2)

is iterated with

(3)

to find *direction vector* .
The cardinality constraint has been moved to the objective as a linear regularization.

The term in (2) is necessary to determine absolute value
because vector can take zero values in its entries.

By initializing direction vector to ,
the first iteration of problem (2) is a 1-norm problem; *i.e.*,

(4)

Subsequent iterations of problem (2) engaging cardinality term
can be interpreted as corrections to this 1-norm problem leading to a -norm solution.

Iteration (2) (3) always converges to a locally optimal solution by virtue of
a monotonically nonincreasing objective sequence.

There can be no proof of global optimality, defined by an optimal objective equal to 0.

Local solutions are therefore detected by nonzero optimal objective.

Heuristics for reinitializing direction vector can lead to a globally optimal solution.

### equivalent

Several simple equivalents to linear programs (2) (3) are easily devised,
but their geometrical interpretation is not as apparent: *e.g.*, equivalent in the limit

For a coded numerical example, see Candes.m