nidantasnu

2022-07-07

Could someone kindly suggest a method of solving the following constrained (equality and inequality) system of equations in the least squares fashion?

$\underset{x}{min}\frac{1}{2}\Vert Ax-b{\Vert}_{2}^{2}$

such that

$\begin{array}{c}{x}_{i}+{x}_{j}=1\phantom{\rule{1em}{0ex}}\text{for a certain}i,j\\ 0{x}_{k}{\tau}_{k}\phantom{\rule{1em}{0ex}}\text{where}k\ne i,j\end{array}$

${\tau}_{k}$ is the upper bound for ${x}_{k}$.

$\underset{x}{min}\frac{1}{2}\Vert Ax-b{\Vert}_{2}^{2}$

such that

$\begin{array}{c}{x}_{i}+{x}_{j}=1\phantom{\rule{1em}{0ex}}\text{for a certain}i,j\\ 0{x}_{k}{\tau}_{k}\phantom{\rule{1em}{0ex}}\text{where}k\ne i,j\end{array}$

${\tau}_{k}$ is the upper bound for ${x}_{k}$.

Elias Flores

Beginner2022-07-08Added 24 answers

You can solve this problem using the projected gradient method (or an accelerated projected gradient method). Let $S$ be the set of all $x$ that satisfy the given constraints. The gradient of the objective function $f$ is

$\mathrm{\nabla}f(x)={A}^{T}(Ax-b)$

and the projected gradient iteration is

${x}^{+}={P}_{S}(x-t\mathrm{\nabla}f(x)).$

${x}^{+}={P}_{S}(x-t\mathrm{\nabla}f(x)).$

The function ${P}_{S}$ projects onto $S$. This projection step requires solving a linear algebra subproblem. (Note that the second set of constraints can be handled independently of the first set of constraints. If the linear constraints are described more explicitly, we can give more detail.) The projected gradient iteration will converge if the step size $t>0$ is sufficiently small.

$\mathrm{\nabla}f(x)={A}^{T}(Ax-b)$

and the projected gradient iteration is

${x}^{+}={P}_{S}(x-t\mathrm{\nabla}f(x)).$

${x}^{+}={P}_{S}(x-t\mathrm{\nabla}f(x)).$

The function ${P}_{S}$ projects onto $S$. This projection step requires solving a linear algebra subproblem. (Note that the second set of constraints can be handled independently of the first set of constraints. If the linear constraints are described more explicitly, we can give more detail.) The projected gradient iteration will converge if the step size $t>0$ is sufficiently small.

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