PEPit.primitive_steps.linear_optimization_step

PEPit.primitive_steps.linear_optimization_step(dir, ind)

This routine outputs the result of a minimization problem with linear objective (whose direction is provided by dir) on the domain of the (closed convex) indicator function ind. That is, it outputs a solution to

\[\arg\min_{\text{ind}(x)=0} \left< \text{dir};\, x \right>,\]

One can notice that \(x\) is solution of this problem if and only if

\[- \text{dir} \in \partial \text{ind}(x).\]

Parameters

• dir (Point) – direction of optimization

• ind (ConvexIndicatorFunction) – convex indicator function

Returns

• x (Point) – the optimal point.

• gx (Point) – the (sub)gradient of ind on x.

• fx (Expression) – the function value of ind on x.