Interiorfeasible

Interiorfeasible is a term used in optimization to describe a feasible point that lies strictly inside the feasible region with respect to the inequality constraints. In problems with inequality constraints g_i(x) ≤ 0 and equality constraints h_j(x) = 0, a point x* is feasible if it satisfies all constraints. It is interior-feasible (also called strictly feasible) if, in addition, every inequality is satisfied strictly: g_i(x*) < 0 for all i, while the equalities hold exactly.

This distinction matters because interior-feasible points lie away from the boundary defined by the inequality constraints. They are central to interior-point methods, which generate iterates that remain in the interior of the feasible set to improve numerical stability and convergence properties. If any inequality constraint is active at x* (g_i(x*) = 0), the point lies on the boundary of the feasible region rather than in its interior.

In convex optimization, the existence of an interior-feasible point often relates to Slater's condition, which asserts the presence of a strictly feasible point for the inequality constraints. This condition helps guarantee strong duality and well-behaved dual problems.

Example: consider the problem minimize f(x) subject to g_1(x) ≤ 0, g_2(x) ≤ 0, and h(x) = 0. If a point x* satisfies h(x*) = 0 and g_1(x*) < 0, g_2(x*) < 0, then x* is interior-feasible. If either g_1(x*) = 0 or g_2(x*) = 0, the point is feasible but lies on the boundary with respect to those constraints.