Projective Cutting-Planes

Given a polytope $\mathcal P$, an interior point ${x}\in\mathcal P$, and a direction ${d}\in\mathbb{R}^n$, the projection of ${x}$ along ${d}$ asks to find the maximum step length $t^*$ such that ${x}+t^*{d}\in\mathcal P$; we say ${x}+t^*{d}$ is the pierce point obtained by projection. In [D. Porumbel, Math. Program., 155 (2016), pp. 147--197], we solely explored the idea of projecting the origin $0_n$ along integer directions only, focusing on dual polytopes $\mathcal P$ in Column Generation models. This work addresses a more general projection subproblem, considering arbitrary interior points ${x}\in\mathcal P$ and arbitrary noninteger directions ${d}\in\mathbb{R}^n$, in areas beyond Column Generation.The projection subproblem generalizes the separation subproblem of the well-known Cutting-Planes. We propose a new algorithm, Projective Cutting-Planes, that relies on this projection subproblem to optimize over polytopes $\mathcal P$ with prohibitively many constraints. At each iteration, this new algorithm selects a point ${x}_{new}$ on the segment joining the points ${x}$ and ${x}+t^*{d}$ determined at the previous iteration. Then, it projects ${x}_{new}$ along the direction ${d}_{new}$ pointing towards the current optimal (outer) solution (of the current outer approximation of $\mathcal P$), so as to generate a new pierce point ${x}_{new}+t^*_{new} {d}_{new}$ and a new constraint of $\mathcal P$. By reoptimizing the linear program enriched with this new constraint, the algorithm finds a new current optimal (outer) solution and moves to the next iteration by updating ${x}={x}_{new}$ and ${d}={d}_{new}$. Compared to Cutting-Planes, the main advantage of Projective Cutting-Planes is that it has a built-in functionality to generate a feasible inner solution ${new}+t^*{d}$ at each iteration. These inner solutions converge iteratively to an optimal solution ${opt}(\mathcal P)$, and so Projective Cutting-Planes is more similar to an interior point method than to the Simplex method. Numerical experiments in different optimization settings confirm the potential of the proposed ideas.