ABSTRACT. Many research projects start from a problem and identify the most effective method to solve it. This work emerged the other way around. We considered an existing method [1] and searched for (new) problems it may solve efficiently. The new method aims to improve the widely-used Cutting-Planes, by upgrading its well-established separation sub-problem to a projection sub- problem: given some feasible x in some polytope P and direction d, what is the maximum t such that x + t · d ∈ P ? This question is easy if one can list all constraints of P , but it is (much) harder in a Cutting-Planes context where P has prohibitively-many constraints. This projection sub-problem is more difficult to implement than the standard separation. So why should we bother, why should we care? Can’t say we should. But there may be some benefits down the road. Long story short, the new algorithm Projective Cutting-Planes generates interior points (feasible solutions) along the iterations. The classical Cutting-Planes proceeds only by removing infeasibility. The new one proceeds both by removing infeasibility and by producing feasibility. Suppose one has to choose one of the following constraints: (1) 10x 1 + 20x 2 + 6x 3 ≤ 30 (2) x 1 + 2x 2 + x 3 ≤ 3 When x ∈ R 3+ , constraint (2) implies (1). Yet, by applying the separation logic on x = [1 1 1], constraint (1) seems tighter because 10 + 20 + 6 − 30 > 1 + 2 + 1 − 3, i.e., the separation sub- problem returns (1) because it seems more violated by [1 1 1]. The projection sub-problem [0 0 0] → [1 1 1] leads to the pierce point [0.75 0.75 0.75] and returns constraint (2) which is tight for this pierce point. Constraint (1) is not tight because it reduces to 36 · 34 ≤ 30 or 27 ≤ 30. When we first submitted [2], we initially solved robust linear programming by comparing the classical Cutting-Planes and the new Projective Cutting-Planes. We generated in both cases only one cut per iteration, because we only wanted to compare the two methods. Pointing out that the most effective Cutting-Planes for this problem uses multiple cuts per iteration, a referee asked as if we could also adapt Projective Cutting Planes for a multi-cut setting. The answer was yes, and – a bit to our surprise – the multi-cuts variant of Projective Cutting-Planes was really faster. For example, it solved in 8 seconds the instance stocfor3, an instance that required thousands of seconds for all other Cutting Planes mentioned above. References [1] Daniel Porumbel, Projective Cutting-Planes, SIAM Journal on Optimization, 30(1), 2020 [2] Daniel Porumbel, Projective Cutting-Planes for Robust Linear Programming and Cutting Stock Problems, INFORMS Journal of Computing, DOI 10.1287/ijoc.2022.1160