Semidefinite programming by Projective Cutting-Planes

\(\) Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is still significantly harder to solve than a similar-size Linear Program (LP). It is well-known that a semidefinite program can be written as an … Read more

Cutting Planes by Projecting Interior Points onto Polytope Facets

Given a point x inside a polytope P and a direction d, the projection of x along d asks to find the maximum step length t such that x+td is feasible; we say x+td is a pierce point because it belongs to the boundary of P. We address this projection sub-problem with arbitrary interior points … Read more

Ray Projection for Optimizing Polytopes with Prohibitively Many Constraints in Set-Covering Column Generation

A recurrent task in mathematical programming requires optimizing polytopes with prohibitively-many constraints, e.g., the primal polytope in cutting-plane methods or the dual polytope in Column Generation (CG). This paper is devoted to the ray projection technique for optimizing such polytopes: start from a feasible solution and advance on a given ray direction until intersecting a … Read more