On strong duality, theorems of the alternative, and projections in conic\n optimization

A conic program is the problem of optimizing a linear function over a closed\nconvex cone intersected with an affine preimage of another cone. We analyse\nthree constraint qualifications, namely a Closedness CQ, Slater CQ, and\nBoundedness CQ (also called Clark-Duffin theorem), that are sufficient for\nachieving strong duality and show that the first implies the second which\nimplies the third, and also give a more general form of the third CQ for conic\nproblems. Furthermore, two consequences of strong duality are presented, the\nfirst being a theorem of the alternative on almost feasibility (also called\nweak infeasibility), and the second being an explicit description of the\nprojection of conic sets onto linear subspaces, akin to using projection cones\nfor polyhedral sets.\n