Convex optimization linear programming. Many optimization problems can be reformulated as convex minimization problems. For example, the problem of "maximizing" a "concave function" formula_1 can be re-formulated equivalently as a problem of minimizing the convex function formula_2. An optimization problem, which aims to find a point formula_3 such that formula_4 for all formula_5 in some feasible set formula_6, is convex if and only if formula_6 and the objective function formula_1 satisfy the following two properties: Perhaps more conveniently, the convex problem can be phrased in the more shorthand general definition of optimization: finding some formula_19 such that for some feasible set formula_6
Compound refractive lens the order of meters, by using many lenses in series. To minimize the absorption the lens must be made from a material such as aluminium, beryllium, or lithium. CRLs were first demonstrated in the mid-1990s by a group of scientists at the ESRF. They drilled holes in an aluminium block, and achieved focusing in two dimensions. For X-rays a concave lens focuses the X-rays because the index of refraction is slightly below unity. In a CRL of this type the walls between the cylindrical holes act as concave lenses for X-rays traveling perpendicular to the axis of the drilled cylinders.