Convex optimization of optimization problem is convex. "Standard form" is the usual and perhaps most intuitive form of describing a convex minimization problem. It consists of the following three parts: A convex minimization problem in standard form is thus written as In reality, this form of problem is exactly equivalent to a problem constrained by only equalities. You can rephrase the problem without equality constraints by replacing each equality constraint formula_52 with a pair of inequality constraints formula_64 and formula_65, and the previous analysis holds. This fact is why formula_53 being both convex and concave, i.e. both formula_53 and formula_68 are convex,
Lens (optics) real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, "S" then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a "real image" on the retina.