Many convex optimization methods are conceived of and analyzed in a largely separate fashion. In contrast to this traditional separation, this manuscript points out and demonstrates the utility of an important but largely unremarked common thread running through many prominent optimization methods. Specifically, we show that methods such as successive orthogonal projection, gradient descent, projected gradient descent, the proximal point method, forward-backward splitting, the alternating direction method of multipliers, and under- or over-relaxed variants of the preceding all involve updates that are of a common type --- namely, the updates satisfy a property known as pseudocontractivity. Moreover, since the property of pseudocontractivity is preserved under both composition and convex combination, updates constructed via these operations from pseudocontractive updates are themselves pseudocontractive. Having demonstrated that pseudocontractive updates are to be found in many optimization methods, we then provide an initial example of the type of unified analysis that becomes possible in the settings where the property of pseudocontractivity is found to hold. Specifically, we prove a novel bound satisfied by the norm of the difference in iterates of pseudocontractive updates and we then use this bound to establish an @math worst-case convergence rate on the error criterion @math for any method involving pseudocontractive updates (where @math is the number of iterations).
An electrically-operated shutter provided with a change-over device or arrangement for effecting daylight and flash photography automatically shifting from a daylight mode to a flash photography mode as a function of the detected brightness level of the field being photographed. One embodiment provides for manually presetting the camera for flash photography which is then automatically carried out with an automatic change-over from a daylight photography mode to a flash photography mode. When carrying out daylight photography the power source energy is used in control and in energizing an electromagnet that controls the exposure termination under control of a delay circuit. When taking flash exposure the delay circuit is not employed and a fixed exposure time is used. The power supply is then used to energize the flashbulb and the accuracy of the shutter timing is unaffected as might be the case if the power source were energizing both the flashbulb and driving or energizing the shutter. The same shutter-operating elements are used in both modes of operation and the change-over arrangement eliminates the use of the delay circuit when the brightness level is below a predetermined level so that flash photography becomes necessary.